On the modularity of elliptic curves over $\mathbf{Q}$: Wild $3$-adic exercises

Abstract

We complete the proof that every elliptic curve over the rational numbers is modular.

Introduction

In this paper, building on work of Wiles Wi and of Taylor and Wiles TW, we will prove the following two theorems (see §2.2).

Theorem A.

If $E_{/\mathbf{Q}}$ is an elliptic curve, then $E$ is modular.

Theorem B.

If ${\overline{\rho }}:\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \rightarrow {\operatorname {GL}}_2(\mathbf{F}_5)$ is an irreducible continuous representation with cyclotomic determinant, then ${\overline{\rho }}$ is modular.

We will first remind the reader of the content of these results and then briefly outline the method of proof.

If $N$ is a positive integer, then we let $\Gamma _1(N)$ denote the subgroup of $\operatorname {SL}_2(\mathbf{Z})$ consisting of matrices that modulo $N$ are of the form

$$\begin{equation*} \left( \begin{array}{cc} {1} & {*} \\{0} & {1} \end{array} \right). \end{equation*}$$

The quotient of the upper half plane by $\Gamma _1(N)$, acting by fractional linear transformations, is the complex manifold associated to an affine algebraic curve $Y_1(N)_{/\mathbf{C}}$. This curve has a natural model $Y_1(N)_{/\mathbf{Q}}$, which for $N >3$ is a fine moduli scheme for elliptic curves with a point of exact order $N$. We will let $X_1(N)$ denote the smooth projective curve which contains $Y_1(N)$ as a dense Zariski open subset.

Recall that a cusp form of weight $k\geq 1$ and level $N\geq 1$ is a holomorphic function $f$ on the upper half complex plane $\mathfrak{H}$ such that

•

for all matrices$$\begin{equation*} \left( \begin{array}{cc} {a} & {b} \\{c} & {d} \end{array} \right) \in \Gamma _1(N) \end{equation*}$$

and all $z \in \mathfrak{H}$, we have $f((az+b)/(cz+d))=(cz+d)^k f(z)$;

•

and $|f(z)|^2 (\operatorname {Im}z)^k$ is bounded on $\mathfrak{H}$.

The space $S_k(N)$ of cusp forms of weight $k$ and level $N$ is a finite-dimensional complex vector space. If $f \in S_k(N)$, then it has an expansion

$$\begin{equation*} f(z) = \sum _{n=1}^\infty c_n(f) e^{2\pi i nz} \end{equation*}$$

and we define the $L$-series of $f$ to be

$$\begin{equation*} L(f,s) = \sum _{n=1}^\infty c_n(f)/n^s. \end{equation*}$$

For each prime $p \nmid N$ there is a linear operator $T_p$ on $S_k(N)$ defined by

$$\begin{equation*} (f|T_p)(z) = p^{-1} \sum _{i=0}^{p-1} f((z+i)/p)+ p^{k-1}(cpz+d)^{-k} f((apz+b)/(cpz+d)) \end{equation*}$$

for any

$$\begin{equation*} \left( \begin{array}{cc} {a} & {b} \\{c} & {d} \end{array} \right) \in \operatorname {SL}_2(\mathbf{Z}) \end{equation*}$$

with $c \equiv 0 \bmod N$ and $d \equiv p \bmod N$. The operators $T_p$ for $p \nmid N$ can be simultaneously diagonalised on the space $S_k(N)$ and a simultaneous eigenvector is called an eigenform. If $f$ is an eigenform, then the corresponding eigenvalues, $a_p(f)$, are algebraic integers and we have $c_p(f)=a_p(f)c_1(f)$.

Let $\lambda$ be a place of the algebraic closure of $\mathbf{Q}$ in $\mathbf{C}$ above a rational prime $\ell$ and let $\overline{\mathbf{Q}}_\lambda$ denote the algebraic closure of $\mathbf{Q}_\ell$ thought of as a $\overline{\mathbf{Q}}$ algebra via $\lambda$. If $f \in S_k(N)$ is an eigenform, then there is a unique continuous irreducible representation

$$\begin{equation*} \rho _{f,\lambda }:\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \longrightarrow {\operatorname {GL}}_2(\overline{\mathbf{Q}}_\lambda ) \end{equation*}$$

such that for any prime $p\nmid Nl$, $\rho _{f,\lambda }$ is unramified at $p$ and $\operatorname {tr}\rho _{f,\lambda }(\operatorname {Frob}_p)=a_p(f)$. The existence of $\rho _{f,\lambda }$ is due to Shimura if $k=2$ Sh2, to Deligne if $k>2$ De and to Deligne and Serre if $k=1$ DS. Its irreducibility is due to Ribet if $k>1$ Ri and to Deligne and Serre if $k=1$ DS. Moreover $\rho$ is odd in the sense that $\det \rho$ of complex conjugation is $-1$. Also, $\rho _{f,\lambda }$ is potentially semi-stable at $\ell$ in the sense of Fontaine. We can choose a conjugate of $\rho _{f,\lambda }$ which is valued in ${\operatorname {GL}}_2({\mathcal{O}}_{\overline{\mathbf{Q}}_\lambda })$, and reducing modulo the maximal ideal and semi-simplifying yields a continuous representation

$$\begin{equation*} {\overline{\rho }}_{f,\lambda }:\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \longrightarrow {\operatorname {GL}}_2(\overline{\mathbf{F}}_\ell ), \end{equation*}$$

which, up to isomorphism, does not depend on the choice of conjugate of $\rho _{f,\lambda }$.

Now suppose that $\rho :G_\mathbf{Q}\rightarrow {\operatorname {GL}}_2(\overline{\mathbf{Q}}_\ell )$ is a continuous representation which is unramified outside finitely many primes and for which the restriction of $\rho$ to a decomposition group at $\ell$ is potentially semi-stable in the sense of Fontaine. To $\rho |_{\operatorname {Gal}(\overline{\mathbf{Q}}_\ell /\mathbf{Q}_\ell )}$ we can associate both a pair of Hodge-Tate numbers and a Weil-Deligne representation of the Weil group of $\mathbf{Q}_\ell$. We define the conductor $N(\rho )$ of $\rho$ to be the product over $p \neq \ell$ of the conductor of $\rho |_{\operatorname {Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)}$ and of the conductor of the Weil-Deligne representation associated to $\rho |_{\operatorname {Gal}(\overline{\mathbf{Q}}_\ell /\mathbf{Q}_\ell )}$. We define the weight $k(\rho )$ of $\rho$ to be $1$ plus the absolute difference of the two Hodge-Tate numbers of $\rho |_{\operatorname {Gal}(\overline{\mathbf{Q}}_\ell /\mathbf{Q}_\ell )}$. It is known by work of Carayol and others that the following two conditions are equivalent:

•

$\rho \sim \rho _{f,\lambda }$ for some eigenform $f$ and some place $\lambda |\ell$;

•

$\rho \sim \rho _{f,\lambda }$ for some eigenform $f$ of level $N(\rho )$ and weight $k(\rho )$ and some place $\lambda |\ell$.

When these equivalent conditions are met we call $\rho$ modular. It is conjectured by Fontaine and Mazur that if $\rho :G_\mathbf{Q}\rightarrow {\operatorname {GL}}_2(\overline{\mathbf{Q}}_\ell )$ is a continuous irreducible representation which satisfies

•

$\rho$ is unramified outside finitely many primes,

•

$\rho |_{\operatorname {Gal}(\overline{\mathbf{Q}}_\ell /\mathbf{Q}_\ell )}$ is potentially semi-stable with its smaller Hodge-Tate number $0$,

•

and, in the case where both Hodge-Tate numbers are zero, $\rho$ is odd,

then $\rho$ is modular FM.

Next consider a continuous irreducible representation ${\overline{\rho }}:\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \rightarrow {\operatorname {GL}}_2(\overline{\mathbf{F}}_\ell )$. Serre Se2 defines the conductor $N({\overline{\rho }})$ and weight $k({\overline{\rho }})$ of ${\overline{\rho }}$. We call ${\overline{\rho }}$ modular if ${\overline{\rho }}\sim {\overline{\rho }}_{f,\lambda }$ for some eigenform $f$ and some place $\lambda |\ell$. We call ${\overline{\rho }}$ strongly modular if moreover we may take $f$ to have weight $k({\overline{\rho }})$ and level $N({\overline{\rho }})$. It is known from work of Mazur, Ribet, Carayol, Gross, Coleman, Voloch and others that for $\ell \geq 3$, ${\overline{\rho }}$ is strongly modular if and only if it is modular (see Di1). Serre has conjectured that all odd, irreducible ${\overline{\rho }}$ are strongly modular Se2.

Now consider an elliptic curve $E_{/\mathbf{Q}}$. Let $\rho _{E,\ell }$ (resp. ${\overline{\rho }}_{E,\ell }$) denote the representation of $\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ on the $\ell$-adic Tate module (resp. the $\ell$-torsion) of $E(\overline{\mathbf{Q}})$. Let $N(E)$ denote the conductor of $E$. It is known that the following conditions are equivalent:

(1)

The $L$-function $L(E,s)$ of $E$ equals the $L$-function $L(f,s)$ for some eigenform $f$.

(2)

The $L$-function $L(E,s)$ of $E$ equals the $L$-function $L(f,s)$ for some eigenform $f$ of weight $2$ and level $N(E)$.

(3)

For some prime $\ell$, the representation $\rho _{E,\ell }$ is modular.

(4)

For all primes $\ell$, the representation $\rho _{E,\ell }$ is modular.

(5)

There is a non-constant holomorphic map $X_1(N)(\mathbf{C}) \rightarrow E(\mathbf{C})$ for some positive integer $N$.

(6)

There is a non-constant morphism $X_1(N(E)) \rightarrow E$ which is defined over $\mathbf{Q}$.

The implications (2) $\Rightarrow$ (1), (4) $\Rightarrow$ (3) and (6) $\Rightarrow$ (5) are tautological. The implication (1) $\Rightarrow$ (4) follows from the characterisation of $L(E,s)$ in terms of $\rho _{E,\ell }$. The implication (3) $\Rightarrow$ (2) follows from a theorem of Carayol Ca1. The implication (2) $\Rightarrow$ (6) follows from a construction of Shimura Sh2 and a theorem of Faltings Fa. The implication (5) $\Rightarrow$ (3) seems to have been first noticed by Mazur Maz. When these equivalent conditions are satisfied we call $E$ modular.

It has become a standard conjecture that all elliptic curves over $\mathbf{Q}$ are modular, although at the time this conjecture was first suggested the equivalence of the conditions above may not have been clear. Taniyama made a suggestion along the lines (1) as one of a series of problems collected at the Tokyo-Nikko conference in September 1955. However his formulation did not make clear whether $f$ should be a modular form or some more general automorphic form. He also suggested that constructions as in (5) and (6) might help attack this problem at least for some elliptic curves. In private conversations with a number of mathematicians (including Weil) in the early 1960’s, Shimura suggested that assertions along the lines of (5) and (6) might be true (see Sh3 and the commentary on [1967a] in We2). The first time such a suggestion appears in print is Weil’s comment in We1 that assertions along the lines of (5) and (6) follow from the main result of that paper, a construction of Shimura and from certain “reasonable suppositions” and “natural assumptions”. That assertion (1) is true for CM elliptic curves follows at once from work of Hecke and Deuring. Shimura Sh1 went on to check assertion (5) for these curves.

Our approach to Theorem A is an extension of the methods of Wiles Wi and of Taylor and Wiles TW. We divide the proof into three cases.

(1)

If ${\overline{\rho }}_{E,5}|_{\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q}(\sqrt {5}))}$ is irreducible, we show that $\rho _{E,5}$ is modular.

(2)

If ${\overline{\rho }}_{E,5}|_{\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q}(\sqrt {5}))}$ is reducible, but ${\overline{\rho }}_{E,3}|_{\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q}(\sqrt {-3}))}$ is absolutely irreducible, we show that $\rho _{E,3}$ is modular.

(3)

If ${\overline{\rho }}_{E,5}|_{\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q}(\sqrt {5}))}$ is reducible and ${\overline{\rho }}_{E,3}|_{\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q}(\sqrt {-3}))}$ is absolutely reducible, then we show that $E$ is isogenous to an elliptic curve with $j$-invariant $0$, $(11/2)^3$, or $-5(29)^3/2^5$ and so (from tables of modular elliptic curves of low conductor) is modular.

In each of cases (1) and (2) there are two steps. First we prove that ${\overline{\rho }}_{E,\ell }$ is modular and then that $\rho _{E,\ell }$ is modular. In case (1) this first step is our Theorem B and in case (2) it is a celebrated theorem of Langlands and Tunnell L, T. In fact, in both cases $E$ obtains semi-stable reduction over a tame extension of $\mathbf{Q}_\ell$ and the deduction of the modularity of $\rho _{E,\ell }$ from that of ${\overline{\rho }}_{E,\ell }$ was carried out in CDT by an extension of the methods of Wi and TW. In the third case we have to analyse the rational points on some modular curves of small level. This we did, with Elkies’ help, in CDT.

It thus only remained to prove Theorem B. Let ${\overline{\rho }}$ be as in that theorem. Twisting by a quadratic character, we may assume that ${\overline{\rho }}|_{\operatorname {Gal}( \overline{\mathbf{Q}}_3/\mathbf{Q}_3)}$ falls into one of the following cases (see §2.2):

(1)

${\overline{\rho }}$ is unramified at $3$.

(2)

${\overline{\rho }}(I_3)$ has order $5$.

(3)

${\overline{\rho }}(I_3)$ has order $4$.

(4)

${\overline{\rho }}(I_3)$ has order $12$ and ${\overline{\rho }}|_{\operatorname {Gal}(\overline{\mathbf{Q}}_3/\mathbf{Q}_3)}$ has conductor $27$.

(5)

${\overline{\rho }}(I_3)$ has order $3$.

(6)

${\overline{\rho }}|_{\operatorname {Gal}(\overline{\mathbf{Q}}_3/\mathbf{Q}_3)}$ is induced from a character $\chi : \operatorname {Gal}(\overline{\mathbf{Q}}_3/\mathbf{Q}_3(\sqrt {-3})) \to \mathbf{F}_{25}^\times$ such that $\chi (-1)=-1$ and$$\begin{equation*} \chi (\sqrt {-3})=\chi (1+3\sqrt {-3})-\chi (1-3\sqrt {-3}), \end{equation*}$$

where we use the Artin map (normalised to take uniformisers to arithmetic Frobenius) to identify $\chi$ with a character of $\mathbf{Q}_3(\sqrt {-3})^\times$.

We will refer to these as the $f=1,3,9,27, 81$ and $243$ cases respectively.

Using the technique of Minkowski and Klein (i.e. the observation that the moduli space of elliptic curves with full level $5$ structure has genus $0$; see for example Kl), Hilbert irreducibility and some local computations of Manoharmayum Man, we find an elliptic curve $E_{/\mathbf{Q}}$ with the following properties (see §2.2):

•

${\overline{\rho }}_{E,5} \sim {\overline{\rho }}$,

•

${\overline{\rho }}_{E,3}$ is surjective onto ${\operatorname {GL}}_2(\mathbf{F}_3)$,

•

and

(1)

in the $f=1$ case, either ${\overline{\rho }}_{E,3}|_{I_3} \otimes \mathbf{F}_9 \sim \omega _2\oplus \omega _2^3$ or$$\begin{equation*} {\overline{\rho }}_{E,3}|_{I_3} \sim \left( \begin{array}{cc} {\omega } & {*} \\{0} & {1} \end{array} \right) \end{equation*}$$

and is peu ramifié;

(2)

in the $f=3$ case,$$\begin{equation*} {\overline{\rho }}_{E,3}|_{I_3} \sim \left( \begin{array}{cc} {\omega } & {*} \\{0} & {1} \end{array} \right); \end{equation*}$$

(3)

in the $f=9$ case, ${\overline{\rho }}_{E,3}|_{I_3} \otimes \mathbf{F}_9 \sim \omega _2\oplus \omega _2^3$;

(4)

in the $f=27$ case,$$\begin{equation*} {\overline{\rho }}_{E,3}|_{I_3} \sim \left( \begin{array}{cc} {\omega } & {*} \\{0} & {1} \end{array} \right) \end{equation*}$$

and is très ramifié;

(5)

in the $f=81$ case,$$\begin{equation*} {\overline{\rho }}_{E,3}|_{I_3} \sim \left( \begin{array}{cc} {1} & {*} \\{0} & {\omega } \end{array} \right) \end{equation*}$$

and is très ramifié;

(6)

in the $f=243$ case,$$\begin{equation*} {\overline{\rho }}_{E,3}|_{\operatorname {Gal}(\overline{\mathbf{Q}}_3/\mathbf{Q}_3)} \sim \left( \begin{array}{cc} {\omega } & {*} \\{0} & {1} \end{array} \right) \end{equation*}$$

is non-split over $\overline{\mathbf{Q}}_3^{\ker {\overline{\rho }}}$ and is très ramifié.

(We are using the terms très ramifié and peu ramifié in the sense of Serre Se2. We are also letting $\omega$ denote the $\bmod 3$ cyclotomic character and $\omega _2$ the second fundamental character $I_3 \rightarrow \mathbf{F}_{9}^\times$, i.e.

$$\begin{equation*} \omega _2(\sigma ) \equiv \sigma (\sqrt [8]{3})/\sqrt [8]{3} \bmod \sqrt [8]{3}. \end{equation*}$$

We will often use the equality $\omega =\omega ^{-1}$ without further remark.) We emphasise that for a general elliptic curve over $\mathbf{Q}$ with ${\overline{\rho }}_{E,5} \cong {\overline{\rho }}$, the representation ${\overline{\rho }}_{E,3}$ does not have the above form, rather we are placing a significant restriction on $E$.

In each case our strategy is to prove that $\rho _{E,3}$ is modular and so deduce that ${\overline{\rho }}\sim {\overline{\rho }}_{E,5}$ is modular. Again we use the Langlands-Tunnell theorem to see that ${\overline{\rho }}_{E,3}$ is modular and then an analogue of the arguments of Wi and TW to conclude that $\rho _{E,3}$ is modular. This was carried out in Di2 in the cases $f=1$ and $f=3$, and in CDT in the case $f=9$. (In these cases the particular form of ${\overline{\rho }}_{E,3}|_{I_3}$ is not important.) This leaves the cases $f=27$, $81$ and $243$, which are complicated by the fact that $E$ now only obtains good reduction over a wild extension of $\mathbf{Q}_3$. In these cases our argument relies essentially on the particular form we have obtained for ${\overline{\rho }}_{E,3}|_{\operatorname {Gal}(\overline{\mathbf{Q}}_3/\mathbf{Q}_3)}$ (depending on ${\overline{\rho }}_{E,5}|_{I_3}$). We do not believe that our methods for deducing the modularity of $\rho _{E,3}$ from that of ${\overline{\rho }}_{E,3}$ would work without this key simplification. It seems to be a piece of undeserved good fortune that for each possibility for ${\overline{\rho }}|_{I_3}$ we can find a choice for ${\overline{\rho }}_{E,3}|_{\operatorname {Gal}(\overline{\mathbf{Q}}_3/\mathbf{Q}_3)}$ for which our methods work.

Following Wiles, to deduce the modularity of $\rho _{E,3}$ from that of ${\overline{\rho }}_{E,3}$, we consider certain universal deformations of ${\overline{\rho }}_{E,3}$ and identify them with certain modular deformations which we realise over certain Hecke algebras. The key problem is to find the right local condition to impose on these deformations at the prime $3$. As in CDT we require that the deformations lie in the closure of the characteristic zero points which are potentially Barsotti-Tate (i.e. come from a $3$-divisible group over the ring of integers of a finite extension of $\mathbf{Q}_3$) and for which the associated representation of the Weil group (see for example Appendix B of CDT) is of some specified form. That one can find suitable conditions on the representation of the Weil group at $3$ for the arguments of TW to work seems to be a rare phenomenon in the wild case. It is here we make essential use of the fact that we need only treat our specific pairs $({\overline{\rho }}_{E,5},{\overline{\rho }}_{E,3})$.

Our arguments follow closely the arguments of CDT. There are two main new features. Firstly, in the $f=243$ case, we are forced to specify the restriction of our representation of the Weil group completely, rather than simply its restriction to the inertia group as we have done in the past. Secondly, in the key computation of the local deformation rings, we now make use of a new description (due to Breuil) of finite flat group schemes over the ring of integers of any $p$-adic field in terms of certain (semi-)linear algebra data (see Br2 and the summary Br1). This description enables us to make these computations. As the persistent reader will soon discover they are lengthy and delicate, particularly in the case $f=243$. It seems miraculous to us that these long computations with finite flat group schemes in §7, §8 and §9 give answers completely in accord with predictions made from much shorter computations with the local Langlands correspondence and the modular representation theory of ${\operatorname {GL}}_2(\mathbf{Q}_3)$ in §3. We see no direct connection, but cannot help thinking that some such connection should exist.

Notation

In this paper $\ell$ denotes a rational prime. In §1.1, §4.1, §4.2 and §4.3 it is arbitrary. In the rest of §1 and in §5 we only assume it is odd. In the rest of the paper we only consider $\ell =3$.

If $F$ is a field we let $\overline{F}$ denote a separable closure, $F^{\operatorname {ab}}$ the maximal subextension of $\overline{F}$ which is abelian over $F$ and $G_F$ the Galois group $\operatorname {Gal}(\overline{F}/F)$. If $F_0$ is a $p$-adic field (i.e. a finite extension of $\mathbf{Q}_p$) and $F'/F_0$ a (possibly infinite) Galois extension, then we let $I_{F'/F_0}$ denote the inertia subgroup of $\operatorname {Gal}(F'/F_0)$. We also let $I_{F_0}$ denote $I_{\overline{F}_0 /F_0}$, $\operatorname {Frob}_{F_0}\in G_{F_0}/I_{F_0}$ denote the arithmetic Frobenius element and $W_{F_0}$ denote the Weil group of $F_0$, i.e. the dense subgroup of $G_{F_0}$ consisting of elements which map to an integer power of $\operatorname {Frob}_{F_0}$. We will normalise the Artin map of local class field theory so that uniformisers and arithmetic Frobenius elements correspond. (We apologise for this convention, which now seems to us a bad choice. However we feel it is important to stay consistent with CDT.) We let ${\mathcal{O}}_{F_0}$ denote the ring of integers of $F_0$, $\wp _{F_0}$ the maximal ideal of ${\mathcal{O}}_{F_0}$ and $k_{F_0}$ the residue field ${\mathcal{O}}_{F_0}/\wp _{F_0}$. We write simply $G_p$ for $G_{\mathbf{Q}_p}$, $I_p$ for $I_{\mathbf{Q}_p}$ and $\operatorname {Frob}_p$ for $\operatorname {Frob}_{\mathbf{Q}_p}$. We also let $\mathbf{Q}_{p^n}$ denote the unique unramified degree $n$ extension of $\mathbf{Q}_p$ in $\overline{\mathbf{Q}}_p$. If $k$ is any perfect field of characteristic $p$ we also use $\operatorname {Frob}_p$ to denote the $p^{th}$-power automorphism of $k$ and its canonical lift to the Witt vectors $W(k)$.

We write $\epsilon$ for the $\ell$-adic cyclotomic character and sometimes $\omega$ for the reduction of $\epsilon$ modulo $\ell$. We write $\omega _2$ for the second fundamental character $I_\ell \rightarrow \mathbf{F}_{\ell ^2}^\times$, i.e.

$$\begin{equation*} \omega _2(\sigma ) \equiv \sigma (\ell ^{1/(\ell ^2-1)})/\ell ^{1/(\ell ^2-1)} \bmod \ell ^{1/(\ell ^2-1)}. \end{equation*}$$

We also use $\omega$ and $\omega _2$ to denote the Teichmuller lifts of $\omega$ and $\omega _2$.

We let ${\mathbf{1}}$ denote the trivial character of a group. We will denote by $V^\vee$ the dual of a vector space $V$.

If $g:A \rightarrow B$ is a homomorphism of rings and if $X_{/\operatorname {Spec}A}$ is an $A$-scheme, then we sometimes write ${^gX}$ for the pullback of $X$ by $\operatorname {Spec}g$. We adopt this notation so that ${}^g({}^hX)={}^{gh}X$. Similarly if $\theta :X \rightarrow Y$ is a morphism of schemes over $A$ we will sometimes write ${^g\theta }$ for the pullback of $\theta$ by $\operatorname {Spec}g$.

By finite flat group scheme we always mean commutative finite flat group scheme. If $F_0$ is a field of characteristic $0$ with fixed algebraic closure $\overline{F}_0$ we use without comment the canonical identification of finite flat $F_0$-group schemes with finite discrete $\operatorname {Gal}(\overline{F}_0/F_0)$-modules, and we will say that such objects correspond. If $R$ is a Dedekind domain with field of fractions $F$ of characteristic $0$, then by a model of a finite flat $F$-group scheme $G$ we mean a finite locally free $R$-group scheme ${\mathcal{G}}$ and an isomorphism $i: G \stackrel{\sim }{\rightarrow }{\mathcal{G}}\times F'$. As in Proposition 2.2.2 of Ra the isomorphism classes of models for $G$ form a lattice ($({\mathcal{G}},i) \geq ({\mathcal{G}}',i')$ if there exists a map of finite flat group schemes ${\mathcal{G}}\rightarrow {\mathcal{G}}'$ compatible with $i$ and $i'$) and we can talk about the inf and sup of two such models. If $R$ is also local we call the model $({\mathcal{G}},i)$ local-local if its special fibre is local-local. When the ring $R$ is understood we sometimes simply refer to $({\mathcal{G}},i)$, or even just ${\mathcal{G}}$, as an integral model of $G$.

We use Serre’s terminology peu ramifié and très ramifié; see Se2.

1. Types

1.1. Types of local deformations

By an $\ell$-type we mean an equivalence class of two-dimensional representations

$$\begin{equation*} \tau : I_{\ell } \rightarrow {\operatorname {GL}}(D) \end{equation*}$$

over $\overline{\mathbf{Q}}_\ell$ which have open kernel and which can be extended to a representation of $W_{\mathbf{Q}_\ell }$. By an extended $\ell$-type we shall simply mean an equivalence class of two-dimensional representations

$$\begin{equation*} \tau ': W_{\mathbf{Q}_\ell } \rightarrow \operatorname {GL}(D') \end{equation*}$$

over $\overline{\mathbf{Q}}_\ell$ with open kernel.

Suppose that $\tau$ is an $\ell$-type and that $K$ is a finite extension of $\mathbf{Q}_\ell$ in $\overline{\mathbf{Q}}_\ell$. Recall from CDT that a continuous representation $\rho$ of $G_\ell$ on a two-dimensional $K$-vector space $M$ is said to be of type $\tau$ if

(1)

$\rho$ is Barsotti-Tate over $F$ for any finite extension $F$ of $\mathbf{Q}_\ell$ such that $\tau |_{I_F}$ is trivial;

(2)

the restriction of $WD(\rho )$ to $I_\ell$ is in $\tau$;

(3)

the character $\epsilon ^{-1} \det \rho$ has finite order prime to $\ell$.

(For the definition of “Barsotti-Tate” and of the representation $WD(\rho )$ associated to a potentially Barsotti-Tate representation, see §1.1 and Appendix B of CDT.) Similarly if $\tau '$ is an extended $\ell$-type, then we say that $\rho$ is of extended type $\tau '$ if

(1)

$\rho$ is Barsotti-Tate over $F$ for any finite extension $F$ of $\mathbf{Q}_\ell$ such that $\tau '|_{I_F}$ is trivial;

(2)

$WD(\rho )$ is equivalent to $\tau '$;

(3)

the character $\epsilon ^{-1} \det \rho$ has finite order prime to $\ell$.

Note that no representation can have extended type $\tau '$ unless $\det \tau '$ is of the form $\chi _1\chi _2$ where $\chi _1$ has finite order prime to $\ell$ and where $\chi _2$ is unramified and takes an arithmetic Frobenius element to $\ell$; see Appendix B of CDT. (Using Theorem 1.4 of Br2, one can show that for $\ell$ odd one obtains equivalent definitions of “type $\tau$” and “extended type $\tau '$” if one weakens the first assumption to simply require that $\rho$ is potentially Barsotti-Tate.)

Now fix a finite extension $K$ of $\mathbf{Q}_\ell$ in $\overline{\mathbf{Q}}_\ell$ over which $\tau$ (resp. $\tau '$) is rational. Let ${\mathcal{O}}$ denote the integers of $K$ and let $k$ denote the residue field of ${\mathcal{O}}$. Let

$$\begin{equation*} {\overline{\rho }}:G_{\ell } \longrightarrow \operatorname {GL}(V) \end{equation*}$$

be a continuous representation of $G_\ell$ on a two-dimensional $k$-vector space $V$ and suppose that $\operatorname {End}_{k[G_\ell ]}V = k$. One then has a universal deformation ring $R_{V,{\mathcal{O}}}$ for ${\overline{\rho }}$ (see, for instance, Appendix A of CDT).

We say that a prime ideal $\mathfrak{p}$ of $R_{V,{\mathcal{O}}}$ is of type $\tau$ (resp. of extended type $\tau '$) if there exist a finite extension $K'$ of $K$ in $\overline{\mathbf{Q}}_\ell$ and an ${\mathcal{O}}$-algebra homomorphism $R_{V,{\mathcal{O}}} \to K'$ with kernel $\mathfrak{p}$ such that the pushforward of the universal deformation of $\rho$ over $R_{V,{\mathcal{O}}}$ to $K'$ is of type $\tau$ (resp. of extended type $\tau '$).

Let $\tau$ be an $\ell$-type and $\tau '$ an irreducible extended $\ell$-type. If there do not exist any prime ideals $\mathfrak{p}$ of type $\tau$ (resp. of extended type $\tau '$), we define $R_{V,{\mathcal{O}}}^D = 0$ (resp. $R_{V,{\mathcal{O}}}^{D'}=0$). Otherwise, define $R_{V,{\mathcal{O}}}^D$ (resp. $R_{V,{\mathcal{O}}}^{D'}$) to be the quotient of $R_{V,{\mathcal{O}}}$ by the intersection of all $\mathfrak{p}$ of type $\tau$ (resp. of extended type $\tau '$). We will sometimes write $R_{V,{\mathcal{O}}}^\tau$ (resp. $R_{V,{\mathcal{O}}}^{\tau '}$) for $R_{V,{\mathcal{O}}}^D$ (resp. $R_{V,{\mathcal{O}}}^{D'}$). We say that a deformation of ${\overline{\rho }}$ is weakly of type $\tau$ (resp. weakly of extended type $\tau '$) if the associated local ${\mathcal{O}}$-algebra map $R_{V,{\mathcal{O}}} \rightarrow R$ factors through the quotient $R_{V,{\mathcal{O}}}^D$ (resp. $R_{V,{\mathcal{O}}}^{D'}$). We say that $\tau$ (resp. $\tau '$) is weakly acceptable for ${\overline{\rho }}$ if either $R_{V,{\mathcal{O}}}^D = 0$ (resp. $R_{V,{\mathcal{O}}}^{D'} = 0$) or there is a surjective local ${\mathcal{O}}$-algebra map ${\mathcal{O}}\lBrack X\rBrack \twoheadrightarrow R_{V,{\mathcal{O}}}^D$ (resp. ${\mathcal{O}}\lBrack X\rBrack \twoheadrightarrow R_{V,{\mathcal{O}}}^{D'}$). We say that $\tau$ (resp. $\tau '$) is acceptable for ${\overline{\rho }}$ if $R_{V,{\mathcal{O}}}^D \ne 0$ (resp. $R_{V,{\mathcal{O}}}^{D'} \ne 0$) and if there is a surjective local ${\mathcal{O}}$-algebra map ${\mathcal{O}}\lBrack X\rBrack \twoheadrightarrow R_{V,{\mathcal{O}}}^D$ (resp. ${\mathcal{O}}\lBrack X\rBrack \twoheadrightarrow R_{V,{\mathcal{O}}}^{D'}$).

If $K'$ is a finite extension of $K$ in $\overline{\mathbf{Q}}_\ell$ with valuation ring ${\mathcal{O}}'$ and residue field $k'$, then ${\mathcal{O}}' \otimes _{{\mathcal{O}}} R_{V,{\mathcal{O}}}^D$ (resp. ${\mathcal{O}}' \otimes _{{\mathcal{O}}} R_{V,{\mathcal{O}}}^{D'}$) is naturally isomorphic to $R_{V\otimes _k k',{\mathcal{O}}'}^{D}$ (resp. $R_{V\otimes _k k', {\mathcal{O}}'}^{D'}$). Thus (weak) acceptability depends only on $\tau$ (resp. $\tau '$) and ${\overline{\rho }}$, and not on the choice of $K$. Moreover $\tau$ (resp. $\tau '$) is acceptable for ${\overline{\rho }}$ if and only if $\tau$ (resp. $\tau '$) is acceptable for ${\overline{\rho }}\otimes _k k'$.

Although it is of no importance for the sequel, we make the following conjecture, part of which we already conjectured as Conjecture 1.2.1 of CDT.

Conjecture 1.1.1.

Suppose that $\tau$ is an $\ell$-type and $\tau '$ an absolutely irreducible extended $\ell$-type. A deformation $\rho :G_\ell \rightarrow \operatorname {GL}(M)$ of ${\overline{\rho }}$ to the ring of integers ${\mathcal{O}}'$ of a finite extension $K'/K$ in $\overline{\mathbf{Q}}_\ell$ is weakly of type $\tau$ (resp. weakly of extended $\ell$-type $\tau '$) if and only if $M$ is of type $\tau$ (resp. of extended type $\tau '$).

If $\tau$ is a tamely ramified $\ell$-type, then we expect that it is frequently the case that $\tau$ is acceptable for residual representations ${\overline{\rho }}$, as in Conjectures 1.2.2 and 1.2.3 of CDT. On the other hand if $\tau$ (resp. $\tau '$) is a wildly ramified $\ell$-type (resp. wildly ramified extended $\ell$-type), then we expect that it is rather rare that $\tau$ (resp. $\tau '$) is acceptable for a residual representation ${\overline{\rho }}$. In this paper we will be concerned with a few wild cases for the prime $\ell =3$ which do turn out to be acceptable.

1.2. Types for admissible representations

From now on we assume that $\ell$ is odd. If $F$ is a finite extension of $\mathbf{Q}_\ell$ we will identify $F^\times$ with $W_F^{\operatorname {ab}}$ via the Artin map. Let $U_0(\ell ^r)$ denote the subgroup of ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ consisting of elements with upper triangular $\bmod \, \ell ^r$ reduction. Also let $\widetilde{U}_0(\ell )$ denote the normaliser of $U_0(\ell )$ in ${\operatorname {GL}}_2(\mathbf{Q}_\ell )$. Thus $\widetilde{U}_0(\ell )$ is generated by $U_0(\ell )$ and by

$$\begin{equation} w_\ell = \left( \begin{array}{cc} 0 & -1 \\\ell & 0 \end{array} \right) .{\tag{1.2.1}} \end{equation}$$

If $\tau$ is an $\ell$-type, set $U_\tau = {\operatorname {GL}}_2(\mathbf{Z}_\ell )$ if $\tau$ is reducible and $U_\tau =U_0(\ell )$ if $\tau$ is irreducible. If $\tau '$ is an extended $\ell$-type with $\tau '|_{I_\ell }$ irreducible, set $U_{\tau '} = \widetilde{U}_0(\ell )$. In this subsection we will associate to an $\ell$-type $\tau$ an irreducible representation $\sigma _\tau$ of $U_\tau$ over $\overline{\mathbf{Q}}_\ell$ with open kernel, and to an extended $\ell$-type $\tau '$ with $\tau '|_{I_\ell }$ irreducible an irreducible representation $\sigma _{\tau '}$ of $U_{\tau '}$ over $\overline{\mathbf{Q}}_\ell$ with open kernel. We need to consider several cases, which we treat one at a time.

First suppose that $\tau = \chi _1|_{I_\ell } \oplus \chi _2|_{I_\ell }$ where each $\chi _i$ is a character of $W_{\mathbf{Q}_\ell }$. Let $a$ denote the conductor of $\chi _1/\chi _2$. If $a=0$, then set

$$\begin{equation*} \sigma _\tau = \operatorname {St}\otimes (\chi _1 \circ \det ) = \operatorname {St}\otimes (\chi _2 \circ \det ), \end{equation*}$$

where $\operatorname {St}$ denotes the Steinberg representation of $\operatorname {PGL}_2(\mathbf{F}_\ell )$. Now suppose that $a>0$. Let $\sigma _\tau$ denote the induction from $U_0(\ell ^a)$ to ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ of the character of $U_0(\ell ^a)$ which sends

$$\begin{equation*} \left( \begin{array}{cc} \alpha & \beta \\\ell ^a\gamma & \delta \end{array} \right) \longmapsto (\chi _1/\chi _2)(\alpha ) \chi _2(\alpha \delta -\ell ^a\beta \gamma ). \end{equation*}$$

This is irreducible and does not depend on the ordering of $\chi _1$ and $\chi _2$.

For the next case, let $F$ denote the unramified quadratic extension of $\mathbf{Q}_\ell$ and $s$ the non-trivial automorphism of $F$ over $\mathbf{Q}_\ell$. Suppose that $\tau$ is the restriction to $I_\ell$ of the induction from $W_F$ to $W_{\mathbf{Q}_\ell }$ of a character $\chi$ of $W_F$ with $\chi \neq \chi ^s$. Let $a$ denote the conductor of $\chi /\chi ^s$, so that $a>0$. Choose a character $\chi '$ of $W_{\mathbf{Q}_\ell }$ such that $\chi '|_{W_F}^{-1}\chi$ has conductor $a$. If $a=1$ we set

$$\begin{equation*} \sigma _\tau = \Theta (\chi '|_{W_F}^{-1}\chi ) \otimes (\chi ' \circ \det ), \end{equation*}$$

where $\Theta (\cdot )$ is the irreducible representation of ${\operatorname {GL}}_2(\mathbf{F}_\ell )$ defined on page 532 of CDT.

To define $\sigma _\tau$ for $a>1$ we will identify ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ with the automorphisms of the $\mathbf{Z}_\ell$-module ${\mathcal{O}}_F$. If $a$ is even, then we let $\sigma _\tau$ denote the induction from ${\mathcal{O}}_F^\times (1+\ell ^{a/2}{\mathcal{O}}_F s)$ to ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ of the character $\varphi$ of ${\mathcal{O}}_F^\times (1+\ell ^{a/2}{\mathcal{O}}_F s)$, where, for $\alpha \in {\mathcal{O}}_F^\times$ and $\beta \in (1+\ell ^{a/2}{\mathcal{O}}_F s)$,

$$\begin{equation*} \varphi (\alpha \beta )= (\chi '|_{W_F}^{-1}\chi )(\alpha ) \chi '(\det \alpha \beta ). \end{equation*}$$

If $a>1$ is odd, then we let $\sigma _\tau$ denote the induction from ${\mathcal{O}}_F^\times (1+\ell ^{(a-1)/2}{\mathcal{O}}_F s)$ to ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ of $\eta$, where $\eta$ is the $\ell$-dimensional irreducible representation of ${\mathcal{O}}_F^\times (1+\ell ^{(a-1)/2}{\mathcal{O}}_F s)$ such that $\eta |_{{\mathcal{O}}_F^\times (1+\ell ^{(a+1)/2}{\mathcal{O}}_F s)}$ is the direct sum of the characters

$$\begin{equation*} \alpha \beta \longmapsto (\chi '|_{W_F}^{-1}\chi \chi '')(\alpha ) \chi '(\det \alpha \beta ) \end{equation*}$$

for $\alpha \in {\mathcal{O}}_F^\times$ and $\beta \in (1+\ell ^{(a+1)/2}{\mathcal{O}}_F s)$, where $\chi ''$ runs over the $\ell$ non-trivial characters of ${\mathcal{O}}_F^\times / \mathbf{Z}_\ell ^\times (1+\ell {\mathcal{O}}_F)$.

Now suppose $\tau '$ is an extended type such that $\tau '|_{I_\ell }$ is irreducible. There is a ramified quadratic extension $F/\mathbf{Q}_\ell$ and a character $\chi$ of $W_F$ such that the induction from $W_F$ to $W_{\mathbf{Q}_\ell }$ of $\chi$ is $\tau '$ (see §2.6 of G). Let $s$ denote the non-trivial field automorphism of $F$ over $\mathbf{Q}_\ell$ and also let $\wp _F$ denote the maximal ideal of the ring of integers ${\mathcal{O}}_F$ of $F$. Let $a$ denote the conductor of $\chi /\chi ^s$, so $a$ is even and $a \geq 2$. We may choose a character $\chi '$ of $W_{\mathbf{Q}_\ell }$ such that $\chi '|_{W_F}^{-1} \chi$ has conductor $a$. We will identify ${\operatorname {GL}}_2(\mathbf{Q}_\ell )$ with the automorphisms of the $\mathbf{Q}_\ell$ vector space $F$. We will also identify $U_0(\ell )$ with the stabiliser of the pair of lattices $\wp _F^{-1} \supset {\mathcal{O}}_F$. We define $\sigma _{\tau '}$ to be the induction from $F^\times (1+\wp _F^{a/2}s)$ to $\widetilde{U}_0(\ell )$ of the character $\varphi$ of $F^\times (1+\wp _F^{a/2}s)$, where

$$\begin{equation*} \varphi (\alpha \beta )= (\chi '|_{W_F}^{-1} \chi \chi '')(\alpha ) \chi ' (\det \alpha \beta ), \end{equation*}$$

with $\alpha \in F^\times$ and $\beta \in (1+\wp _F^{a/2}s)$, where $\chi ''$ is a character of $F^\times /({\mathcal{O}}_F^\times )^2$ defined as follows. Let $\psi$ be a character of $\mathbf{Q}_\ell$ with kernel $\mathbf{Z}_\ell$. Choose $\theta \in F^\times$ such that for $x \in \wp _F^{a-1}$ we have

$$\begin{equation*} (\chi '|_{W_F}^{-1} \chi )(1+x) = \psi (\operatorname {tr}_{F/\mathbf{Q}_\ell } (\theta x)). \end{equation*}$$

We impose the following conditions which determine $\chi ''$:

•

$\chi ''$ is a character of $F^\times /({\mathcal{O}}_F^\times )^2$;

•

$\chi ''|_{{\mathcal{O}}_F^\times }$ is non-trivial;

•

and$$\begin{equation*} \chi ''(-\theta (N_{F/\mathbf{Q}_\ell } \varpi )^{a/2}) = \sum _{x \in \mathbf{Z}/\ell \mathbf{Z}} \psi (x^2/N_{F/\mathbf{Q}_\ell } \varpi ), \end{equation*}$$

where $\varpi$ is a uniformiser in ${\mathcal{O}}_F$.

Finally if $\tau$ is an irreducible $\ell$-type, choose an extended $\ell$-type $\tau '$ which restricts to $\tau$ on $I_\ell$ and set $\sigma _\tau = \sigma _{\tau '}|_{U_0(\ell )}$.

We remark that these definitions are independent of any choices (see G).

Recall that by the local Langlands conjecture we can associate to an irreducible admissible representation $\pi$ of ${\operatorname {GL}}_2(\mathbf{Q}_\ell )$ a two-dimensional representation $WD(\pi )$ of $W_{\mathbf{Q}_\ell }$. (See §4.1 of CDT for the normalisation we use.)

Lemma 1.2.1.

Suppose that $\tau$ is an $\ell$-type and that $\tau '$ is an extended $\ell$-type with $\tau '|_{I_\ell }$ irreducible. Suppose also that $\pi$ is an infinite-dimensional irreducible admissible representation of ${\operatorname {GL}}_2(\mathbf{Q}_\ell )$ over $\overline{\mathbf{Q}}_\ell$. Then:

(1)

$\sigma _\tau$ and $\sigma _{\tau '}$ are irreducible.

(2)

If $WD(\pi )|_{I_\ell } \sim \tau$ (resp. $WD(\pi ) \sim \tau '$), then$$\begin{equation*} \operatorname {Hom}_{U_\tau }(\sigma _\tau ,\pi ) \cong \overline{\mathbf{Q}}_\ell \end{equation*}$$

(resp.$$\begin{equation*} \operatorname {Hom}_{U_{\tau '}}(\sigma _{\tau '},\pi ) \cong \overline{\mathbf{Q}}_\ell ). \end{equation*}$$

(3)

If $WD(\pi )|_{I_\ell } \not \cong \tau$ (resp. $WD(\pi ) \not \cong \tau '$), then$$\begin{equation*} \operatorname {Hom}_{U_\tau }(\sigma _\tau ,\pi ) = (0) \end{equation*}$$

(resp.$$\begin{equation*} \operatorname {Hom}_{U_{\tau '}}(\sigma _{\tau '},\pi ) = (0) ). \end{equation*}$$

Proof.

The case that $\tau$ extends to a reducible representation of $W_{\mathbf{Q}_\ell }$ follows from the standard theory of principal series representations for ${\operatorname {GL}}_2(\mathbf{Q}_\ell )$. The case that $\tau$ is reducible but does not extend to a reducible representation of $W_{\mathbf{Q}_\ell }$ follows from Theorem 3.7 of G. The case of $\tau '$ follows from Theorem 4.6 of G.

Thus, suppose that $\tau$ is an irreducible $\ell$-type and that $\tau '$ is an extension of $\tau$ to an extended $\ell$-type. If $\delta$ denotes the unramified quadratic character of $W_{\mathbf{Q}_\ell }$, then $\tau ' \not \sim \tau ' \otimes \delta$ and so we deduce that

$$\begin{equation*} \sigma _{\tau '} \not \sim \sigma _{\tau ' \otimes \delta } \sim \sigma _{\tau '} \otimes (\delta \circ \det ). \end{equation*}$$

Thus $\sigma _{\tau '}|_{\mathbf{Q}_{\ell }^\times U_0(\ell )}$ is irreducible. It follows that $\sigma _\tau$ is irreducible. The second and third part of the lemma for $\tau$ follow similarly.

■

1.3. Reduction of types for admissible representations

We begin by reviewing some irreducible representations of ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$, $U_0(\ell )$ and $\widetilde{U}_0(\ell )$. Let $\sigma _{1,0}$ denote the standard representation of ${\operatorname {GL}}_2(\mathbf{F}_\ell )$ over $\overline{\mathbf{F}}_\ell$. If $n=0,1,...,\ell -1$ and if $m \in \mathbf{Z}/(\ell -1)\mathbf{Z}$, then we let $\sigma _{n,m} =\operatorname {Symm}^n(\sigma _{1,0}) \otimes \det ^m$. We may think of $\sigma _{n,m}$ as a continuous representation of ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ over $\overline{\mathbf{F}}_\ell$. These representations are irreducible, mutually non-isomorphic and exhaust the irreducible continuous representations of ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ over $\overline{\mathbf{F}}_\ell$.

If $m_1,m_2 \in \mathbf{Z}/(\ell -1)\mathbf{Z}$ we let $\sigma '_{m_1,m_2}$ denote the character of $U_0(\ell )$ over $\overline{\mathbf{F}}_\ell$ determined by

$$\begin{equation*} \left( \begin{array}{cc} a & b \\\ell c & d \end{array} \right) \longmapsto a^{m_1} d^{m_2}. \end{equation*}$$

These representations are irreducible, mutually non-isomorphic, and exhaust the irreducible continuous representations of $U_0(\ell )$ over $\overline{\mathbf{F}}_\ell$.

If $m_1,m_2 \in \mathbf{Z}/(\ell -1)\mathbf{Z}$, $a \in \overline{\mathbf{F}}_\ell ^\times$ and $m_1 \neq m_2$, then we let $\sigma '_{\{ m_1,m_2\}, a}$ denote the representation of $\widetilde{U}_0(\ell )$ over $\overline{\mathbf{F}}_\ell$ obtained by inducing the character of $\mathbf{Q}_\ell ^\times U_0(\ell )$ which restricts to $\sigma '_{m_1,m_2}$ on $U_0(\ell )$ and which sends $-\ell$ to $a$. If $m \in \mathbf{Z}/(\ell -1)\mathbf{Z}$ and $a \in \overline{\mathbf{F}}_\ell ^\times$, then we let $\sigma '_{\{ m \},a}$ denote the character of $\widetilde{U}_0(\ell )$ over $\overline{\mathbf{F}}_\ell$ which restricts to $\sigma '_{m,m}$ on $U_0(\ell )$ and which sends $w_\ell$ to $a$. These representations are irreducible, mutually non-isomorphic and exhaust the irreducible, finite-dimensional, continuous representations of $\widetilde{U}_0(\ell )$ over $\overline{\mathbf{F}}_\ell$.

We will say that a reducible $\ell$-type $\tau$ (resp. irreducible $\ell$-type, resp. extended $\ell$-type $\tau$ with irreducible restriction to $I_\ell$) admits an irreducible representation $\sigma$ of ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ (resp. $U_0(\ell )$, resp. $\widetilde{U}_0(\ell )$) over $\overline{\mathbf{F}}_\ell$, if $\sigma _\tau$ (resp. $\sigma _\tau$, resp. $\sigma _{\tau '}$) contains an invariant ${\mathcal{O}}_{\overline{\mathbf{Q}}_\ell }$-lattice $\Lambda$ and if $\sigma$ is a Jordan-Hölder constituent of $\Lambda \otimes \overline{\mathbf{F}}_\ell$. We will say that $\tau$ (resp. $\tau$, resp. $\tau '$) simply admits $\sigma$ if $\sigma$ is a Jordan-Hölder constituent of $\Lambda \otimes \overline{\mathbf{F}}_\ell$ of multiplicity one.

For each of the $\overline{\mathbf{F}}_\ell$-representations of ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$, $U_0(\ell )$ and $\widetilde{U}_0(\ell )$ just defined, we wish to define notions of “admittance” and “simple admittance” with respect to a continuous representation ${\overline{\rho }}:G_\ell \rightarrow {\operatorname {GL}}_2(\overline{\mathbf{F}}_\ell )$. Let ${\overline{\rho }}$ be a fixed continuous representation $G_\ell \rightarrow {\operatorname {GL}}_2(\overline{\mathbf{F}}_\ell )$.

•

The representation $\sigma _{n,m}$ admits ${\overline{\rho }}$ if either$$\begin{equation*} {\overline{\rho }}|_{I_\ell } \sim \left( \begin{array}{cc} \omega _2^{1-\ell n-m(\ell +1)} & 0 \\0 & \omega _2^{\ell - n-m(\ell +1)} \end{array} \right) \end{equation*}$$

or$$\begin{equation*} {\overline{\rho }}|_{I_\ell } \sim \left( \begin{array}{cc} \omega ^{1-m} & * \\0 & \omega ^{-n-m} \end{array} \right), \end{equation*}$$

which in addition we require to be peu-ramifié in the case $n=0$. (Note that $\sigma _{n,0}$ admits ${\overline{\rho }}$ if and only if the Serre weight (see Se2) of ${\overline{\rho }}^\vee \otimes \omega$ is $n+2$.)

•

The representation $\sigma _{n,m}$ simply admits ${\overline{\rho }}$ if $\sigma _{n,m}$ admits ${\overline{\rho }}$.

•

The representation $\sigma _{m_1,m_2}'$ admits ${\overline{\rho }}$ if either$$\begin{equation*} {\overline{\rho }}|_{I_\ell } \sim \left( \begin{array}{cc} \omega _2^{1-\ell m_i- m_j} & 0 \\0 & \omega _2^{\ell - m_i-\ell m_j} \end{array} \right), \end{equation*}$$

where $\{ m_i,m_j \} = \{ m_1,m_2 \}$ and $m_i \geq m_j$, or$$\begin{equation*} {\overline{\rho }}|_{I_\ell } \sim \left( \begin{array}{cc} \omega ^{1-m_1} & * \\0 & \omega ^{-m_2} \end{array} \right), \end{equation*}$$

or$$\begin{equation*} {\overline{\rho }}|_{I_\ell } \sim \left( \begin{array}{cc} \omega ^{1-m_2} & * \\0 & \omega ^{-m_1} \end{array} \right). \end{equation*}$$

(Note that $\sigma _{m_1,m_2}'$ admits ${\overline{\rho }}$ if and only if some irreducible constituent of $\operatorname {Ind}_{U_0(\ell )}^{{\operatorname {GL}}_2(\mathbf{Z}_\ell )} \sigma _{m_1,m_2}'$ admits ${\overline{\rho }}$.)

•

The representation $\sigma _{m_1,m_2}'$ with $m_1\neq m_2$ simply admits ${\overline{\rho }}$ if either$$\begin{equation*} {\overline{\rho }}|_{I_\ell } \sim \left( \begin{array}{cc} \omega ^{1-m_1} & * \\0 & \omega ^{-m_2} \end{array} \right) \end{equation*}$$

or$$\begin{equation*} {\overline{\rho }}|_{I_\ell } \sim \left( \begin{array}{cc} \omega ^{1-m_2} & * \\0 & \omega ^{-m_1} \end{array} \right). \end{equation*}$$

•

The representation $\sigma _{m,m}'$ simply admits ${\overline{\rho }}$ if$$\begin{equation*} {\overline{\rho }}|_{I_\ell } \sim \left( \begin{array}{cc} \omega ^{1-m} & * \\0 & \omega ^{-m} \end{array} \right) \end{equation*}$$

is très ramifié.

•

The representation $\sigma '_{\{ m_1,m_2\}, a}$ with $m_1 \neq m_2$ admits ${\overline{\rho }}$ if either $\sigma '_{m_1,m_2}$ or $\sigma '_{m_2,m_1}$ admits ${\overline{\rho }}$ and if $(\omega ^{-1} \det {\overline{\rho }})|_{W_{\mathbf{Q}_\ell }}$ equals the central character of $\sigma '_{\{ m_1,m_2\}, a}$. (Note that in this case $\sigma '_{\{ m_1,m_2\}, a}|_{U_0(\ell )}= \sigma _{m_1,m_2}' \oplus \sigma _{m_2,m_1}'$.)

•

The representation $\sigma '_{\{ m_1,m_2\}, a}$ with $m_1\! \neq \! m_2$ simply admits ${\overline{\rho }}$ if $(\omega ^{-1}\! \det {\overline{\rho }})|_{W_{\mathbf{Q}_\ell }}$ equals the central character of $\sigma '_{\{ m_1,m_2\}, a}$ and either$$\begin{equation*} {\overline{\rho }}|_{I_\ell } \sim \left( \begin{array}{cc} \omega ^{1-m_1} & * \\0 & \omega ^{-m_2} \end{array} \right) \end{equation*}$$

or$$\begin{equation*} {\overline{\rho }}|_{I_\ell } \sim \left( \begin{array}{cc} \omega ^{1-m_2} & * \\0 & \omega ^{-m_1} \end{array} \right). \end{equation*}$$

•

The representation $\sigma '_{\{ m \}, a}$ admits ${\overline{\rho }}$ if

–

$\sigma '_{m,m}$ admits ${\overline{\rho }}$,

–

$(\omega ^{-1} \det {\overline{\rho }})|_{W_{\mathbf{Q}_\ell }}$ equals the central character of $\sigma '_{\{ m \}, a}$,

–

and, if$$\begin{equation*} {\overline{\rho }}|_{I_\ell } \sim \left( \begin{array}{cc} \omega ^{1-m} & * \\0 & \omega ^{-m} \end{array} \right) \end{equation*}$$

is très ramifié, then$$\begin{equation*} {\overline{\rho }}\sim \left( \begin{array}{cc} * & * \\0 & \omega ^{-m} \chi \end{array} \right), \end{equation*}$$

where $\chi$ is unramified and sends Frobenius to $-a$.

(Note that $\sigma '_{\{ m \}, a}|_{U_0(\ell )}=\sigma _{m,m}'$.)

•

The representation $\sigma '_{\{ m \}, a}$ simply admits ${\overline{\rho }}$ if $\sigma '_{\{ m \}, a}$ admits ${\overline{\rho }}$.

We remark that the definition of “$\sigma$ admits the Cartier dual of ${\overline{\rho }}$” might look more natural to the reader. We are forced to adopt this version of the definition by some unfortunate choices of normalisations in CDT.

We say that a reducible $\ell$-type $\tau$ (resp. irreducible $\ell$-type $\tau$, resp. extended $\ell$-type $\tau '$ with $\tau '|_{I_\ell }$ irreducible) admits a continuous representation ${\overline{\rho }}:G_\ell \rightarrow {\operatorname {GL}}_2(\overline{\mathbf{F}}_\ell )$ if $\tau$ (resp. $\tau$, resp. $\tau '$) admits an irreducible representation of ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ (resp. $U_0(\ell )$, resp. $\widetilde{U}_0(\ell )$) over $\overline{\mathbf{F}}_\ell$ which in turn admits ${\overline{\rho }}$. We say that $\tau$ (resp. $\tau$, resp. $\tau '$) simply admits ${\overline{\rho }}$ if

•

$\tau$ (resp. $\tau$, resp. $\tau '$) admits a unique irreducible representation $\sigma$ of ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ (resp. $U_0(\ell )$, resp. $\widetilde{U}_0(\ell )$) over $\overline{\mathbf{F}}_\ell$ which admits ${\overline{\rho }}$,

•

$\tau$ (resp. $\tau$, resp. $\tau '$) simply admits $\sigma$,

•

and $\sigma$ simply admits ${\overline{\rho }}$.

Note that the concept of “simply admits” is strictly stronger than the concept “admits”.

The starting point for this work was the following conjecture, of which a few examples will be verified in §2.1.

Conjecture 1.3.1.

Let $k$ be a finite subfield of $\overline{\mathbf{F}}_\ell$, ${\overline{\rho }}:G_\ell \rightarrow {\operatorname {GL}}_2(k)$ a continuous representation, $\tau$ an $\ell$-type and $\tau '$ an extended $\ell$-type with irreducible restriction to $I_\ell$. Suppose that $\det \tau$ and $\det \tau '$ are tamely ramified, that the centraliser of the image of ${\overline{\rho }}$ is $k$ and that the image of $\tau$ is not contained in the centre of ${\operatorname {GL}}_2(\overline{\mathbf{Q}}_\ell )$.

(1)

$\tau$ (resp. $\tau '$) admits ${\overline{\rho }}$ if and only if $R_{V,{\mathcal{O}}}^D\neq (0)$ (resp. $R_{V,{\mathcal{O}}}^{D'} \neq (0)$), i.e. if and only if there is a finite extension $K'$ of $\mathbf{Q}_\ell$ in $\overline{\mathbf{Q}}_\ell$ and a continuous representation $\rho :G_\ell \rightarrow {\operatorname {GL}}_2({\mathcal{O}}_{K'})$ which reduces to ${\overline{\rho }}$ and has type $\tau$ (resp. has extended type $\tau '$).

(2)

$\tau$ (resp. $\tau '$) simply admits ${\overline{\rho }}$ if and only if $\tau$ (resp. $\tau '$) is acceptable for ${\overline{\rho }}$.

We remark that to check if $\tau$ or $\tau '$ simply admits ${\overline{\rho }}$ is a relatively straightforward computation. On the other hand to show that $\tau$ or $\tau '$ is acceptable for ${\overline{\rho }}$ is at present a non-trivial undertaking. (The reader who doubts us might like to compare §3 with §4, §5, §6, §7, §8 and §9. All the latter sections are devoted to verifying some very special cases of this conjecture.)

1.4. The main theorems

With these definitions, we can state our two main theorems. The proofs very closely parallel the proof of Theorem 7.1.1 of CDT.

Theorem 1.4.1.

Let $\ell$ be an odd prime, $K$ a finite extension of $\mathbf{Q}_\ell$ in $\overline{\mathbf{Q}}_\ell$ and $k$ the residue field of $K$. Let

$$\begin{equation*} \rho :G_\mathbf{Q}\longrightarrow {\operatorname {GL}}_2(K) \end{equation*}$$

be an odd continuous representation ramified at only finitely many primes. Assume that its reduction

$$\begin{equation*} {\overline{\rho }}:G_\mathbf{Q}\longrightarrow {\operatorname {GL}}_2(k) \end{equation*}$$

is absolutely irreducible after restriction to $\mathbf{Q}(\sqrt {(-1)^{(\ell -1)/2} \ell })$ and is modular. Further, suppose that

•

${\overline{\rho }}|_{G_\ell }$ has centraliser $k$,

•

$\rho |_{G_\ell }$ is potentially Barsotti-Tate with $\ell$-type $\tau$,

•

$\tau$ admits ${\overline{\rho }}$,

•

and $\tau$ is weakly acceptable for ${\overline{\rho }}$.

Then $\rho$ is modular.

Proof.

Note that the existence of $\rho$ shows that $\tau$ is acceptable for ${\overline{\rho }}$. Now the proof is verbatim the proof of Theorem 7.1.1 of CDT (see §1.3, §1.4, §3, §4, §5 and §6 of that paper, and the corrigendum at the end of this paper), with the following exceptions.

•

On page 539 one should take $U_{S,\ell }=U_\tau$, $V_{S,\ell }=\ker \sigma _\tau$ and $\sigma _{S,l}=\sigma _\tau$.

•

In the proof of Lemma 5.1.1 one must use Lemma 1.2.1 of this paper, in addition to the results recalled in §4 of CDT.

•

On page 546 replace “Setting $S=T({\overline{\rho }}) \cup \{ r\}$ ...” to the end of the first paragraph by the following. (Again the key component of this argument is very similar to the main idea of Kh.)

“Set $S=T({\overline{\rho }}) \cup \{ r\}$; $U_S' = \prod _p U_{S,p}'$ where $U_{S,p}'=U_1(p^{c_p})$ if $p \in T({\overline{\rho }})$ and $U_{S,p}'=U_{S,p}$ otherwise; $V_S' = \prod _p V_{S,p}'$ where $V_{S,p}'=U_1(p^{c_p})$ if $p \in T({\overline{\rho }})$ and $V_{S,p}'=V_{S,p}$ otherwise; and $L_S'=\operatorname {Hom}_{{\mathcal{O}}[U_S'/V_S']} (M_\ell ,H^1(X_{V_S'},{\mathcal{O}}))[I_S']$. Then $\Gamma =\operatorname {SL}_2(\mathbf{Z}) \cap (U_S' {\operatorname {GL}}_2(\mathbf{Z}_\ell ))$ satisfies the hypotheses of Theorem 6.1.1. Furthermore$$\begin{equation*} H^1(Y_{U_S'{\operatorname {GL}}_2(\mathbf{Z}_\ell )},{\mathcal{F}}_M) \cong H^1(\Gamma ,L_n \otimes k) \end{equation*}$$

as a $\widetilde{\operatorname {\mathbf{T}}}_S'$-module, where $M$ is the module for $U_{S,\ell }= {\operatorname {GL}}_2(\mathbf{Z}_\ell )$ defined by the action of ${\operatorname {GL}}_2(\mathbf{F}_\ell )$ on $L_n \otimes k$. Therefore $\mathfrak{m}_S$ is in the support of $H^1(Y_{U_S'{\operatorname {GL}}_2(\mathbf{Z}_\ell )},{\mathcal{F}}_M)$.

We now drop the special assumption on ${\overline{\rho }}|_{I_\ell }$ made in the last paragraph. Twisting we see that if $\sigma$ is an irreducible representation of $\operatorname {GL}_2(\mathbf{Z}_\ell )$ over $\overline{\mathbf{F}}_\ell$ admitting ${\overline{\rho }}|_{G_\ell }$, then$$\begin{equation*} H^1(Y_{U_S'\operatorname {GL}_2(\mathbf{Z}_\ell )},{\mathcal{F}}_{\sigma ^\vee })_{\mathfrak{m}_S} \neq (0). \end{equation*}$$

Moreover if $\tau$ is irreducible and if $\sigma '$ is an irreducible representation of $U_0(\ell )$ over $\overline{\mathbf{F}}_\ell$ which admits ${\overline{\rho }}|_{G_\ell }$, then we see using the definition of admits and Lemmas 3.1.1 and 6.1.2 of CDT that$$\begin{equation*} H^1(Y_{U_S'},{\mathcal{F}}_{\sigma ^\vee })_{\mathfrak{m}_S} \cong H^1(Y_{U_S'\operatorname {GL}_2(\mathbf{Z}_\ell )}, {\mathcal{F}}_{\operatorname {Ind}_{U_0(\ell )}^{\operatorname {GL}_2(\mathbf{Z}_\ell )}\sigma ^\vee })_{\mathfrak{m}_S} \neq (0). \end{equation*}$$

It follows from the definition of admits and Lemma 6.1.2 of CDT that $\mathfrak{m}_S$ is in the support of $H^1(Y_{U_S'},{\mathcal{F}}_{\operatorname {Hom}_{{\mathcal{O}}}(M_\ell ,{\mathcal{O}})})$, so $L_S'$ is non-zero. Using the fact that Lemma 5.1.1 holds with $U_S'$ replacing $U_S$ and $\sigma _\ell$ replacing $\sigma _S$ and the discussion on page 541 we conclude that ${\operatorname {\mathcal{N}}}_S$ is non-empty.”

■

Theorem 1.4.2.

Let $\ell$ be an odd prime, $K$ a finite extension of $\mathbf{Q}_\ell$ in $\overline{\mathbf{Q}}_\ell$ and $k$ the residue field of $K$. Let

$$\begin{equation*} \rho :G_\mathbf{Q}\longrightarrow {\operatorname {GL}}_2(K) \end{equation*}$$

be an odd continuous representation ramified at only finitely many primes. Assume that its reduction

$$\begin{equation*} {\overline{\rho }}:G_\mathbf{Q}\longrightarrow {\operatorname {GL}}_2(k) \end{equation*}$$

is absolutely irreducible after restriction to $\mathbf{Q}(\sqrt {(-1)^{(\ell -1)/2} \ell })$ and is modular. Further, suppose that

•

${\overline{\rho }}|_{G_\ell }$ has centraliser $k$,

•

$\rho |_{G_\ell }$ is potentially Barsotti-Tate with extended $\ell$-type $\tau '$,

•

$\tau '$ admits ${\overline{\rho }}$,

•

and $\tau '$ is weakly acceptable for ${\overline{\rho }}$.

Then $\rho$ is modular.

Proof.

The existence of $\rho$ shows that $\tau '$ is in fact acceptable for ${\overline{\rho }}$. Again the proof now follows very closely that of Theorem 7.1.1 of CDT. In this case we have to make the following changes. All references are to CDT unless otherwise indicated.

•

On page 539 one should take $U_{S,\ell }=U_0(\ell )$, $V_{S,\ell }=\ker \sigma _{\tau '}|_{U_0(\ell )}$ and $\sigma _{S,\ell }=\sigma _{\tau '}|_{U_0(\ell )}$. One should also define $\widetilde{U}_S$ to be the group generated by $U_S$ and $w_\ell \in {\operatorname {GL}}_2(\mathbf{Q}_\ell )$ and $\widetilde{\sigma }_S$ to be the extension of $\sigma _S$ to $\widetilde{U}_S$ which restricts to $\sigma _{\tau '}$ on $\widetilde{U}_0(\ell )$.

•

In the statement of Lemma 5.1.1 one should replace $\operatorname {Hom}_{U_S}(\sigma _S, \pi ^\infty )$ by $\operatorname {Hom}_{\widetilde{U}_S}(\widetilde{\sigma }_S,\pi ^\infty )$.

•

In the proof of Lemma 5.1.1 one must use Lemma 1.2.1 above in addition to the results recalled in §4 of CDT.

•

Because $\tau '$ is acceptable for ${\overline{\rho }}$, we know that $\det \tau$ of a Frobenius lift is $\ell \zeta$ for some root of unity $\zeta$. Thus, $\sigma _{\tau '}(\ell ^s)=1$ for some $s>0$. Hence, $\widetilde{\sigma }_S$ factors through the finite group $\widetilde{G}_S=\widetilde{U}_S/V_S \ell ^{s \mathbf{Z}}$, where $\ell \in {\operatorname {GL}}_2(\mathbf{Q}_\ell )$.

•

In §5.3 choose $M_\ell$ so that it is invariant for the action of $\widetilde{U}_0(\ell )/V_{S,\ell } \ell ^{s\mathbf{Z}}$. Also, in the definition of $L_S$ replace $G_S$ by $\widetilde{G}_S$.

•

In the proof of Lemma 5.3.1 replace $U_S$ by $\widetilde{U}_S$ and $\sigma _S$ by $\widetilde{\sigma }_S$.

•

Note that $w_\ell$ acts naturally on $Y_S$ and ${\mathcal{F}}_S$. In Lemma 6.1.3 we should replace the group $H^1_c(Y_S,{\mathcal{F}}_S)$ by $H^1_c(Y_S,{\mathcal{F}}_S)^{w_\ell =1}$ and the group $H^1(Y_S,{\mathcal{F}}_S)$ by $H^1(Y_S,{\mathcal{F}}_S)^{w_\ell =1}$.

•

Replace §6.2 with the proof of the required extension of Proposition 5.4.1 given below.

•

On page 547 the isomorphism$$\begin{equation*} H^1_c(Y_S,{\mathcal{F}}_S) \longrightarrow \operatorname {Hom}_{\mathcal{O}}(H^1(Y_S,{\mathcal{F}}_S),{\mathcal{O}}) \end{equation*}$$

on line 6 is $\tilde{\operatorname {\mathbf{T}}}_S'[w_l]$-linear. In the next line one should not only localise at $\mathfrak{m}$ but restrict to the kernel of $w_\ell -1$. Because $w_\ell ^2=1$ on $H^1(Y_S,{\mathcal{F}}_S)_\mathfrak{m}$ we see that the natural map$$\begin{equation*} H^1(Y_S,{\mathcal{F}}_S)_\mathfrak{m}^{w_\ell =1} \longrightarrow H^1(Y_S,{\mathcal{F}}_S)_\mathfrak{m}/(w_\ell -1) \end{equation*}$$

is an isomorphism, and so the map$$\begin{equation*} L_S \longrightarrow \operatorname {Hom}_{\mathcal{O}}(L_S,{\mathcal{O}}) \end{equation*}$$

is also an isomorphism.

•

On page 547 the groups $H^1(Y_S,{\mathcal{F}}_S)_{\mathfrak{m}_S^{(p)}}$ and $H^1(Y_{S'}, {\mathcal{F}}_{S'})_{\mathfrak{m}_S^{(p)}}$ should be replaced by their maximal subgroups on which $w_\ell =1$.

•

On page 549 one should also define $\widetilde{V}_0$ (resp. $\widetilde{V}_1$) to be the group generated by $V_0$ (resp. $V_1$) and $w_\ell \in {\operatorname {GL}}_2(\mathbf{Q}_\ell )$. Similarly define $\widetilde{\sigma }$ to be $\widetilde{\sigma }_\emptyset \otimes \psi _{r'}^{-2}$.

•

In Lemma 6.4.1 replace $V_0$ by $\widetilde{V}_0$, $V_1$ by $\widetilde{V}_1$ and $\sigma$ by $\widetilde{\sigma }$. In the proof of Lemma 6.4.1 also replace $U_{\{ r,r' \} }$ (resp. $U_{S \cup \{ r,r' \} }$) by $\widetilde{U}_{ \{ r,r' \} }$ (resp. $\widetilde{U}_{ S \cup \{ r,r' \} }$) and $\sigma _{ \{ r,r' \} }$ (resp. $\sigma _{S\cup \{ r,r' \} }$) by $\widetilde{\sigma }_{ \{ r,r' \} }$ (resp. $\widetilde{\sigma }_{S \cup \{ r,r' \} }$).

•

On line 20 of page 550 $M$ should be chosen as a model of $\widetilde{\sigma }$. This is possible because $\ker \widetilde{\sigma }$ has finite index in $\widetilde{V}_0$, because in turn $\sigma _{\tau '}(\ell ^s)=1$ for some $s>0$. One should also set $L_i=H^1(Y_{V_i},{\mathcal{F}}_{M^\vee })_\mathfrak{m}^{w_\ell =1}$. On line 25, we must replace $V_i$ by $\widetilde{V}_i$.

•

In the proof of Lemma 6.4.2, one must replace $V_1$ by $\widetilde{V}_1$ and $\sigma$ by $\widetilde{\sigma }$.

•

In line 2 of the proof of Lemma 6.4.3, to see that $L_1$ is a direct summand of $H^1(Y_{V_1},{\mathcal{F}}_{M^\vee })$ as an ${\mathcal{O}}[\Delta _S]$-module, one needs to note that $H^1(Y_{V_1},{\mathcal{F}}_{M^\vee })_\mathfrak{m}^{w_\ell =1}$ is a direct summand of $H^1(Y_{V_1},{\mathcal{F}}_{M^\vee })_\mathfrak{m}$, because $w_\ell ^2=1$ on $H^1(Y_{V_1}, {\mathcal{F}}_{M^\vee })_\mathfrak{m}$.

•

On line 12 of page 551 replace $R_{V,{\mathcal{O}}}^{\emptyset ,D}$ by $R_{V,{\mathcal{O}}}^{\emptyset ,D'}$.

Proof of extension of Proposition 5.4.1 of CDT.

Let $\Theta = \bigotimes _{p \in T({\overline{\rho }})} M_p$.

First suppose that $\tau '$ admits $\sigma '_{ \{ m_1,m_2 \} , a}$ with $m_1 \neq m_2$ and that $\sigma _{ \{ m_1,m_2 \} , a}'$ admits ${\overline{\rho }}$. As in the proof of Theorem 1.4.1 (especially §6.2 of CDT as modified above), we have

$$\begin{equation*} H^1(Y_{\{ r \} }, {\mathcal{F}}_{\Theta ^\vee } \otimes {\mathcal{F}}_{(\sigma _{m_1,m_2}')^\vee } )_{\mathfrak{m}_{ \{ r \} }'} \neq (0). \end{equation*}$$

On the other hand

$$\begin{equation*} \begin{array}{rcl} H^1(Y_{\{ r \} }, {\mathcal{F}}_{\Theta ^\vee } \otimes {\mathcal{F}}_{(\sigma _{m_1,m_2}')^\vee } )_{\mathfrak{m}_{ \{ r \} }'} & \cong & H^1(Y_{\{ r \} }, {\mathcal{F}}_{\Theta ^\vee } \otimes {\mathcal{F}}_{(\sigma _{m_1,m_2}')^\vee })_{\mathfrak{m}_{ \{ r \} }'}^{w_\ell ^2=a} \\& \cong & H^1(Y_{\{ r \} }, {\mathcal{F}}_{\Theta ^\vee } \otimes {\mathcal{F}}_{(\sigma _{\{m_1,m_2\}, a }')^\vee })_{\mathfrak{m}_{ \{ r \} }'}^{w_\ell =1}. \end{array} \end{equation*}$$

Thus, using the definition of “admits” and Lemma 6.1.2 of CDT, we see that

$$\begin{equation*} H^1(Y_{\{ r \} }, {\mathcal{F}}_{\{ r \} })^{w_\ell =1}_{\mathfrak{m}_{ \{ r \} }'} \neq (0), \end{equation*}$$

so ${\operatorname {\mathcal{N}}}_\emptyset = {\operatorname {\mathcal{N}}}_{ \{ r \} } \neq \emptyset$.

Next suppose that $\tau '$ admits $\sigma _{ \{ m \} , a}'$ which in turn admits ${\overline{\rho }}$. Assume that ${\overline{\rho }}|_{G_\ell }$ is irreducible or peu ramifié. By twisting we may reduce to the case $m=0$. As in the proof of Theorem 1.4.1 (especially §6.2 of CDT as modified above), we have

$$\begin{equation*} H^1(Y_{U_{ \{ r \} }{\operatorname {GL}}_2(\mathbf{Z}_\ell )}, {\mathcal{F}}_{\Theta ^\vee })_{\mathfrak{m}_{ \{ r \} }'} \neq (0). \end{equation*}$$

Thus

$$\begin{equation*} H^1(Y_{U_{ \{ r \} }{\operatorname {GL}}_2(\mathbf{Z}_\ell )}, {\mathcal{F}}_{\Theta ^\vee } )_{\mathfrak{m}_{ \{ r \} }'}^{w_\ell ^2=\widetilde{a}^2} \neq (0), \end{equation*}$$

where $\widetilde{a}$ is the Teichmüller lift of $a$. Using the embedding

$$\begin{equation*} \widetilde{a}+w_\ell : H^1(Y_{U_{ \{ r \} }{\operatorname {GL}}_2(\mathbf{Z}_\ell )}, {\mathcal{F}}_{\Theta ^\vee }) \otimes \overline{\mathbf{Q}}_\ell \hookrightarrow H^1(Y_{\{ r \} }, {\mathcal{F}}_{\Theta ^\vee }) \otimes \overline{\mathbf{Q}}_\ell , \end{equation*}$$

we deduce that

$$\begin{equation*} H^1(Y_{\{ r \} }, {\mathcal{F}}_{\Theta ^\vee })_{\mathfrak{m}_{ \{ r \} }'}^{w_\ell =\widetilde{a}} \neq (0), \end{equation*}$$

and so

$$\begin{equation*} H^1(Y_{\{ r \} }, {\mathcal{F}}_{\Theta ^\vee } \otimes {\mathcal{F}}_{(\sigma _{ \{ 0 \} , a}')^\vee } )_{\mathfrak{m}_{ \{ r \} }'}^{w_\ell =1} \neq (0). \end{equation*}$$

Thus, using the definition of “admits” and Lemma 6.1.2 of CDT, we see that

$$\begin{equation*} H^1(Y_{\{ r \} }, {\mathcal{F}}_{\{ r \} })^{w_\ell =1}_{\mathfrak{m}_{ \{ r \} }'} \neq (0), \end{equation*}$$

and so ${\operatorname {\mathcal{N}}}_\emptyset = {\operatorname {\mathcal{N}}}_{ \{ r \} } \neq \emptyset$.

Finally suppose that $\tau '$ admits $\sigma _{ \{ m \} , a}$ which in turn admits ${\overline{\rho }}$, and that ${\overline{\rho }}|_{G_\ell }$ is reducible and très ramifié. By twisting we may reduce to the case $m=0$. Note that ${\overline{\rho }}_{I_\ell }(\operatorname {Frob}_\ell ) =-a$. As in the proof of Theorem 1.4.1 (especially §6.2 of CDT as modified above), we have

$$\begin{equation*} H^1(Y_{ \{ r \} }, {\mathcal{F}}_{\Theta ^\vee })_{\mathfrak{m}_{ \{ r \} }'} \neq (0). \end{equation*}$$

Suppose that $\pi$ is a cuspidal automorphic representation which contributes to $H^1(Y_{ \{ r \} }, {\mathcal{F}}_{\Theta ^\vee })_{\mathfrak{m}_{ \{ r \} }'}$, so $\pi$ is a cuspidal automorphic representation of ${\operatorname {GL}}_2(\mathbf{A})$ such that $\pi _\infty$ is the lowest discrete series with trivial infinitesimal character, $\rho _\pi$ is a lift of ${\overline{\rho }}$ of type $(\{ r \}, 1)$, and hence of type $(\emptyset , 1)$, and $\dim \pi _\ell ^{U_0(\ell )}=1$. As $\epsilon ^{-1} \det \rho _\pi$ has order prime to $\ell$, we see that $w_\ell ^2$ acts on $\pi _\ell ^{U_0(\ell )}$ by the Teichmüller lift of $a^2$. As $\pi _\ell$ has a $U_0(\ell )$-fixed vector but no ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$-fixed vector, we see that $1+U_\ell w_\ell ^{-1}=0$ on $\pi _\ell ^{U_0(\ell )}$. On the other hand, the eigenvalue of $U_\ell$ on $\pi _\ell ^{U_0(\ell )}$ reduces to $-a$. Thus, $w_\ell$ acts on $\pi _\ell ^{U_0(\ell )}$ by the Teichmüller lift of $a$, so $w_\ell$ acts on $H^1(Y_{ \{ r \} }, {\mathcal{F}}_{\Theta ^\vee })_{\mathfrak{m}_{ \{ r \} }'}$ by the Teichmüller lift of $a$. We deduce that

$$\begin{equation*} H^1(Y_{ \{ r \} }, {\mathcal{F}}_{\Theta ^\vee } \otimes {\mathcal{F}}_{(\sigma _{0,0}')^\vee } )^{w_\ell =a}_{\mathfrak{m}_{ \{ r \} }'} \neq (0). \end{equation*}$$

Using the definition of “admits” and Lemma 6.1.2 of CDT, we see that

$$\begin{equation*} H^1(Y_{\{ r \} }, {\mathcal{F}}_{\{ r \} })^{w_\ell =1}_{\mathfrak{m}_{ \{ r \} }'} \neq (0), \end{equation*}$$

so ${\operatorname {\mathcal{N}}}_\emptyset = {\operatorname {\mathcal{N}}}_{ \{ r \} } \neq \emptyset$.

■

2. Examples and applications

2.1. Examples

Now we will specialise to the case $\ell =3$. Fix an element $\zeta \in {\operatorname {GL}}_2(\mathbf{Z}_3)$ with $\zeta ^3=1$ but $\zeta \neq 1$. The following definitions, which concern isomorphism classes of $2$-dimensional representations into $\operatorname {GL}_2(\overline{\mathbf{Q}}_3)$, do not depend on this choice. We will consider the following $\ell$-types. (These are in fact, up to twist, a complete list of the wildly ramified types which can arise from elliptic curves over $\mathbf{Q}_3$, or, in the case of conductor 243, the extended types. We will not need this fact. Rather the justification for studying these particular types can be found in §2.2. More detailed information about the fixed fields of these types can be found in §6.)

•

$\tau _1$ corresponds to the order $3$ homomorphism$$\begin{equation*} \mathbf{Z}_3^\times \longrightarrow \mathbf{Z}_3[\zeta ]^\times \end{equation*}$$

determined by$$\begin{equation*} \begin{array}{rcl} -1 & \longmapsto & 1 \\4 & \longmapsto & \zeta . \end{array} \end{equation*}$$

•

$\tau _{-1}$ corresponds to the order $3$ homomorphism$$\begin{equation*} \mathbf{Z}_3[\sqrt {-1}]^\times \longrightarrow \mathbf{Z}_3[\zeta ]^\times \end{equation*}$$

determined by$$\begin{equation*} \begin{array}{rcl} \sqrt [4]{-1} & \longmapsto & 1 \\4 & \longmapsto & 1 \\1+3\sqrt {-1} & \longmapsto & \zeta . \end{array} \end{equation*}$$

•

$\tau _3$ is the unique $3$-type such that $\tau _{3}|_{I_{\mathbf{Q}_3(\sqrt {3})}}$ corresponds to the order $6$ homomorphism$$\begin{equation*} \mathbf{Z}_3[\sqrt {3}]^\times \longrightarrow \mathbf{Z}_3[\zeta ]^\times \end{equation*}$$

determined by$$\begin{equation*} \begin{array}{rcl} -1 & \longmapsto & -1 \\4 & \longmapsto & 1 \\1+\sqrt {3} & \longmapsto & \zeta . \end{array} \end{equation*}$$

•

$\tau _{-3}$ is the unique $3$-type such that $\tau _{-3}|_{I_{\mathbf{Q}_3 (\sqrt {-3})}}$ corresponds to the order $6$ homomorphism$$\begin{equation*} \mathbf{Z}_3[\sqrt {-3}]^\times \longrightarrow \mathbf{Z}_3[\zeta ]^\times \end{equation*}$$

determined by$$\begin{equation*} \begin{array}{rcl} -1 & \longmapsto & -1 \\4 & \longmapsto & 1 \\1+3\sqrt {-3} & \longmapsto & 1 \\1+\sqrt {-3} & \longmapsto & \zeta . \end{array} \end{equation*}$$

For $i \in \mathbf{Z}/3\mathbf{Z}$, we will also consider the unique extended $3$-types $\tau _{i}'$ whose restrictions to $G_{\mathbf{Q}_3(\sqrt {-3})}$ correspond to the homomorphisms

$$\begin{equation*} \mathbf{Q}_3(\sqrt {-3})^\times \rightarrow \mathbf{Q}_3(\zeta )^\times \end{equation*}$$

determined by

$$\begin{equation} \begin{array}{rcl} \sqrt {-3} & \longmapsto & \zeta -\zeta ^{-1} \\-1 & \longmapsto & -1 \\4 & \longmapsto & 1 \\1+3\sqrt {-3} & \longmapsto & \zeta \\1+\sqrt {-3} & \longmapsto & \zeta ^i. \end{array}{\tag{2.1.1}} \end{equation}$$

Subsequent sections of this paper will be devoted to checking the following special cases of Conjecture 1.3.1.

Lemma 2.1.1.

Suppose that ${\overline{\rho }}:G_3 \rightarrow {\operatorname {GL}}_2(\mathbf{F}_3)$ and

$$\begin{equation*} {\overline{\rho }}|_{I_3} \sim \left( \begin{array}{cc} 1 & * \\0 & \omega \end{array} \right) \end{equation*}$$

is très ramifié. Both $\tau _1$ and $\tau _{-1}$ simply admit ${\overline{\rho }}$.

Theorem 2.1.2.

Suppose that ${\overline{\rho }}:G_3 \rightarrow {\operatorname {GL}}_2(\mathbf{F}_3)$ and

$$\begin{equation*} {\overline{\rho }}|_{I_3} \sim \left( \begin{array}{cc} 1 & * \\0 & \omega \end{array} \right) \end{equation*}$$

is très ramifié. Both $\tau _1$ and $\tau _{-1}$ are weakly acceptable for ${\overline{\rho }}$.

Lemma 2.1.3.

Suppose that ${\overline{\rho }}:G_3 \rightarrow {\operatorname {GL}}_2(\mathbf{F}_3)$ and

$$\begin{equation*} {\overline{\rho }}|_{I_3} \sim \left( \begin{array}{cc} \omega & * \\0 & 1 \end{array} \right) \end{equation*}$$

is très ramifié. Both $\tau _3$ and $\tau _{-3}$ simply admit ${\overline{\rho }}$.

Theorem 2.1.4.

Suppose that ${\overline{\rho }}:G_3 \rightarrow {\operatorname {GL}}_2(\mathbf{F}_3)$ and

$$\begin{equation*} {\overline{\rho }}|_{I_3} \sim \left( \begin{array}{cc} \omega & * \\0 & 1 \end{array} \right) \end{equation*}$$

is très ramifié. Both $\tau _3$ and $\tau _{-3}$ are weakly acceptable for ${\overline{\rho }}$.

Lemma 2.1.5.

Let $i \in \mathbf{Z}/3\mathbf{Z}$. Suppose that ${\overline{\rho }}:G_3 \rightarrow {\operatorname {GL}}_2(\mathbf{F}_3)$ and

$$\begin{equation*} {\overline{\rho }}\sim \left( \begin{array}{cc} \omega & * \\0 & 1 \end{array} \right) \end{equation*}$$

is très ramifié. The extended $3$-type $\tau _{i}'$ simply admits ${\overline{\rho }}$.

Theorem 2.1.6.

Let $i \in \mathbf{Z}/3\mathbf{Z}$. Suppose that ${\overline{\rho }}:G_3 \rightarrow {\operatorname {GL}}_2(\mathbf{F}_3)$ and

$$\begin{equation*} {\overline{\rho }}\sim \left( \begin{array}{cc} \omega & * \\0 & 1 \end{array} \right) \end{equation*}$$

is très ramifié. Then $\tau _{i}'$ is weakly acceptable for ${\overline{\rho }}$.

We remark that in Theorems 2.1.2, 2.1.4 and 2.1.6 we could replace “weakly acceptable” by “acceptable”. This can be shown by using elliptic curves to construct explicit liftings of the desired type. For Theorems 2.1.2 and 2.1.4 the results of Man suffice for this. For Theorem 2.1.6 a slightly more refined analysis along the lines of §2.3 is required.

We also remark that it was Lemmas 2.1.1, 2.1.3, 2.1.5 and Conjecture 1.3.1 which originally suggested to us that we try to prove Theorems 2.1.2, 2.1.4 and 2.1.6.

2.2. Applications

Conditional on the results stated in §2.1, which we will prove below, we prove the following results.

Theorem 2.2.1.

Any continuous absolutely irreducible representation ${\overline{\rho }}: G_\mathbf{Q}\rightarrow {\operatorname {GL}}_2(\mathbf{F}_5)$ with cyclotomic determinant is modular.

Proof.

Choose an element ${\overline{\zeta }}\in {\operatorname {GL}}_2(\mathbf{F}_5)$ with ${\overline{\zeta }}^3=1$ but ${\overline{\zeta }}\neq 1$. (The following classification will be independent of the choice of ${\overline{\zeta }}$.) Then up to equivalence and twisting by a quadratic character, one of the following possibilities can be attained.

(1)

${\overline{\rho }}$ is tamely ramified at $3$.

(2)

${\overline{\rho }}|_{G_3}$ is given by the character$$\begin{equation*} \mathbf{Q}_3^\times \longrightarrow \mathbf{F}_5({\overline{\zeta }})^\times \end{equation*}$$

determined by$$\begin{equation*} \begin{array}{rcl} 3 & \longmapsto & {\overline{\zeta }}^i ({\overline{\zeta }}- {\overline{\zeta }}^{-1}) \\-1 & \longmapsto & 1 \\4 & \longmapsto & {\overline{\zeta }}, \end{array} \end{equation*}$$

where $i \in \mathbf{Z}/3\mathbf{Z}$.

(3)

${\overline{\rho }}|_{G_{\mathbf{Q}_3(\sqrt {-1})}}$ is given by the character$$\begin{equation*} \mathbf{Q}_3(\sqrt {-1})^\times \longrightarrow \mathbf{F}_5({\overline{\zeta }})^\times \end{equation*}$$

determined by$$\begin{equation*} \begin{array}{rcl} 3 & \longmapsto & 2 \\\sqrt [4]{-1} & \longmapsto & 1 \\4 & \longmapsto & 1 \\1+3 \sqrt {-1} & \longmapsto & {\overline{\zeta }}. \end{array} \end{equation*}$$

(4)

${\overline{\rho }}|_{G_{\mathbf{Q}_3(\sqrt {3})}}$ is given by the character$$\begin{equation*} \mathbf{Q}_3(\sqrt {3})^\times \longrightarrow \mathbf{F}_5({\overline{\zeta }})^\times \end{equation*}$$

determined by$$\begin{equation*} \begin{array}{rcl} \sqrt {3} & \longmapsto & {\overline{\zeta }}- {\overline{\zeta }}^{-1} \\-1 & \longmapsto & -1 \\4 & \longmapsto & 1 \\1+\sqrt {3} & \longmapsto & {\overline{\zeta }}. \end{array} \end{equation*}$$

(5)

${\overline{\rho }}|_{G_{\mathbf{Q}_3(\sqrt {-3})}}$ is given by the character$$\begin{equation*} \mathbf{Q}_3(\sqrt {-3})^\times \longrightarrow \mathbf{F}_5({\overline{\zeta }})^\times \end{equation*}$$

determined by$$\begin{equation*} \begin{array}{rcl} \sqrt {-3} & \longmapsto & {\overline{\zeta }}- {\overline{\zeta }}^{-1} \\-1 & \longmapsto & -1 \\4 & \longmapsto & 1 \\1+3\sqrt {-3} & \longmapsto & 1 \\1+\sqrt {-3} & \longmapsto & {\overline{\zeta }}. \end{array} \end{equation*}$$

(6)

${\overline{\rho }}|_{G_{\mathbf{Q}_3(\sqrt {-3})}}$ is given by the character$$\begin{equation*} \mathbf{Q}_3(\sqrt {-3})^\times \longrightarrow \mathbf{F}_5({\overline{\zeta }})^\times \end{equation*}$$

determined by$$\begin{equation*} \begin{array}{rcl} \sqrt {-3} & \longmapsto & {\overline{\zeta }}- {\overline{\zeta }}^{-1} \\-1 & \longmapsto & -1 \\4 & \longmapsto & 1 \\1+3\sqrt {-3} & \longmapsto & {\overline{\zeta }}\\1+\sqrt {-3} & \longmapsto & {\overline{\zeta }}^i, \end{array} \end{equation*}$$

where $i \in \mathbf{Z}/3\mathbf{Z}$.

To see that one of these cases can be attained, use the following facts, all of which are easy to verify.

•

A subgroup of ${\operatorname {GL}}_2(\mathbf{F}_5)$ with a non-trivial normal subgroup of $3$-power order is, up to conjugation, contained in the normaliser of $\mathbf{F}_5({\overline{\zeta }})^\times$.

•

The intersection of $\operatorname {SL}_2(\mathbf{F}_5)$ with the normaliser of ${\overline{\zeta }}$ in ${\operatorname {GL}}_2(\mathbf{F}_5)$ is generated by ${\overline{\zeta }}$ and an element $\alpha$ such that $\alpha ^2=-1$ and $\alpha {\overline{\zeta }}\alpha ^{-1} = {\overline{\zeta }}^{-1}$.

•

If $\beta \in \mathbf{F}_5({\overline{\zeta }})^\times$, $\det \beta =3$, and $\alpha \beta \alpha ^{-1}=-\beta$, then $\beta = \pm ({\overline{\zeta }}- {\overline{\zeta }}^{-1})$.

In each case, we may choose an elliptic curve $E_{1/\mathbf{Q}_3}$ such that the representation ${\overline{\rho }}_{E_1,5}$ of $G_3$ on $E_1[5](\overline{\mathbf{Q}}_3)$ is isomorphic to ${\overline{\rho }}|_{G_3}$ and such that the representation ${\overline{\rho }}_{E_1,3}$ of $G_3$ on $E_1[3](\overline{\mathbf{Q}}_3)$ has the following form (where we use the same numbering as above).

(1)

We place no restriction on ${\overline{\rho }}_{E_1,3}$.

(2)

The restriction of ${\overline{\rho }}_{E_1,3}$ to $I_3$ has the form$$\begin{equation*} \left( \begin{array}{cc} 1 & * \\0 & \omega \end{array} \right) \end{equation*}$$

and is très ramifié. (Use Theorem 5.3.2 of Man.)

(3)

The restriction of ${\overline{\rho }}_{E_1,3}$ to $I_3$ has the form$$\begin{equation*} \left( \begin{array}{cc} 1 & * \\0 & \omega \end{array} \right) \end{equation*}$$

and is très ramifié. (Use Theorem 5.3.2 of Man.)

(4)

The restriction of ${\overline{\rho }}_{E_1,3}$ to $I_3$ has the form$$\begin{equation*} \left( \begin{array}{cc} \omega & * \\0 & 1 \end{array} \right) \end{equation*}$$

and is très ramifié. (Use §5.4 of Man.)

(5)

The restriction of ${\overline{\rho }}_{E_1,3}$ to $I_3$ has the form$$\begin{equation*} \left( \begin{array}{cc} \omega & * \\0 & 1 \end{array} \right) \end{equation*}$$

and is très ramifié. (Use §5.4 of Man.)

(6)

${\overline{\rho }}_{E_1,3}$ has the form$$\begin{equation*} \left( \begin{array}{cc} \omega & * \\0 & 1 \end{array} \right), \end{equation*}$$

is très ramifié and remains indecomposable when restricted to the splitting field of ${\overline{\rho }}$. (Use Corollary 2.3.2 below.)

In each case choose such an $E_1$ and fix an isomorphism $\alpha : \mathbf{F}_5^2 \stackrel{\sim }{\rightarrow }E_1[5](\overline{\mathbf{Q}}_3)$, such that the Weil pairing on $E_1[5]$ corresponds to the standard alternating pairing on $\mathbf{F}_5^2$, following the conventions in §1 of SBT. The pair $(E_1, \alpha )$ defines a $\mathbf{Q}_3$-rational point on the smooth curve denoted $X_{\overline{\rho }}$ in SBT. We can find a $3$-adic open set ${\operatorname {\mathcal{U}}}\subset X_{\overline{\rho }}(\mathbf{Q}_3)$ containing $(E_1,\alpha )$ such that if $(E_2,\beta )$ defines a point in ${\operatorname {\mathcal{U}}}$, then $E_2[3] \cong E_1[3]$ as $\mathbf{F}_3[G_3]$-modules.

Using Ekedahl’s version of the Hilbert Irreducibility Theorem (see Theorem 1.3 of E) and the argument of §1 of SBT we may find an elliptic curve $E_{/\mathbf{Q}}$ and an $\mathbf{F}_5[G_\mathbf{Q}]$-module isomorphism $\beta$ of ${\overline{\rho }}$ with $E[5](\overline{\mathbf{Q}})$ such that (see also §2 of Man)

•

under $\beta$, the standard alternating pairing on $\mathbf{F}_5^2$ and the Weil pairing on $E[5]$ agree;

•

the representation ${\overline{\rho }}_{E,3}$ of $G_\mathbf{Q}$ on $E[3](\overline{\mathbf{Q}})$ is surjective onto $\operatorname {Aut}(E[3](\overline{\mathbf{Q}}))$;

•

and $(E,\beta )$ defines a point of ${\operatorname {\mathcal{U}}}$.

Corresponding to the six types of ${\overline{\rho }}$ considered above, Proposition B.4.2 of CDT ensures that the representation $\rho _{E,3}$ of $G_\mathbf{Q}$ on the $3$-adic Tate module of $E$ is

(1)

either, up to quadratic twist, ordinary in the sense of Wiles Wi or potentially Barsotti-Tate of some tamely ramified type;

(2)

potentially Barsotti-Tate of type $\tau _1$;

(3)

potentially Barsotti-Tate of type $\tau _{-1}$;

(4)

potentially Barsotti-Tate of type $\tau _3$;

(5)

potentially Barsotti-Tate of type $\tau _{-3}$;

(6)

potentially Barsotti-Tate of extended type $\tau _{i}'$.

In the first case, $E$ is modular by Theorem 7.2.1 of CDT. In the other cases we will simply write $\tau$ for the type/extended type. We see that ${\overline{\rho }}_{E,3}(G_3)$ has centraliser $\mathbf{F}_3$ and the results of §2.1 show that $\tau$ admits ${\overline{\rho }}_{E,3}$ and that $\tau$ is weakly acceptable for ${\overline{\rho }}_{E,3}$. Moreover ${\overline{\rho }}_{E,3}|_{\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q}(\sqrt {-3}))}$ is absolutely irreducible and, by the Langlands-Tunnell theorem (see Wi), modular. Thus by Theorems 1.4.1 and 1.4.2 we see that $\rho _{E,3}$ is modular. We deduce that $E$ is modular, so ${\overline{\rho }}\cong {\overline{\rho }}_{E,5}$ is modular.

■

Combining this theorem with Theorem 7.2.4 of CDT we immediately obtain the following corollary.

Theorem 2.2.2.

Every elliptic curve defined over the rational numbers is modular.

2.3. An extension of a result of Manoharmayum

The following facts follow at once from Man, particularly the classification given just before Theorem 5.4.2 of that paper. Consider elliptic curves $E$ over $\mathbf{Q}_3$ with minimal Weierstrass equation $Y^2 = X^3 + AX + B$, where

$$\begin{equation*} A \equiv B + 3 \equiv 0 \bmod 9, \end{equation*}$$

so ${\overline{\rho }}_{E,3}$ has the form $\left( \begin{array}{cc} {\omega } & {*} \\{0} & {1} \end{array} \right)$ and is très ramifié. This leaves three possibilities for the equivalence class of ${\overline{\rho }}_{E,3}$. Fix ${\overline{\zeta }}$ in ${\operatorname {GL}}_2(\mathbf{F}_5)$ with ${\overline{\zeta }}^3=1$ but ${\overline{\zeta }}\neq 1$. The action of $G_{\mathbf{Q}_3 (\sqrt {-3})}$ on $E[5](\overline{\mathbf{Q}}_3)$ is via a representation of the form

$$\begin{equation*} \begin{array}{rcl} \sqrt {-3} & \longmapsto & \delta ({\overline{\zeta }}- {\overline{\zeta }}^{-1})\\-1 & \longmapsto & -1\\4 & \longmapsto & 1\\1+\sqrt {-3} & \longmapsto & {\overline{\zeta }}^i\\1+3\sqrt {-3} & \longmapsto & {\overline{\zeta }},\end{array} \end{equation*}$$

for some $\delta = \pm 1$ and some $i \in \mathbf{Z}/3\mathbf{Z}$. All nine possibilities for the pair $({\overline{\rho }}_{E,3},i)$ satisfying these conditions can arise for some such choice of $A$ and $B$.

Lemma 2.3.1.

With the above notation and assumptions, we have $\delta =1$.

Proof.

Let $F= \mathbf{Q}_3(\sqrt {-3},\beta ,\alpha )$, where $\beta ^2 = -\sqrt {-3}$ and

$$\begin{equation*} \alpha ^3 + A\alpha + B = 9\sqrt {-3}. \end{equation*}$$

$F$ is a totally ramified abelian extension of $\mathbf{Q}_3(\sqrt {-3})$ of degree 6, with uniformiser $\varpi = \alpha /\beta$. The change of coordinates $Y \mapsto \varpi ^{15}Y$, $X \mapsto \varpi ^{10}X + \alpha$ shows that $E$ has good reduction over $F$, and the reduction is isomorphic to

$$\begin{equation*} Y^2 = X^3 - X - 1. \end{equation*}$$

The arithmetic Frobenius of $W_F$ therefore has trace 3 on $E[5]$. Since

$$\begin{equation*} N_{F/\mathbf{Q}_3(\sqrt {-3})}(\varpi ) \equiv \sqrt {-3}(1-3\sqrt {-3}) \bmod 9 \sqrt {-3}, \end{equation*}$$

we conclude that

$$\begin{equation*} \operatorname {tr}\delta ({\overline{\zeta }}-{\overline{\zeta }}^{-1}) {\overline{\zeta }}^{-1} = 3, \end{equation*}$$

so $\delta =1$.

■

Twisting by quadratic characters we immediately deduce the following corollary.

Corollary 2.3.2.

Let ${\overline{\rho }}_3:G_3 \rightarrow {\operatorname {GL}}_3(\mathbf{F}_3)$ have the form

$$\begin{equation*} \left( \begin{array}{cc} {\omega } & {*} \\{0} & {1} \end{array} \right) \,\,\, {\text{or} } \,\,\, \left( \begin{array}{cc} {1} & {*} \\{0} & {\omega } \end{array} \right) \end{equation*}$$

and be très ramifié. Let ${\overline{\rho }}_5:G_3 \rightarrow {\operatorname {GL}}_2(\mathbf{F}_5)$ have cyclotomic determinant and restriction to $G_{\mathbf{Q}_3(\sqrt {-3})}$ given by a character

$$\begin{equation*} \mathbf{Q}_3(\sqrt {-3})^\times \longrightarrow \mathbf{F}_5({\overline{\zeta }})^\times \end{equation*}$$

determined by

$$\begin{equation*} \begin{array}{rcl} \sqrt {-3} & \longmapsto & ({\overline{\zeta }}- {\overline{\zeta }}^{-1})\\-1 & \longmapsto & -1\\4 & \longmapsto & 1\\1+\sqrt {-3} & \longmapsto & {\overline{\zeta }}^i\\1+3\sqrt {-3} & \longmapsto & {\overline{\zeta }},\end{array} \end{equation*}$$

for some $i \in \mathbf{Z}/3\mathbf{Z}$. There is an elliptic curve $E_{/\mathbf{Q}_3}$, with $E[3](\overline{\mathbf{Q}}_3) \sim {\overline{\rho }}_3$ and $E[5](\overline{\mathbf{Q}}_3) \sim {\overline{\rho }}_5$. In particular, the action of $I_3$ on $T_5E$ factors through a finite group and so $E$ has potentially good reduction.

3. Admittance

In this section we will check Lemmas 2.1.1, 2.1.3 and 2.1.5. We freely use the terminology introduced in §1.2 and §1.3.

3.1. The case of $\tau _1$

In this case $\sigma _{\tau _1}$ is the induction from $U_0(9)$ to ${\operatorname {GL}}_2(\mathbf{Z}_3)$ of a character of order $3$. Its reduction modulo a prime above $3$ has the same Jordan-Hölder constituents as the reduction modulo $3$ of $\operatorname {Ind}_{U_0(9)}^{\operatorname {GL}_2(\mathbf{Z}_3)} {\mathbf{1}}$. Using Brauer characters, we see that the reduction modulo $3$ of $\operatorname {Ind}_{U_0(9)}^{U_0(3)} {\mathbf{1}}$ has Jordan-Hölder constituents $\sigma _{0,0}'$, $\sigma _{0,0}'$ and $\sigma _{1,1}'$. Thus, $\tau _1$ admits $\sigma _{0,0}$, $\sigma _{2,0}$, $\sigma _{0,1}$ and $\sigma _{2,1}$, the latter two simply. Lemma 2.1.1 follows in this case.

3.2. The case of $\tau _{-1}$

Let $U$ denote the subgroup of ${\operatorname {GL}}_2(\mathbf{Z}_3)$ consisting of matrices

$$\begin{equation*} \left( \begin{array}{cc} a&b \\c&d \end{array} \right) \end{equation*}$$

with $a \equiv d \bmod 3$ and $b+c \equiv 0 \bmod 3$, so $\sigma _{\tau _{-1}}$ is the induction from $U$ to ${\operatorname {GL}}_2(\mathbf{Z}_3)$ of a character of order $3$. Upon reduction modulo a prime above $3$ this will have the same Jordan-Hölder constituents as the reduction modulo $3$ of $\operatorname {Ind}_U^{\operatorname {GL}_2(\mathbf{Z}_3)} {\mathbf{1}}$. If $\psi$ denotes the non-trivial character of $\mathbf{F}_3^\times$ and $\phi$ a character of $\mathbf{F}_9^\times$ of order $4$, then this latter induction splits up as the sum of the representations of ${\operatorname {GL}}_2(\mathbf{Z}_3) \rightarrow \hspace{-.86em} \rightarrow {\operatorname {GL}}_2(\mathbf{F}_3)$ denoted $1$, $\operatorname {sp}_\psi$ and $\Theta (\phi )$ in §3.1 of CDT. By Lemma 3.1.1 of CDT we see that $\tau _{-1}$ admits $\sigma _{0,0}$, $\sigma _{2,1}$ and $\sigma _{0,1}$, the latter two simply. Lemma 2.1.1 follows in this case.

3.3. The case of $\tau _{\pm 3}$

Let $U$ denote the subgroup of ${\operatorname {GL}}_2(\mathbf{Z}_3)$ consisting of matrices

$$\begin{equation*} \left( \begin{array}{cc} a&b \\c&d \end{array} \right) \end{equation*}$$

with $a \equiv d \bmod 3$ and $c \equiv 0 \bmod 3$. Then $\sigma _{\tau _{\pm 3}}$ is the induction from $U$ to $U_0(3)$ of a character of order $3$. Upon reduction modulo a prime above $3$ this will have the same Jordan-Hölder constituents as the reduction modulo $3$ of $\operatorname {Ind}_U^{U_0(3)} {\mathbf{1}}$. Thus, $\tau _{\pm 3}$ simply admits $\sigma _{0,0}'$ and $\sigma _{1,1}'$. Lemma 2.1.3 follows.

3.4. The case of $\tau _{i}'$

Let $\chi$ be the character of $\mathbf{Q}_3(\sqrt {-3})^\times$ as in (2.1.1). Let $\psi$ be a character of $\mathbf{Q}_3$ with kernel $\mathbf{Z}_3$ and which sends $1/3$ to $\zeta$. If $x \in (3 \sqrt {-3}) \mathbf{Z}_3[\sqrt {-3}]$ we have

$$\begin{equation*} \chi (1+x) = \psi (\operatorname {tr}_{\mathbf{Q}_3(\sqrt {-3})/\mathbf{Q}_3} (-x \sqrt {-3}/54)). \end{equation*}$$

We deduce that if $\chi ''$ is the character used to define $\sigma _{\tau _{i}'}$ in §1.2, then $\chi ''(\sqrt {-3})= (\zeta -\zeta ^{-1})^{-1}$.

Let $U$ denote the subgroup of ${\operatorname {GL}}_2(\mathbf{Z}_3)$ consisting of matrices

$$\begin{equation*} \left( \begin{array}{cc} a&b \\3c&d \end{array} \right) \end{equation*}$$

with $a \equiv d \bmod 3$ and $b+c \equiv 0 \bmod 3$. Let $\widetilde{U}$ be the group generated by $w_3$ (see (1.2.1)) and $U$, so $\sigma _{\tau _{i}'}$ is the representation of $\widetilde{U}_0(3)$ induced from a character of $\widetilde{U}$ which sends $w_3$ to $1$ and has order $3$ when restricted to $U$. Thus, the Jordan-Hölder constituents of the reduction of $\sigma _{\tau _{i}'}$ modulo a prime above $3$ are the same as the Jordan-Hölder constituents of the reduction modulo $3$ of $\operatorname {Ind}_{\widetilde{U}}^{\widetilde{U}_0(3)} {\mathbf{1}}$.

Let $V$ denote the subgroup of ${\operatorname {GL}}_2(\mathbf{Z}_3)$ consisting of matrices

$$\begin{equation*} \left( \begin{array}{cc} a&b \\3c&d \end{array} \right) \end{equation*}$$

with $a \equiv d \bmod 3$. Let $\widetilde{V}$ be the group generated by $w_3$ and $V$, and let $\nu$ denote the character of $\widetilde{V}/V$ which sends $w_3$ to $-1$. We have

$$\begin{equation*} \operatorname {Ind}_{\widetilde{U}}^{\widetilde{V}}({\mathbf{1}}) \sim {\mathbf{1}}\oplus \operatorname {Ind}_{V\mathbf{Q}_3^\times }^{\widetilde{V}} \eta , \end{equation*}$$

where $\eta$ is a non-trivial character of $V/U=(V \mathbf{Q}_3^\times )/(U \mathbf{Q}_3^\times )$. The reduction modulo a prime above $3$ of this ($3$-dimensional) representation has the same Jordan-Hölder constituents as the reduction modulo $3$ of ${\mathbf{1}}\oplus {\mathbf{1}}\oplus \nu$. Thus, $\tau _{i}'$ admits $\sigma _{\{0\},1}'$, $\sigma _{\{1\},1}'$, $\sigma _{\{0\},-1}'$ and $\sigma _{\{1\},-1}'$, the latter two simply. Lemma 2.1.5 follows.

4. New deformation problems

In this section we begin the proof of Theorems 2.1.2, 2.1.4 and 2.1.6. One could approach this directly by using the results of Br2 to attempt to describe $R_{V,{\mathcal{O}}}^D$ (resp. $R_{V,{\mathcal{O}}}^{D'}$). At least one of the authors of this paper (Taylor) thinks that such an approach holds out more promise of attacking the non-acceptable case, and another author (Breuil) has indeed made several computations along these lines. However in the present case it seems to be easier to proceed less directly.

To this end we will use ad hoc arguments to define deformation problems, which will be represented by ${\mathcal{O}}$-algebras $S$ such that

•

$\dim _k \mathfrak{m}_S/(\wp _K, \mathfrak{m}_S^2)\leq 1$,

•

and the map $R_{V,{\mathcal{O}}} \rightarrow \hspace{-.86em} \rightarrow R_{V,{\mathcal{O}}}^D$ (resp. $R_{V,{\mathcal{O}}}^{D'}$) factors through $S$.

An important advantage of this approach is that to calculate $\mathfrak{m}_S/(\wp _K, \mathfrak{m}_S^2)$ one need only work in the category of finite flat group schemes killed by a prime. Breuil modules (see §5) for finite flat group schemes killed by an odd prime are significantly simpler than the general case (of prime power torsion). This is particularly true when we also use descent data. On the other hand, to suitably define the new deformation problems is rather delicate. That is what we will do in this section.

4.1. Some generalities on group schemes

In this section, and in §4.2, $\ell$ will again be an arbitrary rational prime. Moreover $R$ will denote a complete discrete valuation ring with fraction field $F'$ of characteristic zero and perfect residue field $k$ of characteristic $\ell$. We will let $\Gamma$ denote a finite group of continuous automorphisms of $R$ and we will let $F_0$ denote the subfield of $F'$ consisting of elements fixed by $\Gamma$. Thus $F'/F_0$ will be finite and Galois with group $\Gamma$. In our applications of these results it suffices to consider the case where $F_0$ is a finite extension of $\mathbf{Q}_3$ (although we will occasionally pass to the completion of the maximal unramified extension of $F$).

Lemma 4.1.1.

Let ${\mathcal{G}}$ be a finite flat $R$-group scheme. Scheme theoretic closure gives a bijection between subgroup schemes of ${\mathcal{G}}\times F'$ and finite flat closed subgroup schemes of ${\mathcal{G}}$.

(See for instance §1.1 of Co.)

Lemma 4.1.2.

Let ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ be finite flat group schemes over $R$ which have local-local closed fibre. Suppose that ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ are the only finite flat $R$-group schemes with local-local closed fibre which have generic fibres ${\mathcal{G}}_1 \times F'$ and ${\mathcal{G}}_2 \times F'$ respectively. Suppose also that we have an exact sequence of finite flat $R$-group schemes

$$\begin{equation*} (0) \longrightarrow {\mathcal{G}}_1 \longrightarrow {\mathcal{G}}\longrightarrow {\mathcal{G}}_2 \longrightarrow (0). \end{equation*}$$

Then ${\mathcal{G}}$ is the unique finite flat $R$-group scheme with local-local closed fibre and with generic fibre ${\mathcal{G}}\times F$.

Proof.

Let $\mathcal{G}_+$ and $\mathcal{G}_-$ denote the maximal and minimal local-local models for ${\mathcal{G}}\times F$. The proof that these exist follows the proof of Proposition 2.2.2 of Ra and uses the fact that the Cartier dual of a local-local finite flat group scheme is local-local. We must show that the canonical map $\mathcal{G}_+ \to \mathcal{G}_-$ is an isomorphism. The scheme-theoretic closure of ${\mathcal{G}}_1 \times F$ in $\mathcal{G}_\pm$ must be isomorphic to $\mathcal{G}_1$ (by uniqueness), so we have closed immersions $\mathcal{G}_1 \hookrightarrow \mathcal{G}_\pm$ extending ${\mathcal{G}}_1 \times F \hookrightarrow {\mathcal{G}}_\pm \times F$. Similarly $\mathcal{G}_\pm /\mathcal{G}_1$ must be isomorphic to $\mathcal{G}_2$. This gives a commutative diagram with exact rows:

$$\begin{equation*} \begin{array}{ccccccc} 0 \rightarrow &\mathcal{G}_1&\rightarrow &\mathcal{G}_+&\rightarrow &\mathcal{G}_2&\rightarrow 0\\&\downarrow & &\downarrow & & \downarrow & \\0 \rightarrow &\mathcal{G}_1&\rightarrow &\mathcal{G}_-&\rightarrow &\mathcal{G}_2&\rightarrow 0\\\end{array} \end{equation*}$$

The vertical maps ${\mathcal{G}}_1 \rightarrow {\mathcal{G}}_1$ and ${\mathcal{G}}_2 \rightarrow {\mathcal{G}}_2$ induce isomorphisms on the generic fibre and hence are isomorphisms. This is because some power of them is the identity on the generic fibre and hence is the identity. Working in the abelian category of fppf abelian sheaves over $\operatorname {Spec}R$, the middle map must also be an isomorphism.

■

When ${\mathcal{G}}$ has $\ell$-power order, we will let $D({\mathcal{G}})$ denote the classical (contravariant) Dieudonné module of ${\mathcal{G}}\times k$. It is a $W(k)$-module equipped with a Frobenius operator $\mathbf{F}$ and a Verschiebung operator $\mathbf{V}$. We have $\mathbf{F}\mathbf{V}=\mathbf{V}\mathbf{F}=\ell$ and for all $x \in W(k)$, $\mathbf{F}x=(\operatorname {Frob}_\ell x)\mathbf{F}$ and $\mathbf{V}x=(\operatorname {Frob}_\ell ^{-1}x)\mathbf{V}$.

If ${\mathcal{G}}$ is a finite flat $R$-group scheme, then by descent data for ${\mathcal{G}}$ over $F_0$ we mean a collection $\{ [g] \}$ of group scheme isomorphisms over $R$

$$\begin{equation*} [g]: {\mathcal{G}}\stackrel{\sim }{\longrightarrow }{^g{\mathcal{G}}} \end{equation*}$$

for $g \in \Gamma$ such that for all $g,h \in \Gamma$ we have

$$\begin{equation*} [gh]= ({^g[h]}) \circ [g]. \end{equation*}$$

Note that this is not descent data in the sense of Grothendieck, since $R/ R^{\Gamma }$ might be ramified. However, $\operatorname {Spec}F'/\operatorname {Spec}F_0$ is étale, so by étale descent we obtain a finite flat group scheme $({\mathcal{G}},\{ [g]\})_{F_0}$ over $F_0$ together with an isomorphism

$$\begin{equation*} ({\mathcal{G}},\{ [g]\})_{F_0} \times _{F_0} F' \cong {\mathcal{G}}\times _R F' \end{equation*}$$

compatible with descent data. We also obtain a natural left action of $\Gamma$ on the Dieudonné module $D({\mathcal{G}})$, semi-linear with respect to the $W(k)$-module structure and commuting with $\mathbf{F}$ and $\mathbf{V}$. We refer to the pair $({\mathcal{G}},\{ [g]\})$ as an $R$-group scheme with descent data relative to $F_0$. One defines morphisms of such objects to be morphisms of $R$-group schemes which commute with the descent data. By a closed finite flat subgroup scheme with descent data we mean a closed finite flat subgroup scheme such that the descent data on the ambient scheme takes the subscheme to itself. Quotients by such subobjects are defined in the obvious way. Thus we obtain an additive category with a notion of short exact sequence. Suppose that $G$ is a finite flat $F_0$-group scheme. By a model with descent data (or simply model) for $G$ over $R$ we shall mean a triple $({\mathcal{G}},\{ [g]\},i)$, where $({\mathcal{G}},\{ [g]\})$ is an $R$-group scheme with descent data relative to $F_0$ and where $i: ({\mathcal{G}},\{ [g]\})_{F_0} \stackrel{\sim }{\rightarrow }G$. Sometimes we will suppress $i$ from the notation. It is easy to check that isomorphism classes of models admitting descent data for $G$ over $R$ form a sublattice of the lattice of models for $G_{/F'}$ over $R$. The following lemma follows without difficulty from Lemma 4.1.1.

Lemma 4.1.3.

Let $F'/F_0$ be a finite Galois extension as above, and let $({\mathcal{G}}, \{ [g] \})$ be a finite flat $R$-group scheme with descent data relative to $F_0$. Base change from $F_0$ to $F'$, followed by scheme theoretic closure, gives a bijection between subgroup schemes of $({\mathcal{G}}, \{ [g]\})_{F_0}$ and closed finite flat subgroup schemes with descent data in $({\mathcal{G}}, \{ [g]\})$.

We let ${\mathcal{F}}{{\mathcal{F}}}_{F'}$ denote the category of finite flat group schemes over $R$ and ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0}$ the category of finite flat group schemes over $R$ with descent data over $F_0$. Let $W(k)[\mathbf{F},\mathbf{V}][\Gamma ]$ denote the (non-commutative) $W(k)$-algebra generated by elements $\mathbf{F}$, $\mathbf{V}$ and $[g]$ for $g \in \Gamma$ satisfying

•

$[gh]=[g][h]$ for all $g, h \in \Gamma$;

•

$[g]\mathbf{F}=\mathbf{F}[g]$ and $[g]\mathbf{V}=\mathbf{V}[g]$ for all $g \in \Gamma$;

•

$\mathbf{F}\mathbf{V}=\mathbf{V}\mathbf{F}=\ell$;

•

$[g]x=(gx)[g]$ for all $x \in W(k)$ and $g \in \Gamma$;

•

$\mathbf{F}x=(\operatorname {Frob}_\ell x) F$ and $\mathbf{V}x=(\operatorname {Frob}_\ell ^{-1}x)\mathbf{V}$ for all $x \in W(k)$.

If ${\mathcal{I}}$ is a two-sided ideal in $W(k)[\mathbf{F},\mathbf{V}][\Gamma ]$, we will let ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$ denote the full subcategory of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0}$ consisting of objects $({\mathcal{G}},\{[g]\})$ such that ${\mathcal{I}}$ annihilates $D({\mathcal{G}})$. If $({\mathcal{G}},\{ [g]\})$ is an object of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$ and if $({\mathcal{H}},\{[g]\}) \subset ({\mathcal{G}},\{[g]\})$ is a closed finite flat subgroup scheme with descent data, then $({\mathcal{H}},\{[g]\})$ and $({\mathcal{G}},\{[g]\})/({\mathcal{H}},\{[g]\})$ are again objects of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$.

Lemma 4.1.4.

For ${\mathcal{I}}$ a two-sided ideal of the ring $W(k)[\mathbf{F},\mathbf{V}][\Gamma ]$, choose objects $({\mathcal{G}}_1, \{ [g] \})$ and $({\mathcal{G}}_2, \{ [g] \})$ in ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$ so that $({\mathcal{G}}_1, \{ [g] \})_{F_0} \cong ({\mathcal{G}}_2,\{ [g] \})_{F_0}$. Let $G$ denote the base change of this $F_0$-group scheme to $F'$, so $G$ has canonical descent data relative to $F'/F_0$. Then the $\sup$ and $\inf$ of ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ in the lattice of integral models for $G$ are stable under the descent data on $G$ and with this descent data are objects of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$.

Proof.

By uniqueness of the inf and sup, they are stable under the descent data on the generic fibre. It follows from Raynaud’s construction of the inf and sup (Proposition 2.2.2 of Ra) in terms of subgroup schemes and quotients of ${\mathcal{G}}_1 \times {\mathcal{G}}_2$ that the sup and inf are objects of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$.

■

Corollary 4.1.5.

Let ${\mathcal{I}}$ be a two-sided ideal of the ring $W(k)[\mathbf{F},\mathbf{V}][\Gamma ]$. Let

$$\begin{equation} (0) \longrightarrow G_1 \longrightarrow G \longrightarrow G_2 \longrightarrow (0) {\tag{4.1.1}} \end{equation}$$

be an exact sequence of finite flat group schemes over $F_0$. Let $({\mathcal{G}}_1,\{ [g]\})$ and $({\mathcal{G}}_2,\{ [g]\})$ be objects of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$ such that $({\mathcal{G}}_1,\{ [g]\})_{F_0} \cong G_1$ and $({\mathcal{G}}_2,\{ [g]\})_{F_0} \cong G_2$. Suppose that for all objects $({\mathcal{G}},\{ [g]\})$ of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$ with $({\mathcal{G}},\{ [g]\})_{F_0} \cong G$, the filtration on $({\mathcal{G}},\{ [g]\})$ induced by the filtration on $G$ has subobject isomorphic to $({\mathcal{G}}_1,\{ [g]\})$ and quotient isomorphic to $({\mathcal{G}}_2,\{ [g]\})$ (without any assumed compatibility with (4.1.1$))$. Then there is at most one model for $G$ in ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$.

Proof.

By Lemma 4.1.4, it suffices to prove that if $({\mathcal{G}}_+,\{ [g]\},i_+)$ and $({\mathcal{G}}_-,\{ [g]\},i_-)$ are two such models with a morphism between them, then the morphism between them must be an isomorphism. In such a case we have a commutative diagram with exact rows:

$$\begin{equation*} \begin{array}{ccccccc} 0 \rightarrow &\mathcal{G}_1&\rightarrow &\mathcal{G}_+&\rightarrow &\mathcal{G}_2&\rightarrow 0\\&\downarrow & &\downarrow & & \downarrow & \\0 \rightarrow &\mathcal{G}_1&\rightarrow &\mathcal{G}_-&\rightarrow &\mathcal{G}_2&\rightarrow 0\\\end{array} \end{equation*}$$

The vertical maps ${\mathcal{G}}_1 \rightarrow {\mathcal{G}}_1$ and ${\mathcal{G}}_2 \rightarrow {\mathcal{G}}_2$ induce isomorphisms on the generic fibre and hence are isomorphisms. This is because some power of them is the identity on the generic fibre and hence is the identity. Working in the abelian category of fppf abelian sheaves over $\operatorname {Spec}R$, the middle map must also be an isomorphism.

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4.2. Filtrations

We keep the notation and assumptions of the previous section. Let $\Sigma$ be a finite non-empty set of objects $({\mathcal{G}}_i,\{ [g]\})$ of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,({\mathcal{I}},\ell )}$. (Note the $\ell$ in the subscript $({\mathcal{I}},\ell )$, which denotes the two-sided ideal generated by ${\mathcal{I}}$ and $\ell$.) Suppose that

$$\begin{equation} \begin{split} \operatorname {Hom}(({\mathcal{G}}_i,\{ [g]\}),({\mathcal{G}}_j,\{ [g]\}))& = \operatorname {Hom}(({\mathcal{G}}_i,\{ [g]\})_{F_0},({\mathcal{G}}_j,\{ [g]\})_{F_0})\\ & = \left\{\begin{array}{ll} 0&\text{if $i \ne j$},\\\text{finite field}&\text{if $i = j$} \end{array}\right. \end{split}{\tag{4.2.1}} \end{equation}$$

(in particular, the objects in $\Sigma$ are non-zero and pairwise non-isomorphic). By a $\Sigma$-filtration on a finite flat $F_0$-group scheme $G$ we mean an increasing filtration $\operatorname {Fil}^j G$ such that for all $j$ the graded piece $\operatorname {Fil}^j G /\operatorname {Fil}^{j-1} G$ is isomorphic to $({\mathcal{G}}_{i(j)},\{ [g]\})_{F_0}$ for a (unique) $({\mathcal{G}}_{i(j)},\{ [g]\}) \in \Sigma$. The following lemma is proved by the standard Jordan-Hölder argument.

Lemma 4.2.1.

If $G$ is a finite flat $F_0$-group scheme which admits a $\Sigma$-filtration and if $H$ is a quotient or subobject of $G$ which admits a $\Sigma$-filtration, then any $\Sigma$-filtration of $H$ can be extended to a $\Sigma$-filtration of $G$. In addition, all $\Sigma$-filtrations of $G$ have the same length and the same set of successive quotients (with multiplicities).

We say that an object $(\mathcal{G},\{[g]\})$ of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$ is weakly filtered by $\Sigma$ if there is some increasing filtration $\operatorname {Fil}^j(\mathcal{G},\{[g]\})$ of $(\mathcal{G},\{[g]\})$ by closed subobjects such that for all $j$, the graded piece

$$\begin{equation*} \operatorname {Fil}^j (\mathcal{G},\{[g]\}) /\operatorname {Fil}^{j-1} (\mathcal{G},\{[g]\}) \end{equation*}$$

is isomorphic to an element of $\Sigma$. We say that an object $(\mathcal{G},\{[g]\})$ of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$ is strongly filtered by $\Sigma$ if $(\mathcal{G},\{[g]\})$ is weakly filtered by $\Sigma$ and if for every $\Sigma$-filtration of $(\mathcal{G},\{[g]\})_{F_0}$ the corresponding filtration of $(\mathcal{G},\{[g]\})$ satisfies

$$\begin{equation*} \operatorname {Fil}^j (\mathcal{G},\{[g]\}) /\operatorname {Fil}^{j-1} (\mathcal{G},\{[g]\}) \end{equation*}$$

is isomorphic to an element of $\Sigma$ for all $j$. The following lemma follows at once from the definitions and from Lemma 4.2.1.

Lemma 4.2.2.

(1)

If $({\mathcal{G}},\{ [g] \})$ and $({\mathcal{G}}',\{ [g] \})$ are objects of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$ which are weakly filtered by $\Sigma$, then $({\mathcal{G}},\{ [g] \}) \times ({\mathcal{G}}',\{ [g] \})$ is also weakly filtered by $\Sigma$.

(2)

Let $({\mathcal{G}},\{ [g] \})$ and $({\mathcal{G}}',\{ [g] \})$ be objects of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$ with $({\mathcal{G}}',\{ [g] \})$ a closed subobject or quotient of $({\mathcal{G}},\{ [g] \})$. Suppose that $({\mathcal{G}},\{ [g] \})$ is strongly filtered by $\Sigma$ and that $({\mathcal{G}}',\{ [g]\})_{F_0}$ admits a $\Sigma$-filtration. Then $({\mathcal{G}}',\{ [g]\})$ is strongly filtered by $\Sigma$.

If any object of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$ which is weakly filtered by $\Sigma$ is strongly filtered by $\Sigma$, then we will let ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}},\Sigma }$ denote the full subcategory of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$ consisting of objects which are weakly (and therefore strongly) filtered by $\Sigma$.

Lemma 4.2.3.

Suppose that any object of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$ which is weakly filtered by $\Sigma$ is strongly filtered by $\Sigma$. Let $G$ be a finite flat $F_0$-group scheme. If $({\mathcal{G}}_1,\{ [g]\})$ and $({\mathcal{G}}_2,\{ [g]\})$ are two objects of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}},\Sigma }$ with isomorphisms

$$\begin{equation*} i_j: G \stackrel{\sim }{\longrightarrow }({\mathcal{G}}_j,\{ [g]\})_{F_0} \end{equation*}$$

for $j=1,2$, then there is a unique isomorphism

$$\begin{equation*} \phi : ({\mathcal{G}}_1,\{ [g]\}) \stackrel{\sim }{\longrightarrow }({\mathcal{G}}_2,\{ [g]\}) \end{equation*}$$

such that on the generic fibre $i_2=\phi \circ i_1$.

Proof.

It follows from Raynaud’s construction of sup and inf that the sup and inf of $(({\mathcal{G}}_1,\{ [g] \}),i_1)$ and $(({\mathcal{G}}_2,\{ [g] \}),i_2)$ are again objects of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}},\Sigma }$. Thus we may suppose that there exists a map $\phi : ({\mathcal{G}}_1,\{ [g]\}) \rightarrow ({\mathcal{G}}_2,\{ [g]\})$ such that on the generic fibre $i_2=\phi \circ i_1$. We will argue by induction on the rank of $G$ that $\phi$ is an isomorphism.

If $({\mathcal{G}}_1,\{ [g]\})$ is isomorphic to an element of $\Sigma$, then the result follows by our assumption on $\Sigma$.

If $({\mathcal{G}}_1,\{ [g]\})$ is not isomorphic to an element of $\Sigma$, then choose an exact sequence

$$\begin{equation*} (0) \longrightarrow ({\mathcal{G}}_{11},\{ g\}) \longrightarrow ({\mathcal{G}}_1,\{ g\}) \longrightarrow ({\mathcal{G}}_{12},\{ g\}) \longrightarrow (0), \end{equation*}$$

where $({\mathcal{G}}_{11},\{ g\})$ and $({\mathcal{G}}_{12},\{ g\})$ are weakly filtered by $\Sigma$. Let $({\mathcal{G}}_{21},\{ [g]\})$ denote the closed subobject of $({\mathcal{G}}_2,\{[g]\})$ corresponding to $({\mathcal{G}}_{11},\{ [g]\})_{F_0}$ and define $({\mathcal{G}}_{22},\{[g]\})=({\mathcal{G}}_2,\{[g]\})/({\mathcal{G}}_{21},\{ [g]\})$. Then we have a commutative diagram with exact rows

$$\begin{equation*} \begin{array}{ccccccc} 0 \rightarrow &\mathcal{G}_{11}&\rightarrow &\mathcal{G}_1&\rightarrow &\mathcal{G}_{12}&\rightarrow 0\\&\downarrow & &\downarrow & & \downarrow & \\0 \rightarrow &\mathcal{G}_{21}&\rightarrow &\mathcal{G}_2&\rightarrow &\mathcal{G}_{22}&\rightarrow 0 \end{array} \end{equation*}$$

compatible with descent data, where the central vertical arrow is $\phi$ and where by inductive hypothesis the outside vertical arrows are isomorphisms. Working in the abelian category of fppf abelian sheaves over $\operatorname {Spec}R$, we see that $\phi$ is an isomorphism.

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The following lemma and its corollary give criteria for the equivalence of the notions of being weakly filtered by $\Sigma$ and of being strongly filtered by $\Sigma$.

Lemma 4.2.4.

Fix ${\mathcal{I}}$ and $\Sigma$ as above. Suppose that for any pair of (possibly equal) elements $(\mathcal{G}',\{ [g] \})$ and $(\mathcal{G}'',\{ [g] \})$ in $\Sigma$, the natural map

$$\begin{equation*} \operatorname {Ext}^1_{{\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,({\mathcal{I}},\ell )}}((\mathcal{G}'',\{ [g] \}),(\mathcal{G}',\{ [g] \})) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_\ell [G_{F_0}]}((\mathcal{G}'',\{ [g] \})_{F_0}, (\mathcal{G}',\{ [g] \})_{F_0}) \end{equation*}$$

is injective. Then any object $(\mathcal{G},\{ [g] \})$ of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$ which is weakly filtered by $\Sigma$ is also strongly filtered by $\Sigma$.

Proof.

For brevity, we say “weakly/strongly filtered” rather than “weakly/strongly filtered by $\Sigma$” since the data $\Sigma$ is fixed for the entire proof. Also, we omit the specification of descent data from the notation, but it should not be forgotten.

Suppose $\mathcal{G}$ is weakly filtered. In order to prove that $\mathcal{G}$ is strongly filtered, we argue by induction on the length of a $\Sigma$-filtration of $\mathcal{G}_{F_0}$, this length being well-defined by Lemma 4.2.1. The case of length $\le 1$ is clear. Otherwise, by the definition of being weakly filtered, there is a short exact sequence of finite flat $R$-group schemes (with descent data relative to $F_0$)

$$\begin{equation*} (0) \longrightarrow \mathcal{G}' \longrightarrow \mathcal{G}\longrightarrow \mathcal{G}'' \longrightarrow (0) \end{equation*}$$

with $\mathcal{G}' \in \Sigma$ and $\mathcal{G}''$ weakly filtered (and hence, by inductive hypothesis, strongly filtered). Let $\mathcal{H}$ be any closed subgroup scheme of $\mathcal{G}$ (with compatible descent data relative to $F_0$) such that $\mathcal{H}_{F_0} \simeq \mathcal{G}_{i_0,F_0}$ for some $\mathcal{G}_{i_0} \in \Sigma$ and such that $(\mathcal{G}/\mathcal{H})_{F_0}$ admits a $\Sigma$-filtration. We need to prove (in the category of finite flat group schemes with descent data relative to $F'/F_0$) that

•

$\mathcal{H}\simeq \mathcal{G}_{i_0}$,

•

and $\mathcal{G}/\mathcal{H}$ is weakly filtered.

If the composite map

$$\begin{equation*} \mathcal{H}\hookrightarrow \mathcal{G}\longrightarrow \mathcal{G}'' \end{equation*}$$

is zero, then $\mathcal{H}= \mathcal{G}'$ as closed subgroup schemes of $\mathcal{G}$ (with descent data) and likewise $\mathcal{G}/\mathcal{H}= \mathcal{G}''$, so we are done. The interesting case is when the composite map is non-zero. The map ${\mathcal{G}}_{i_0,F_0} \cong \mathcal{H}_{F_0} \rightarrow \mathcal{G}''_{F_0}$ is then non-zero and therefore must be a closed immersion by the assumption (4.2.1) on $\Sigma$ and a devissage with respect to a $\Sigma$-filtration of $\mathcal{G}''_{F_0}$. We conclude that the map of generic fibre étale group schemes $\mathcal{H}\times F' \rightarrow \mathcal{G}'' \times F'$ is a closed immersion.

Taking scheme-theoretic closures, we obtain a closed subgroup scheme $\mathcal{H}'' \hookrightarrow \mathcal{G}''$ (with unique compatible descent data over $F_0$) fitting into a commutative diagram of group schemes with descent data

$$\begin{equation*} \begin{matrix} &\mathcal{H}&\stackrel{\alpha }{\longrightarrow }&\mathcal{H}''&\\ &\downarrow &&\downarrow &\\ (0)\longrightarrow \mathcal{G}'\longrightarrow &\mathcal{G}&\longrightarrow &\mathcal{G}''& \longrightarrow (0) \end{matrix} \end{equation*}$$

in which the lower row is short exact, the vertical maps are closed immersions and the top map $\mathcal{H}\rightarrow \mathcal{H}''$ induces an isomorphism on generic fibres. By Lemma 4.2.1 we may extend $\mathcal{H}_{F_0} \hookrightarrow \mathcal{G}''_{F_0}$ to a $\Sigma$-filtration on $\mathcal{G}''_{F_0}$ and so, because ${\mathcal{G}}''$ is strongly filtered by induction, we may extend $\mathcal{H}'' \hookrightarrow \mathcal{G}''$ to a $\Sigma$-filtration. In particular $\mathcal{H}''$ is isomorphic to an object in $\Sigma$ and $\mathcal{G}''/\mathcal{H}''$ is strongly filtered.

Pulling back the short exact sequence

$$\begin{equation*} 0 \longrightarrow \mathcal{G}' \longrightarrow \mathcal{G}\longrightarrow \mathcal{G}'' \longrightarrow 0 \end{equation*}$$

by $\mathcal{H}'' \rightarrow \mathcal{G}''$, we get a diagram

$$\begin{equation*} \begin{matrix} &&&\mathcal{H}&&&\\ &&&\downarrow &&&\\ 0 \longrightarrow &\mathcal{G}'&\longrightarrow &\mathcal{G}\times _{\mathcal{G}''} \mathcal{H}''&\longrightarrow &\mathcal{H}''&\longrightarrow 0\\ \end{matrix} \end{equation*}$$

in which the row is a short exact sequence of fppf abelian sheaves and all of the terms are finite flat group schemes (for the middle, this follows from the flatness of $\mathcal{G}\rightarrow \mathcal{G}''$). Thus, this bottom row is a short exact sequence of finite flat group schemes (with descent data). As $\mathcal{H}_{F_0} \stackrel{\sim }{\rightarrow }\mathcal{H}''_{F_0}$, the sequence

$$\begin{equation*} 0 \longrightarrow \mathcal{G}'_{F_0} \longrightarrow (\mathcal{G}\times _{\mathcal{G}''} \mathcal{H}'')_{F_0} \longrightarrow \mathcal{H}''_{F_0} \longrightarrow 0 \end{equation*}$$

is split. In particular $(\mathcal{G}\times _{\mathcal{G}''} \mathcal{H}'')_{F_0}$ and hence $\mathcal{G}\times _{\mathcal{G}''} \mathcal{H}''$ are killed by $\ell$. By the hypothesis of the lemma

$$\begin{equation*} 0 \longrightarrow \mathcal{G}' \longrightarrow \mathcal{G}\times _{\mathcal{G}''} \mathcal{H}'' \longrightarrow \mathcal{H}'' \longrightarrow 0 \end{equation*}$$

is also split, i.e. we have an isomorphism

$$\begin{equation*} {\mathcal{G}}\times _{{\mathcal{G}}''} {\mathcal{H}}'' \cong {\mathcal{G}}' \times _R {\mathcal{H}}'' \end{equation*}$$

such that ${\mathcal{G}}' \hookrightarrow {\mathcal{G}}\times _{{\mathcal{G}}''} {\mathcal{H}}''$ corresponds to injection to the first factor of ${\mathcal{G}}' \times _R {\mathcal{H}}''$ and ${\mathcal{G}}\times _{{\mathcal{G}}''} {\mathcal{H}}'' \rightarrow \hspace{-.86em} \rightarrow {\mathcal{H}}''$ corresponds to projection onto the second factor. By our hypotheses on $\Sigma$ we can find a morphism $\phi :{\mathcal{H}}'' \rightarrow {\mathcal{G}}'$ extending

$$\begin{equation*} {\mathcal{H}}_{F_0}'' \stackrel{\sim }{\longleftarrow } {\mathcal{H}}_{F_0} \hookrightarrow {\mathcal{G}}_{F_0}' \times {\mathcal{H}}_{F_0}'' \stackrel{{\mathrm{pr}}}{\rightarrow \hspace{-.86em} \rightarrow } {\mathcal{G}}_{F_0}'. \end{equation*}$$

Then our closed immersion ${\mathcal{H}}\hookrightarrow {\mathcal{G}}' \times _R {\mathcal{H}}''$ factors as

$$\begin{equation*} {\mathcal{H}}\longrightarrow {\mathcal{H}}'' \stackrel{\phi \times 1}{\longrightarrow } {\mathcal{G}}' \times _R {\mathcal{H}}''. \end{equation*}$$

As ${\mathcal{H}}\rightarrow {\mathcal{G}}' \times _R {\mathcal{H}}''$ is a closed immersion, $\alpha :{\mathcal{H}}\rightarrow {\mathcal{H}}''$ must be a closed immersion and hence an isomorphism. Thus ${\mathcal{H}}$ is isomorphic to an object in $\Sigma$.

Now we turn to the proof that $\mathcal{G}/\mathcal{H}$ is weakly filtered. Since $\alpha :\mathcal{H}\rightarrow \mathcal{H}''$ is an isomorphism, it is clear that the natural map

$$\begin{equation*} \mathcal{H}\times _R \mathcal{G}' \longrightarrow \mathcal{G}\times _{{\mathcal{G}}''} {\mathcal{H}}'' \end{equation*}$$

is an isomorphism, and hence that

$$\begin{equation*} {\mathcal{H}}\times _R {\mathcal{G}}' \longrightarrow {\mathcal{G}} \end{equation*}$$

is a closed immersion. Thus, the finite flat group scheme $\mathcal{G}/(\mathcal{H}\times \mathcal{G}')$ makes sense and the natural map

$$\begin{equation*} \mathcal{G}/(\mathcal{H}\times \mathcal{G}') \longrightarrow \mathcal{G}''/\mathcal{H}'' \end{equation*}$$

is an isomorphism (as one sees by using the universal properties of quotients to construct an inverse map). We therefore arrive at a short exact sequence

$$\begin{equation*} 0 \longrightarrow \mathcal{G}' \longrightarrow \mathcal{G}/\mathcal{H}\longrightarrow \mathcal{G}''/\mathcal{H}'' \longrightarrow 0 \end{equation*}$$

(compatible with descent data). Since $\mathcal{G}''/\mathcal{H}''$ is strongly filtered, as we noted above, and $\mathcal{G}' \in \Sigma$, it follows that $\mathcal{G}/\mathcal{H}$ is weakly filtered.

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Corollary 4.2.5.

Fix ${\mathcal{I}}$ and $\Sigma$ as above. Suppose that $\Sigma =\{ ({\mathcal{G}},\{ [g] \}) \}$ is a singleton. Suppose also that we have a short exact sequence

$$\begin{equation*} (0) \longrightarrow ({\mathcal{G}}_1,\{ [g]\}) \longrightarrow ({\mathcal{G}},\{ [g]\}) \longrightarrow ({\mathcal{G}}_2,\{ [g]\}) \longrightarrow (0) \end{equation*}$$

in ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$, where for any $i,j$ (possibly equal)

$$\begin{align*} \operatorname {Hom}(({\mathcal{G}}_i,\{ [g]\}),({\mathcal{G}}_j,\{ [g]\})) &= \operatorname {Hom}(({\mathcal{G}}_i,\{ [g]\})_{F_0},({\mathcal{G}}_j,\{ [g]\})_{F_0}) \\ &= \left\{\begin{array}{ll} 0&\text{if $i \ne j$},\\\text{finite field}&\text{if $i = j$}, \end{array}\right. \end{align*}$$

and the natural map

$$\begin{equation} \operatorname {Ext}^1_{{\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,({\mathcal{I}},\ell )}}((\mathcal{G}_i,\{ [g] \}),(\mathcal{G}_j,\{ [g] \})) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_\ell [G_{F_0}]}((\mathcal{G}_i,\{ [g] \})_{F_0}, (\mathcal{G}_j,\{ [g] \})_{F_0}) {\tag{4.2.2}} \end{equation}$$

is injective. Then any object $(\mathcal{H},\{ [g] \})$ of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/F_0,{\mathcal{I}}}$ which is weakly filtered by $\Sigma$ is also strongly filtered by $\Sigma$.

Proof.

As $(\mathcal{H},\{[g]\})$ is weakly filtered by $\Sigma$, it is weakly filtered by $\{ ({\mathcal{G}}_1,\{ [g]\}), ({\mathcal{G}}_2,\{ [g]\}) \}$, and so by Lemma 4.2.4 is strongly filtered by $\{ ({\mathcal{G}}_1,\{ [g]\}), ({\mathcal{G}}_2,\{ [g]\}) \}$. Any $\Sigma$-filtration of $(\mathcal{H},\{[g]\})_{F_0}$ extends to a $\{ ({\mathcal{G}}_1,\{ [g]\}), ({\mathcal{G}}_2,\{ [g]\}) \}$-filtration of $(\mathcal{H},\{[g]\})_{F_0}$, which in turn gives rise to a $\{ ({\mathcal{G}}_1,\{ [g]\}), ({\mathcal{G}}_2,\{ [g]\}) \}$-filtration of $(\mathcal{H},\{[g]\})$. By the injectivity of (4.2.2) we see that this yields a $\Sigma$-filtration of $({\mathcal{H}},\{ [g]\})$ that induces to our chosen $\Sigma$-filtration of $({\mathcal{H}},\{ [g]\})_{F_0}$.

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4.3. Generalities on deformation theory

Again in this section $\ell$ denotes an arbitrary rational prime. We let $K$ denote a finite extension of $\mathbf{Q}_\ell$, ${\mathcal{O}}$ the ring of integers $K$, $\wp _K$ the maximal ideal of ${\mathcal{O}}$ and $k$ its residue field. Note that $k$ has a different meaning from the previous two sections. Let $V$ be a two-dimensional $k$-vector space and ${\overline{\rho }}:G_\ell \rightarrow \operatorname {Aut}_k(V)$ a continuous representation. Suppose that the centraliser of $G_\ell$ in $\operatorname {End}_k(V)$ is $k$. Let $\psi :G_\ell \rightarrow {\mathcal{O}}^\times$ denote a continuous character such that $(\psi \bmod \wp _K) \cong \det {\overline{\rho }}$. Let ${\operatorname {\mathcal{S}}}({\overline{\rho }})$ denote the full subcategory of the category of finite length (discrete) ${\mathcal{O}}$-modules with a continuous ${\mathcal{O}}$-linear action of $G_\ell$ consisting of objects which admit a finite filtration so that each successive quotient is isomorphic to $V$. Because $\operatorname {End}_{k[G_\ell ]}(V)=k$, it follows from the usual Jordan-Hölder argument that ${\operatorname {\mathcal{S}}}({\overline{\rho }})$ is an abelian category.

Let ${\operatorname {\mathcal{S}}}$ be a full subcategory of ${\operatorname {\mathcal{S}}}({\overline{\rho }})$ stable under isomorphisms and which is closed under finite products, ${\operatorname {\mathcal{S}}}({\overline{\rho }})$-subobjects and ${\operatorname {\mathcal{S}}}({\overline{\rho }})$-quotients, and which contains $V$. We will consider the following set-valued functors on the category of complete noetherian local ${\mathcal{O}}$-algebras $R$ with finite residue field $k$.

•

$\operatorname {\mathcal{D}}_{V,{\mathcal{O}}}(R)$ is the set of conjugacy classes of continuous representations $\rho :G_\ell \rightarrow {\operatorname {GL}}_2(R)$ such that $\rho \bmod \mathfrak{m}_R$ is conjugate to ${\overline{\rho }}$.

•

$\operatorname {\mathcal{D}}_{V,{\mathcal{O}}}^\psi (R)$ is the set of conjugacy classes of continuous representations $\rho :G_\ell \rightarrow {\operatorname {GL}}_2(R)$ such that $\rho \bmod \mathfrak{m}_R$ is conjugate to ${\overline{\rho }}$ and $\det \rho = \psi$.

•

$\operatorname {\mathcal{D}}_{V,{\mathcal{O}}}^{\operatorname {\mathcal{S}}}(R)$ is the set of conjugacy classes of continuous representations $\rho :G_\ell \rightarrow {\operatorname {GL}}_2(R)$ such that $\rho \bmod \mathfrak{m}_R$ is conjugate to ${\overline{\rho }}$ and such that for each open ideal $\mathfrak{a}\subset R$ the action $\rho$ makes $(R/\mathfrak{a})^2$ into an object of ${\operatorname {\mathcal{S}}}$.

•

$\operatorname {\mathcal{D}}_{V,{\mathcal{O}}}^{\psi ,{\operatorname {\mathcal{S}}}}(R)$ is the set of conjugacy classes of continuous representations $\rho :G_\ell \rightarrow {\operatorname {GL}}_2(R)$ such that $\rho \bmod \mathfrak{m}_R$ is conjugate to ${\overline{\rho }}$, such that $\det \rho = \psi$, and such that for each open ideal $\mathfrak{a}\subset R$ the action $\rho$ makes $(R/\mathfrak{a})^2$ into an object of ${\operatorname {\mathcal{S}}}$.

Each of these deformation problems is representable by objects which we will denote $R_{V,{\mathcal{O}}}$, $R_{V,{\mathcal{O}}}^\psi$, $R_{V,{\mathcal{O}}}^{\operatorname {\mathcal{S}}}$ and $R_{V,{\mathcal{O}}}^{\psi ,{\operatorname {\mathcal{S}}}}$, respectively.

Recall that the following sets are in natural ($k$-linear) bijection with each other.

•

$(\mathfrak{m}_{R_{V,{\mathcal{O}}}}/(\wp _K, \mathfrak{m}_{R_{V,{\mathcal{O}}}}^2))^\vee$.

•

The set of deformations of ${\overline{\rho }}$ to $k[\varepsilon ]/(\varepsilon ^2)$.

•

$\operatorname {Ext}^1_{k[G_\ell ]}(V,V)$.

•

$H^1(G_\ell , \operatorname {ad}{\overline{\rho }})$.

These bijections give rise to an isomorphism

$$\begin{equation*} (\mathfrak{m}_{R_{V,{\mathcal{O}}}^\psi }/(\wp _K, \mathfrak{m}_{R_{V,{\mathcal{O}}}^\psi }^2))^\vee \cong H^1(G_\ell , \operatorname {ad}^0 {\overline{\rho }}), \end{equation*}$$

as well as bijections between

•

$(\mathfrak{m}_{R_{V,{\mathcal{O}}}^{\operatorname {\mathcal{S}}}}/(\wp _K, \mathfrak{m}_{R_{V,{\mathcal{O}}}^{\operatorname {\mathcal{S}}}}^2))^\vee$,

•

the set of deformations of ${\overline{\rho }}$ to $k[\varepsilon ]/(\varepsilon ^2)$ which make $(k[\varepsilon ]/(\varepsilon ^2))^2$ into an object of ${\operatorname {\mathcal{S}}}$,

•

$\operatorname {Ext}^1_{k[G_\ell ],{\operatorname {\mathcal{S}}}}(V,V)$, i.e. $\operatorname {Ext}^1$ in the category of discrete $k[G_\ell ]$-modules which are also objects of ${\operatorname {\mathcal{S}}}$,

•

the subgroup $H^1_{\operatorname {\mathcal{S}}}(G_\ell , \operatorname {ad}{\overline{\rho }}) \subset H^1(G_\ell , \operatorname {ad}{\overline{\rho }})$ corresponding to $\operatorname {Ext}^1_{k[G_\ell ],{\operatorname {\mathcal{S}}}}(V,V)$.

We will set $H^1_{\operatorname {\mathcal{S}}}(G_\ell , \operatorname {ad}^0 {\overline{\rho }})=H^1_{\operatorname {\mathcal{S}}}(G_\ell , \operatorname {ad}{\overline{\rho }}) \cap H^1(G_\ell , \operatorname {ad}^0 {\overline{\rho }})$, so that we get an isomorphism

$$\begin{equation*} (\mathfrak{m}_{R_{V,{\mathcal{O}}}^{\psi ,{\operatorname {\mathcal{S}}}}}/(\wp _K, \mathfrak{m}_{R_{V,{\mathcal{O}}}^{\psi ,{\operatorname {\mathcal{S}}}}}^2))^\vee \cong H^1_{\operatorname {\mathcal{S}}}(G_\ell , \operatorname {ad}^0 {\overline{\rho }}). \end{equation*}$$

4.4. Reduction steps for Theorem 2.1.2

We now begin the proof of Theorem 2.1.2. Making an unramified twist we may suppose that ${\overline{\rho }}$ has the form

$$\begin{equation*} \left( \begin{array}{cc} {1} & {*} \\{0} & {\omega } \end{array} \right). \end{equation*}$$

We may also suppose that ${\mathcal{O}}=\mathbf{Z}_3$.

Let $F_1=F_1'$ denote a totally ramified cubic Galois extension of $\mathbf{Q}_3$. Let $F_{-1}'$ denote the unique cubic extension of $\mathbf{Q}_3(\sqrt {-1})$ such that $F_{-1}'/\mathbf{Q}_3$ is Galois but not abelian, and let $F_{-1}$ denote a cubic subfield of $F_{-1}'$, so $F_{-1}'/F_{-1}$ is unramified.

Let ${\operatorname {\mathcal{S}}}_{\pm 1}$ denote the full subcategory ${\operatorname {\mathcal{S}}}({\overline{\rho }})$ consisting of $\mathbf{Z}_3[G_3]$-modules $X$ for which there exists a finite flat ${\mathcal{O}}_{F_{\pm 1}'}$-group scheme $({\mathcal{G}},\{ [g] \})$ with descent data for $F_{\pm 1}'/\mathbf{Q}_3$ such that $X \cong ({\mathcal{G}},\{ [g]\})_{\mathbf{Q}_3}(\overline{\mathbf{Q}}_3)$ as a $\mathbf{Z}_3[G_3]$-module. By Lemma 4.1.3 we see that ${\operatorname {\mathcal{S}}}_{\pm 1}$ is closed under finite products, subobjects and quotients. Using Tate’s theorem on the uniqueness of extensions of $3$-divisible groups from $F_{\pm 1}'$ to ${\mathcal{O}}_{F_{\pm 1}'}$ (Theorem 4 of T), we see that the map $R_{V,\mathbf{Z}_3} \rightarrow \hspace{-.86em} \rightarrow R_{V,\mathbf{Z}_3}^{\tau _{\pm 1}}$ factors through $R_{V,\mathbf{Z}_3}^{\epsilon ,{\operatorname {\mathcal{S}}}_{\pm 1}}$. Thus, Theorem 2.1.2 follows from the following result which we will prove in §7.

Theorem 4.4.1.

$\dim H^1_{{\operatorname {\mathcal{S}}}_{\pm 1}}(G_3,\operatorname {ad}^0 {\overline{\rho }}) \leq 1$.

4.5. Reduction steps for Theorem 2.1.4

We now begin the proof of Theorem 2.1.4. Making an unramified twist, we may suppose that ${\overline{\rho }}$ has the form

$$\begin{equation*} \left( \begin{array}{cc} {\omega } & {*} \\{0} & {1} \end{array} \right). \end{equation*}$$

We may also suppose that ${\mathcal{O}}=\mathbf{Z}_3$.

Let $F_{\pm 3}'$ denote the degree $12$ abelian extension of $\mathbf{Q}_3(\sqrt {\pm 3})$ with norm subgroup in $\mathbf{Q}_3(\sqrt {\pm 3})^\times$ topologically generated by $\pm 3$, $4$ and $1+3 \sqrt {\pm 3}$. Note that $F_{\pm 3}'/\mathbf{Q}_3$ is Galois. We have an isomorphism

$$\begin{equation*} \operatorname {Gal}(F_{\pm 3}'/\mathbf{Q}_3(\sqrt {\pm 3})) \cong C_2 \times C_2 \times C_3. \end{equation*}$$

Let $\gamma _4^2 \in I_{F_{\pm 3}'/\mathbf{Q}_3(\sqrt {\pm 3})}$ be the unique element of order $2$. (In later applications this will be the square of an element of order $4$ in $\operatorname {Gal}(F_{\pm 3}'/\mathbf{Q}_3)$.) We also let $F_{\pm 3}$ denote the fixed field of a Frobenius lift of order $2$, so $F_{\pm 3}/\mathbf{Q}_3$ is totally ramified.

We will let ${\mathcal{I}}_{\pm 3}$ denote the two-sided ideal of $W(\mathbf{F}_9) [\mathbf{F},\mathbf{V}][\operatorname {Gal}(F_{\pm 3}'/\mathbf{Q}_3)]$ generated by

•

$\mathbf{F}+\mathbf{V}$,

•

and $[\gamma _4^2]+1$.

Let ${\operatorname {\mathcal{S}}}_{\pm 3}$ denote the full subcategory of ${\operatorname {\mathcal{S}}}({\overline{\rho }})$ consisting of objects $X$ for which we can find an object $({\mathcal{G}},\{ [g] \})$ of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F_{\pm 3}'/\mathbf{Q}_3,{\mathcal{I}}_{\pm 3} }$ such that $X \cong ({\mathcal{G}},\{ [g]\})_{\mathbf{Q}_3}(\overline{\mathbf{Q}}_3)$ as a $\mathbf{Z}_3[G_3]$-module. By Lemma 4.1.3, we see that ${\operatorname {\mathcal{S}}}_{\pm 3}$ is closed under finite products, subobjects and quotients.

Now choose a finite extension $K/\mathbf{Q}_3$ and continuous map of rings $f:R_{V,\mathbf{Z}_3} \rightarrow \overline{\mathbf{Q}}_3$ such that the corresponding representation $\rho :G_3 \rightarrow {\operatorname {GL}}_2({\mathcal{O}}_{K})$ is of type $\tau _{\pm 3}$. Let $G$ be the corresponding $3$-divisible group over $\mathbf{Q}_3$. By Tate’s theorem (Theorem 4 of T), the base change of $G$ to $F_{\pm 3}'$ has a unique extension to a $3$-divisible group ${\mathcal{G}}$ over ${\mathcal{O}}_{F'_{\pm 3}}$. By the uniqueness of this extension, it is also equipped with descent data $\{ [g] \}$ relative to $F_{\pm 3}'/\mathbf{Q}_3$ and with an action of ${\mathcal{O}}_{K}$, compatible with the canonical structure on the generic fibre.

Let $\widetilde{\gamma }_2 \in \operatorname {Gal}(\mathbf{Q}_3(\sqrt {\pm 3})^{\operatorname {ab}}/\mathbf{Q}_3(\sqrt {\pm 3}))$ correspond to $\sqrt {\pm 3}$. We will use the notation of Appendix B of CDT (in particular $\operatorname {WD}$ and $D'({\mathcal{G}})$), except that we will write $\mathbf{F}$ and $\mathbf{F}'$ in place of $\phi$ and $\phi '$. Then

•

$\operatorname {WD}(\rho )(\gamma _4^2)=-1$,

•

$\operatorname {WD}(\rho )(\widetilde{\gamma }_2^2)$, but not $\operatorname {WD}(\rho )(\widetilde{\gamma }_2)$, is a scalar,

•

and $\det \operatorname {WD}(\rho )(\widetilde{\gamma }_2) =3$.

Thus $\operatorname {WD}(\rho )(\widetilde{\gamma }_2^2)=-3$. Hence on $D'({\mathcal{G}})\otimes \mathbf{Q}_3$ we have

•

$[\gamma _4^2]=\operatorname {WD}(\rho )(\gamma _4^2)=-1$,

•

and $(\mathbf{F}')^2=[\widetilde{\gamma }_2^2] \operatorname {WD}(\rho )(\widetilde{\gamma }_2^{-2})=-1/3$.

We conclude that on $D({\mathcal{G}})$ we have

•

$[\gamma _4^2]=-1$,

•

$\mathbf{F}^2=-3$,

•

and so $\mathbf{F}=-\mathbf{V}$.

In particular ${\mathcal{I}}_{\pm 3}$ annihilates $D({\mathcal{G}})$ and for all $m \geq 1$ the map $(f \bmod 3^m):R_{V,\mathbf{Z}_3} \rightarrow {\mathcal{O}}_{K}/(3^m)$ factors through $R_{V,\mathbf{Z}_3}^{\epsilon ,{\operatorname {\mathcal{S}}}_{\pm 3}}$. Hence, the map $R_{V,\mathbf{Z}_3} \rightarrow \hspace{-.86em} \rightarrow R_{V,\mathbf{Z}_3}^{\tau _{\pm 3}}$ factors through $R_{V,\mathbf{Z}_3}^{\epsilon ,{\operatorname {\mathcal{S}}}_{\pm 3}}$ and Theorem 2.1.4 follows from the following result which we will prove in §8.

Theorem 4.5.1.

$\dim H^1_{{\operatorname {\mathcal{S}}}_{\pm 3}}(G_3,\operatorname {ad}^0 {\overline{\rho }}) \leq 1$.

4.6. Reduction steps for Theorem 2.1.6

We now begin the proof of Theorem 2.1.6. We may suppose that ${\mathcal{O}}=\mathbf{Z}_3$.

Let $F_i'$ denote the degree $12$ abelian extension of $\mathbf{Q}_3(\sqrt {- 3})$ with norms the subgroup of $\mathbf{Q}_3(\sqrt {- 3})^\times$ topologically generated by $- 3$, $4$, $1+9 \sqrt {- 3}$ and $1+ (1-3\tilde{\imath })\sqrt {-3}$, where $\tilde{\imath }$ is the unique lift of $i$ to $\mathbf{Z}$ with $0 \leq \tilde{\imath }< 3$. Note that $F_i'/\mathbf{Q}_3$ is Galois. We identify

$$\begin{equation*} \operatorname {Gal}(F_i'/\mathbf{Q}_3(\sqrt {-3})) \cong \langle \gamma _2 \rangle \times \langle \gamma _3 \rangle \times \langle \gamma _4^2 \rangle , \end{equation*}$$

where $\gamma _2$ corresponds to $\sqrt {-3}$ and has order $2$, $\gamma _3$ corresponds to $1-3\sqrt {- 3}$ and has order $3$, and $\gamma _4^2$ corresponds to $-1$ and has order $2$. We also let $F_i$ denote the fixed field of $\{ 1, \gamma _2 \}$, so $F_i/\mathbf{Q}_3$ is totally ramified.

We will let ${\mathcal{I}}_i$ denote the two-sided ideal of $W(\mathbf{F}_9) [\mathbf{F},\mathbf{V}][\operatorname {Gal}(F_i'/\mathbf{Q}_3)]$ generated by

•

$\mathbf{F}+\mathbf{V}$,

•

$[\gamma _4^2]+1$,

•

and $([\gamma _3]-[\gamma _3^{-1}])[\gamma _2]-\mathbf{F}$.

We remark that the ideal ${\mathcal{I}}_i$ is unchanged if we change our choice of $\sqrt {-3}$.

In §9 we will prove the following result (and explain the unusual looking notation).

Theorem 4.6.1.

There are objects $({\mathcal{G}},\{ [g]\})_{(2,6)}$, $({\mathcal{G}},\{ [g]\})_{(6,10)}$, $({\mathcal{G}},\{ [g]\})_{(2,10)}$ and $({\mathcal{G}},\{ [g]\})_{(6,6)}$ in the category ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F_i'/\mathbf{Q}_3,{\mathcal{I}}_i}$ with the following properties.

(1)

For $(r,s)\!=\!(2,6)$, $(6,10)$, $(2,10)$ and $(6,6)$ we have ${\overline{\rho }}\cong (({\mathcal{G}},\{ [g]\})_{(r,s)})_{\mathbf{Q}_3}(\overline{\mathbf{Q}}_3)$ as $G_3$-modules.

(2)

For $(r,s)=(2,6)$, $(6,10)$, $(2,10)$ and $(6,6)$ there is a short exact sequence in ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'_i/\mathbf{Q}_3,{\mathcal{I}}_i}$,$$\begin{equation*} (0) \longrightarrow ({\mathcal{G}}_1,\{ [g]\})_{(r,s)} \longrightarrow ({\mathcal{G}},\{ [g]\})_{(r,s)} \longrightarrow ({\mathcal{G}}_2,\{ [g]\})_{(r,s)} \longrightarrow (0), \end{equation*}$$

such that $({\mathcal{G}}_1,\{ [g]\})_{(r,s)}$ and $({\mathcal{G}}_2,\{ [g]\})_{(r,s)}$ have order $3$ and for all $a,b \in \{ 1,2\}$ (possibly equal) the natural map$$\begin{multline*} \operatorname {Ext}^1_{{\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/\mathbf{Q}_3,({\mathcal{I}}_i,3)}}((\mathcal{G}_a,\{ [g] \})_{(r,s)},(\mathcal{G}_b,\{ [g] \}))_{(r,s)}\\ \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_{\mathbf{Q}_3}]}((\mathcal{G}_a,\{ [g] \})_{(r,s),\mathbf{Q}_3}, (\mathcal{G}_b,\{ [g] \})_{(r,s),\mathbf{Q}_3}) \end{multline*}$$

is injective.

(3)

If $k/\mathbf{F}_3$ is a finite field extension and if $({\mathcal{G}},\{ [g]\})$ is an object of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F_i'/\mathbf{Q}_3,{\mathcal{I}}_i}$ with an action of $k$ such that $({\mathcal{G}},\{ [g]\})_{\mathbf{Q}_3}(\overline{\mathbf{Q}}_3)$ is isomorphic to ${\overline{\rho }}\otimes k$, then for some $(r,s)=(2,6)$, $(6,10)$, $(2,10)$ or $(6,6)$ the object $({\mathcal{G}},\{ [g]\})$ of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F_i'/\mathbf{Q}_3,{\mathcal{I}}_i}$ is weakly filtered by $\{ ({\mathcal{G}}_1,\{ [g]\})_{(r,s)}, ({\mathcal{G}}_2,\{ [g]\})_{(r,s)} \}$.

(4)

For $(r,s)=(2,6)$, $(6,10)$ and $(2,10)$ we have $\mathbf{F}=0$ on $D({\mathcal{G}}_{(r,s)})$, while $\mathbf{F}\neq 0$ on $D({\mathcal{G}}_{(6,6)})$.

Note that for all $a,b$ (possibly equal), we must have

$$\begin{align*} \operatorname {Hom}(({\mathcal{G}}_a,\{ [g]\})_{(r,s)},({\mathcal{G}}_b,\{ [g]\})_{(r,s)}) &= \operatorname {Hom}(({\mathcal{G}}_a,\{ [g]\})_{(r,s),\mathbf{Q}_3},({\mathcal{G}}_b,\{ [g]\})_{(r,s),\mathbf{Q}_3}) \\ &= \left\{\begin{array}{ll} 0&\text{if $a \ne b$,}\\\mathbf{F}_3 &\text{if $a = b$.} \end{array}\right. \end{align*}$$

For $(r,s)=(2,6)$, $(6,10)$, $(2,10)$ and $(6,6)$, we let ${\operatorname {\mathcal{S}}}_{i,(r,s)}$ denote the full subcategory of ${\operatorname {\mathcal{S}}}({\overline{\rho }})$ consisting of objects $X$ which are isomorphic to $({\mathcal{H}},\{ [g]\})_{\mathbf{Q}_3}$ for some object $({\mathcal{H}},\{ [g]\})$ of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F_i'/\mathbf{Q}_3, {\mathcal{I}}_i,\{ ({\mathcal{G}},\{ [g]\})_{(r,s)} \} }$. By Lemma 4.2.2, Corollary 4.2.5 and Theorem 4.6.1 we see that ${\operatorname {\mathcal{S}}}_{i,(r,s)}$ is closed under finite products, ${\operatorname {\mathcal{S}}}({\overline{\rho }})$-subobjects and ${\operatorname {\mathcal{S}}}({\overline{\rho }})$-quotients. In §9 we will also prove the following two results.

Theorem 4.6.2.

For $(r,s)=(2,6)$, $(6,10)$, $(2,10)$ and $(6,6)$ we have

$$\begin{equation*} \dim H^1_{{\operatorname {\mathcal{S}}}_{i,(r,s)}}(G_3,\operatorname {ad}^0 {\overline{\rho }}) \leq 1. \end{equation*}$$

Theorem 4.6.3.

For $(r,s)=(2,6)$, $(6,10)$ and $(2,10)$ and for any $N \geq 1$ there exists a continuous representation

$$\begin{equation*} \rho _N:G_{\mathbf{Q}_3} \longrightarrow {\operatorname {GL}}_2(\mathbf{F}_3[[T]]/(T^N)) \end{equation*}$$

such that

•

$\det \rho _N = \epsilon$,

•

for some object $({\mathcal{G}}_N,\{ [g]\})$ of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F_i'/\mathbf{Q}_3,({\mathcal{I}}_i,\mathbf{F}),\{ ({\mathcal{G}},\{ [g]\})_{(r,s)} \} }$ we have$$\begin{equation*} \rho _N \cong ({\mathcal{G}}_N,\{ [g]\})_{\mathbf{Q}_3}(\overline{\mathbf{Q}}_3) \end{equation*}$$

(where $({\mathcal{I}}_i,\mathbf{F})$ denotes the two-sided ideal of $W(\mathbf{F}_9)[\mathbf{F},\mathbf{V}][\operatorname {Gal}(F_i'/\mathbf{Q}_3)]$ generated by ${\mathcal{I}}_i$ and $\mathbf{F}$),

•

and $\rho \bmod (T^2) \not \cong {\overline{\rho }}\otimes k[[T]]/(T^2)$.

(We are not asserting that $\rho _N$ and ${\mathcal{G}}_N$ are independent of the choice of $(r,s)$, though in fact we believe that $\rho _N$ is independent of this choice.)

From these results we can easily draw the following consequence.

Corollary 4.6.4.

For $(r,s)=(2,6)$, $(6,10)$ and $(2,10)$ we have

$$\begin{equation*} R_{V,\mathbf{Z}_3}^{\epsilon ,{\operatorname {\mathcal{S}}}_{i,(r,s)}} \cong \mathbf{F}_3[[T]]. \end{equation*}$$

Proof.

By Theorems 4.6.2 and 4.6.3 we see that $R_{V,\mathbf{Z}_3}^{\epsilon ,{\operatorname {\mathcal{S}}}_{i,(r,s)}}/(3) \cong \mathbf{F}_3[[T]]$ and that if $R$ is an Artinian quotient of $R_{V,\mathbf{Z}_3}^{\epsilon ,{\operatorname {\mathcal{S}}}_{i,(r,s)}}/(3)$ corresponding to a (necessarily unique; see Lemma 4.2.3) object $({\mathcal{G}},\{ [g]\})$ of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F_i'/\mathbf{Q}_3,({\mathcal{I}}_i,3), \{ ({\mathcal{G}},\{ [g]\})_{(r,s)} \} }$, then $\mathbf{F}=0$ on $D({\mathcal{G}})$.

Now suppose $R$ is any Artinian quotient of $R_{V,\mathbf{Z}_3}^{\epsilon ,{\operatorname {\mathcal{S}}}_{i,(r,s)}}$ which corresponds to an object $({\mathcal{G}}, \{ [g]\})$ of the category ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F_i'/\mathbf{Q}_3,{\mathcal{I}}_i,\{ ({\mathcal{G}},\{ [g]\})_{(r,s)} \} }$. Let $G=({\mathcal{G}},\{ [g]\})_{\mathbf{Q}_3}$ and consider the exact sequences

$$\begin{equation*} (0) \longrightarrow G[3] \longrightarrow G \longrightarrow 3G \longrightarrow (0) \end{equation*}$$

and

$$\begin{equation*} (0) \longrightarrow 3G \longrightarrow G \longrightarrow G/3G \longrightarrow (0). \end{equation*}$$

By Lemma 4.2.3, we have exact sequences

$$\begin{equation*} ({\mathcal{G}},\{ [g]\}) \longrightarrow ({\operatorname {\mathcal{K}}},\{ [g]\}) \longrightarrow (0) \end{equation*}$$

and

$$\begin{equation*} (0) \longrightarrow ({\operatorname {\mathcal{K}}},\{ [g]\}) \longrightarrow ({\mathcal{G}},\{ [g]\}) \longrightarrow ({\mathcal{H}},\{ [g]\}) \longrightarrow (0) \end{equation*}$$

in ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F_i'/\mathbf{Q}_3,{\mathcal{I}}_i,\{ ({\mathcal{G}},\{ [g]\})_{(r,s)} \} }$ such that the composite

$$\begin{equation*} ({\mathcal{G}},\{ [g]\}) \rightarrow \hspace{-.86em} \rightarrow ({\operatorname {\mathcal{K}}},\{ [g]\}) \hookrightarrow ({\mathcal{G}},\{ [g]\}) \end{equation*}$$

is multiplication by $3$. In particular we have exact sequences

$$\begin{equation*} (0) \longrightarrow D({\operatorname {\mathcal{K}}}) \longrightarrow D({\mathcal{G}}) \end{equation*}$$

and

$$\begin{equation*} (0) \longrightarrow D({\mathcal{H}}) \longrightarrow D({\mathcal{G}}) \longrightarrow D({\operatorname {\mathcal{K}}}) \longrightarrow (0), \end{equation*}$$

such that the composite

$$\begin{equation*} D({\mathcal{G}}) \rightarrow \hspace{-.86em} \rightarrow D({\operatorname {\mathcal{K}}}) \hookrightarrow D({\mathcal{G}}) \end{equation*}$$

is multiplication by $3$. As $\mathbf{F}=-\mathbf{V}=0$ on $D({\mathcal{H}})$ we see that $\mathbf{F}$ and $\mathbf{V}$ factor through maps $D({\operatorname {\mathcal{K}}}) \rightarrow D({\mathcal{G}})$, i.e. we can write $\mathbf{F}=3\mathbf{F}'$ and $\mathbf{V}=3\mathbf{V}'$ for some endomorphisms $\mathbf{F}'$ and $\mathbf{V}'$ of $D({\mathcal{G}})$. Thus $3=9\mathbf{F}'\mathbf{V}'$ equals zero on $D({\mathcal{G}})/9D({\mathcal{G}})$ and so $D({\operatorname {\mathcal{K}}})=0$. We conclude that ${\operatorname {\mathcal{K}}}=(0)$, so that $3G=(0)$ and $3R=(0)$.

Thus

$$\begin{equation*} R_{V,\mathbf{Z}_3}^{\epsilon ,{\operatorname {\mathcal{S}}}_{i,(r,s)}} = R_{V,\mathbf{Z}_3}^{\epsilon ,{\operatorname {\mathcal{S}}}_{i,(r,s)}}/ (3) = \mathbf{F}_3[[T]]. \end{equation*}$$ ■

We now modify the argument in §4.4. Choose a finite extension $K/\mathbf{Q}_3$ and continuous map of rings $f:R_{V,\mathbf{Z}_3} \rightarrow \overline{\mathbf{Q}}_3$ such that the corresponding representation $\rho :G_3 \rightarrow {\operatorname {GL}}_2({\mathcal{O}}_{K})$ is of extended type $\tau _i'$. Let $G$ be the corresponding $3$-divisible group over $\mathbf{Q}_3$. By Tate’s theorem (Theorem 4 of T) $G$ has a unique extension to a $3$-divisible group ${\mathcal{G}}$ over ${\mathcal{O}}_{F'_i}$. By the uniqueness of this extension, ${\mathcal{G}}$ comes equipped with descent data $\{ [g] \}$ relative to $F_i'/\mathbf{Q}_3$ and with an action of ${\mathcal{O}}_{K}$, compatible with the canonical structure on the generic fibre.

Let $\widetilde{\gamma }_2 \in \operatorname {Gal}(\mathbf{Q}_3(\sqrt {-3})^{\operatorname {ab}}/\mathbf{Q}_3(\sqrt {-3}))$ correspond to $\sqrt {-3}$. We will use the notation of Appendix B of CDT (in particular $\operatorname {WD}$ and $D'({\mathcal{G}})$), except that we will write $\mathbf{F}$ and $\mathbf{F}'$ in place of $\phi$ and $\phi '$. Then

•

$\operatorname {WD}(\rho )(\gamma _4^2)=-1$,

•

$\operatorname {WD}(\rho )(\widetilde{\gamma }_2^2)=-3$,

•

and $\operatorname {WD}(\rho )(\widetilde{\gamma }_2)(\operatorname {WD}(\rho )(\gamma _3) - \operatorname {WD}(\rho )(\gamma _3)^{-1}) =3$.

Thus on $D'({\mathcal{G}}) \otimes \mathbf{Q}_3$ we have

•

$[\gamma _4^2]=\operatorname {WD}(\rho )(\gamma _4^2)=-1$,

•

$(\mathbf{F}')^2=[\widetilde{\gamma }_2^2] \operatorname {WD}(\rho )(\widetilde{\gamma }_2^{-2})=-1/3$,

•

and $[\gamma _2]([\gamma _3]-[\gamma _3^{-1}])=3\mathbf{F}'$.

We conclude that on $D({\mathcal{G}})$ we have

•

$[\gamma _4^2]=-1$,

•

$\mathbf{F}^2=-3$,

•

and $[\gamma _2]([\gamma _3^{-1}]-[\gamma _3])=3\mathbf{F}^{-1}$.

Hence also

•

$\mathbf{F}=-\mathbf{V}$,

•

and $[\gamma _2]([\gamma _3]-[\gamma _3^{-1}])=\mathbf{F}$.

In particular ${\mathcal{I}}_i$ annihilates $D({\mathcal{G}})$.

Thus $({\mathcal{G}}[\wp _K],\{ [g]\})$ is an object of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F_i'/\mathbf{Q}_3,{\mathcal{I}}_i}$ such that $({\mathcal{G}}[\wp _K],\{ [g]\})_{\mathbf{Q}_3}$ corresponds to ${\overline{\rho }}\otimes {\mathcal{O}}_K /\wp _K$. By Theorem 4.6.1 we see that $({\mathcal{G}}[\wp _K],\{ [g]\})$ is weakly filtered by $\{ ({\mathcal{G}}_1,\{ [g]\})_{(r,s)}, ({\mathcal{G}}_2,\{ [g]\})_{(r,s)} \}$ for some $(r,s)=(2,6)$, $(6,10)$, $(2,10)$ or $(6,6)$. We will prove $(r,s)=(6,6)$. By Theorem 4.6.1 and Lemma 4.2.4, $({\mathcal{G}}[\wp _K],\{ [g]\})$ is strongly filtered by $\{ ({\mathcal{G}}_1,\{ [g]\})_{(r,s)}, ({\mathcal{G}}_2,\{ [g]\})_{(r,s)} \}$. As $({\mathcal{G}}[\wp _K],\{ [g]\})_{\mathbf{Q}_3}$ is filtered by ${\overline{\rho }}$, using Theorem 4.6.1, we see that $({\mathcal{G}}[\wp _K],\{ [g]\})$ is weakly filtered by $({\mathcal{G}},\{ [g]\})_{(r,s)}$. For all $m \geq 1$ we have

$$\begin{equation*} ({\mathcal{G}}[\wp _K^m]/{\mathcal{G}}[\wp _K^{m-1}],\{ [g]\}) \stackrel{\sim }{\longrightarrow }({\mathcal{G}}[\wp _K],\{ [g]\}), \end{equation*}$$

so for all $m \geq 1$ the object $({\mathcal{G}}[\wp _K^m],\{ [g]\})$ is also weakly and hence strongly filtered by $({\mathcal{G}},\{ [g]\})_{(r,s)}$ for the same $(r,s)$. Thus, for all $m \geq 1$, the map $(f \bmod p^m):R_{V,\mathbf{Z}_3} \rightarrow {\mathcal{O}}_{K'}/(3^m)$ factors through $R_{V,\mathbf{Z}_3}^{\epsilon ,{\operatorname {\mathcal{S}}}_{i,(r,s)}}$. By Corollary 4.6.4 we see that $(r,s)=(6,6)$, so the map $R_{V,\mathbf{Z}_3} \rightarrow \hspace{-.86em} \rightarrow R_{V,\mathbf{Z}_3}^{\tau _{i}'}$ factors through $R_{V,\mathbf{Z}_3}^{\epsilon ,{\operatorname {\mathcal{S}}}_{i,(6,6)}}$ and Theorem 2.1.6 follows from Theorem 4.6.2.

4.7. Some Galois cohomology

In this section we will begin the proofs of Theorems 4.4.1, 4.5.1 and 4.6.2. We will let ${\operatorname {\mathcal{S}}}$ denote one of the categories ${\operatorname {\mathcal{S}}}_{\pm 1}$, ${\operatorname {\mathcal{S}}}_{\pm 3}$ or ${\operatorname {\mathcal{S}}}_{i, (r,s)}$. We will let $\chi =\omega$ in the cases ${\operatorname {\mathcal{S}}}_{\pm 1}$ and $\chi =1$ otherwise. In all cases

$$\begin{equation*} {\overline{\rho }}\sim \left( \begin{array}{cc} {\chi \omega } & {*} \\{0} & {\chi } \end{array} \right) \end{equation*}$$

is très ramifié.

The maps $\omega \otimes \chi \hookrightarrow {\overline{\rho }}$ and ${\overline{\rho }}\rightarrow \hspace{-.86em} \rightarrow \chi$ induce a commutative diagram with exact rows and columns:

$$\begin{equation*} \begin{array}{ccccccc} &&&&&& (0) \\&&&&&& \downarrow \\&&&&&& \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(\omega \otimes \chi , \omega \otimes \chi ) \\&&&&&& \downarrow \\&&&&\operatorname {Ext}^1_{\mathbf{F}_3[G_3]}({\overline{\rho }},{\overline{\rho }})&\longrightarrow & \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(\omega \otimes \chi , {\overline{\rho }}) \\&&&& \downarrow && \downarrow \\(0) &\longrightarrow & \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(\chi ,\chi )& \longrightarrow & \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}({\overline{\rho }}, \chi )& \longrightarrow & \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(\omega \otimes \chi , \chi )\end{array} \end{equation*}$$

We will let $\theta _0$ denote the composite map

$$\begin{equation*} \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}({\overline{\rho }},{\overline{\rho }}) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(\omega \otimes \chi ,\chi ), \end{equation*}$$

and $\theta _1$ (resp. $\theta _\omega$) the induced mapping

$$\begin{equation*} \ker \theta _0 \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(\chi ,\chi ) \end{equation*}$$

(resp.

$$\begin{equation*} \ker \theta _0 \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(\omega \otimes \chi , \omega \otimes \chi )). \end{equation*}$$

We will also let ${\overline{\theta }}_1$ (resp. ${\overline{\theta }}_\omega$) denote the induced mapping

$$\begin{equation*} \ker \theta _0 \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(\chi ,\chi ) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[I_3]}(\chi ,\chi ) \end{equation*}$$

(resp.

$$\begin{equation*} \ker \theta _0 \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(\omega \otimes \chi , \omega \otimes \chi )\longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[I_3]}(\omega \otimes \chi , \omega \otimes \chi )). \end{equation*}$$

If we reinterpret our $\operatorname {Ext}$-groups as cohomology groups and use the isomorphism ${\overline{\rho }}^\vee \sim {\overline{\rho }}\otimes \omega$, our diagram becomes:

$$\begin{equation*} \begin{array}{ccccccc} &&&&&& (0) \\&&&&&& \downarrow \\&&&&&& H^1(G_3,\mathbf{F}_3) \\&&&&&& \downarrow \\&&&& H^1(G_3,\operatorname {ad}{\overline{\rho }}) &\longrightarrow & H^1(G_3,{\overline{\rho }}\otimes \omega \otimes \chi ) \\&&&& \downarrow && \downarrow \\(0) &\longrightarrow & H^1(G_3,\mathbf{F}_3) & \longrightarrow & H^1(G_3, {\overline{\rho }}\otimes \omega \otimes \chi )& \longrightarrow & H^1(G_3,\omega )\end{array} \end{equation*}$$

Fix a basis of $\mathbf{F}_3^2$ so that ${\overline{\rho }}$ takes the form

$$\begin{equation*} \left( \begin{array}{cc} {\omega \otimes \chi } & {*} \\{0} & {\chi } \end{array} \right). \end{equation*}$$

Then any extension of ${\overline{\rho }}$ by ${\overline{\rho }}$ in characteristic $3$ may be represented by a matrix

$$\begin{equation*} \left( \begin{array}{cc} {{\overline{\rho }}} & {\phi {\overline{\rho }}} \\{0} & {{\overline{\rho }}} \end{array} \right), \end{equation*}$$

where the cocycle

$$\begin{equation*} \phi =\left( \begin{array}{cc} {\phi _{11}} & {\phi _{12}} \\{\phi _{21}} & {\phi _{22}} \end{array} \right) \in Z^1(G_3,\operatorname {ad}{\overline{\rho }}) \end{equation*}$$

represents the class of this extension in $\operatorname {Ext}^1_{\mathbf{F}_3[G_3]}({\overline{\rho }},{\overline{\rho }}) \cong H^1(G_3,\operatorname {ad}{\overline{\rho }})$. Moreover

•

$\theta _0([\phi ])=[\phi _{21}] \in H^1(G_3,\omega )$,

•

if $\phi _{21}\!=\!0$, then $\theta _1([\phi ])\!=\![\phi _{22}]\! \in \! H^1(G_3,\mathbf{F}_3)$ and $\theta _\omega ([\phi ])\!=\![\phi _{11}] \!\in \!H^1(G_3,\mathbf{F}_3)$,

•

and $[\phi ] \in H^1(G_3,\operatorname {ad}^0 {\overline{\rho }})$ if and only if $0=[\phi _{11}+ \phi _{22}] \in H^1(G_3,\mathbf{F}_3)$.

In particular we have $\theta _1=-\theta _\omega$ on $H^1(G_3,\operatorname {ad}^0 {\overline{\rho }}) \cap \ker \theta _0$.

We have an exact sequence

$$\begin{equation*} (0) \longrightarrow {\overline{\rho }}\otimes \chi \longrightarrow \operatorname {ad}^0 {\overline{\rho }}\longrightarrow \omega \longrightarrow (0) , \end{equation*}$$

where the first map sends

$$\begin{equation*} \left( \begin{array}{c} x \\y \end{array} \right) \longmapsto \left( \begin{array}{cc} {-y/2} & {x} \\{0} & {y/2} \end{array} \right) \end{equation*}$$

and the second map sends

$$\begin{equation*} \left( \begin{array}{cc} {a} & {b} \\{c} & {-a} \end{array} \right) \longmapsto c. \end{equation*}$$

Thus we get an exact sequence

$$\begin{equation*} (0) \longrightarrow H^1(G_3,{\overline{\rho }}\otimes \chi ) \longrightarrow H^1(G_3, \operatorname {ad}^0 {\overline{\rho }}) \stackrel{\theta _0}{\longrightarrow } H^1(G_3, \omega ) \end{equation*}$$

and so we may identify $H^1(G_3, \operatorname {ad}^0{\overline{\rho }}) \cap \ker \theta _0$ with $H^1(G_3,{\overline{\rho }}\otimes \chi )$. We also have an exact sequence

$$\begin{equation} (0) \longrightarrow \omega \longrightarrow {\overline{\rho }}\otimes \chi \longrightarrow 1 \longrightarrow (0), {\tag{4.7.1}} \end{equation}$$

which gives rise to an exact sequence

$$\begin{equation*} (0) \longrightarrow \mathbf{F}_3 \longrightarrow H^1(G_3,\omega ) \longrightarrow H^1(G_3,{\overline{\rho }}\otimes \chi ) \longrightarrow H^1(G_3,\mathbf{F}_3) \longrightarrow H^2(G_3,\omega ). \end{equation*}$$

If we identify $H^1(G_3,{\overline{\rho }}\otimes \chi )$ with $H^1(G_3,\operatorname {ad}^0{\overline{\rho }}) \cap \ker \theta _0$, then the latter map $H^1(G_3,{\overline{\rho }}\otimes \chi ) \rightarrow H^1(G_3,\mathbf{F}_3)$ is identified with $\theta _\omega =-\theta _1$.

Lemma 4.7.1.

The sequence

$$\begin{equation*} (0) \longrightarrow \mathbf{F}_3 \longrightarrow H^1(G_3,\omega ) \longrightarrow H^1(G_3,{\overline{\rho }}\otimes \chi ) \longrightarrow H^1(I_3,\mathbf{F}_3) \end{equation*}$$

is exact.

Proof.

The key point is that ${\overline{\rho }}$ is très ramifié (compare with Proposition 6.1 of Di1). It suffices to show that the composite

$$\begin{equation*} H^1(G_{\mathbf{F}_3},\mathbf{F}_3) \longrightarrow H^1(G_3,\mathbf{F}_3) \longrightarrow H^2(G_3,\omega ) \end{equation*}$$

is injective. Suppose that $x \in H^1(G_3,\mathbf{F}_3)$ maps to zero in $H^2(G_3,\omega )$; then by Tate duality $x$ is annihilated by the image of the map $H^0(G_3,\mathbf{F}_3) \rightarrow H^1(G_3,\omega )$ coming from the short exact sequence

$$\begin{equation*} (0) \longrightarrow \omega \longrightarrow ({\overline{\rho }}\otimes \chi )^\vee \otimes \omega \longrightarrow 1 \longrightarrow (0) \end{equation*}$$

Cartier dual to (4.7.1). As $({\overline{\rho }}\otimes \chi )^\vee \otimes \omega$ is très ramifié we see that the image of

$$\begin{equation*} H^0(G_3,\mathbf{F}_3) \longrightarrow H^1(G_3,\omega ) \cong \mathbf{Q}_3^\times /(\mathbf{Q}_3^\times )^3 \end{equation*}$$

is not contained in $\mathbf{Z}_3^\times /(\mathbf{Z}_3^\times )^3$. Thus

$$\begin{equation*} x \in \operatorname {Hom}(\mathbf{Q}_3^\times /\mathbf{Z}_3^\times ,\mathbf{F}_3) \cong H^1(G_{\mathbf{F}_3},\mathbf{F}_3) \subset H^1(G_3,\mathbf{F}_3) \cong \operatorname {Hom}(\mathbf{Q}_3^\times ,\mathbf{F}_3) \end{equation*}$$

must be zero (see Proposition 3 of §1 of Chapter XIV of Se1).

■

Corollary 4.7.2.

The maps

$$\begin{equation*} {\overline{\theta }}_1: H^1(G_3,\operatorname {ad}^0{\overline{\rho }}) \cap \ker \theta _0 \longrightarrow H^1(I_3,\mathbf{F}_3) \end{equation*}$$

and

$$\begin{equation*} {\overline{\theta }}_\omega : H^1(G_3,\operatorname {ad}^0{\overline{\rho }}) \cap \ker \theta _0 \longrightarrow H^1(I_3,\mathbf{F}_3) \end{equation*}$$

have the same kernel and this has dimension $1$ over $\mathbf{F}_3$.

Theorems 4.4.1, 4.5.1 and 4.6.2 now follow from the following results, which we will prove later. One advantage of these new formulations is that, with one exception, they refer only to $\operatorname {Ext}^1_{{\operatorname {\mathcal{S}}}}({\overline{\rho }},{\overline{\rho }})$ and make no mention of the determinant or $\operatorname {ad}^0 {\overline{\rho }}$, concepts which we found tricky to translate into the language of integral models.

Theorem 4.7.3.

(1)

$\theta _0: \operatorname {Ext}^1_{{\operatorname {\mathcal{S}}}_{\pm 1}}({\overline{\rho }},{\overline{\rho }}) \rightarrow H^1(G_3,\omega )$ is the zero map.

(2)

${\overline{\theta }}_\omega : \operatorname {Ext}^1_{{\operatorname {\mathcal{S}}}_{- 1}}({\overline{\rho }},{\overline{\rho }}) \rightarrow H^1(I_3,\mathbf{F}_3)$ is the zero map.

(3)

${\overline{\theta }}_\omega : H^1_{{\operatorname {\mathcal{S}}}_{1}}(G_3,\operatorname {ad}^0{\overline{\rho }}) \rightarrow H^1(I_3,\mathbf{F}_3)$ is the zero map.

Theorem 4.7.4.

(1)

$\theta _0: \operatorname {Ext}^1_{{\operatorname {\mathcal{S}}}_{\pm 3}}({\overline{\rho }},{\overline{\rho }}) \rightarrow H^1(G_3,\omega )$ is the zero map.

(2)

${\overline{\theta }}_\omega : \operatorname {Ext}^1_{{\operatorname {\mathcal{S}}}_{\pm 3}}({\overline{\rho }},{\overline{\rho }}) \rightarrow H^1(I_3,\mathbf{F}_3)$ is the zero map.

Theorem 4.7.5.

Suppose that $i \in \mathbf{Z}/3\mathbf{Z}$ and $(r,s)= (2,6)$, $(6,10)$, $(2,10)$ or $(6,6)$.

(1)

$\theta _0: \operatorname {Ext}^1_{{\operatorname {\mathcal{S}}}_{i, (r,s)}}({\overline{\rho }},{\overline{\rho }}) \rightarrow H^1(G_3,\omega )$ is the zero map.

(2)

Either ${\overline{\theta }}_\omega : \operatorname {Ext}^1_{{\operatorname {\mathcal{S}}}_{i,(r,s)}}({\overline{\rho }},{\overline{\rho }}) \rightarrow H^1(I_3,\mathbf{F}_3)$ or ${\overline{\theta }}_1: \operatorname {Ext}^1_{{\operatorname {\mathcal{S}}}_{i,(r,s)}}({\overline{\rho }},{\overline{\rho }}) \rightarrow H^1(I_3,\mathbf{F}_3)$ is the zero map.

The deduction of Theorems 4.4.1, 4.5.1 and 4.6.2 from these results is immediate.

5. Breuil modules

In this section we recall some results from Br2 (see also the summary Br1) and give some slight extensions of them. Three of the authors apologise to the fourth for the title of this section, but they find that the term “Breuil module” is much more convenient than “filtered $\phi _1$-module”.

Throughout this section, $\ell$ will be an odd rational prime and $R$ will be a complete discrete valuation ring with fraction field $F'$ of characteristic zero and perfect residue field $k$ of characteristic $\ell$.

5.1. Basic properties of Breuil modules

We will fix a choice of uniformiser $\pi$ of $R$ and let

$$\begin{equation*} E_{\pi }(u) = u^e - \ell G_{\pi }(u) \end{equation*}$$

be the Eisenstein polynomial which is the minimal polynomial of $\pi$ over the fraction field of $W(k)$, so $G_{\pi }(u) \in W(k)[u]$ is a polynomial with unit constant term $G_{\pi }(0) \in W(k)^{\times }$ (and degree at most $e - 1$). The $\ell ^{th}$ power map on $k[u]/u^{e\ell }$ is denoted $\phi$, and we define

$$\begin{equation} c_{\pi } = -\phi (G_{\pi }(u)) \in (k[u]/u^{e\ell })^{\times }. {\tag{5.1.1}} \end{equation}$$

It is very important to keep in mind that these definitions, as well as many of the definitions below, depend on the choice of the uniformiser $\pi$.

The category of $\ell$-torsion Breuil modules (or “$\ell$-torsion Breuil modules over $R$”, or simply “Breuil modules” or “Breuil modules over $R$”) is defined to be the category of triples $(\operatorname {\mathcal{M}},\operatorname {\mathcal{M}}_1,\phi _1)$, where

•

$\operatorname {\mathcal{M}}$ is a finite free $k[u]/u^{e\ell }$-module,

•

$\operatorname {\mathcal{M}}_1$ is a $k[u]/u^{e\ell }$-submodule of $\operatorname {\mathcal{M}}$ containing $u^e \operatorname {\mathcal{M}}$,

•

$\phi _1:\operatorname {\mathcal{M}}_1 \rightarrow \operatorname {\mathcal{M}}$ is $\phi$-semi-linear and has image whose $k[u]/u^{e\ell }$-span is all of $\operatorname {\mathcal{M}}$.

(A morphism $(\operatorname {\mathcal{M}},\operatorname {\mathcal{M}}_1,\phi _1) \rightarrow (\operatorname {\mathcal{N}},\operatorname {\mathcal{N}}_1,\psi _1)$ is a morphism $f:\operatorname {\mathcal{M}}\rightarrow \operatorname {\mathcal{N}}$ of $k[u]/u^{e\ell }$-modules such that $f\operatorname {\mathcal{M}}_1 \subset \operatorname {\mathcal{N}}_1$ and $\psi _1 \circ f = f \circ \phi _1$ on $\operatorname {\mathcal{M}}_1$.) We define the rank of $(\operatorname {\mathcal{M}},\operatorname {\mathcal{M}}_1,\phi _1)$ to be the rank of $\operatorname {\mathcal{M}}$ over $k[u]/u^{e\ell }$. Breuil modules form an additive category (not abelian in general) in the obvious manner and this category does not depend on the choice of $\pi$. It is denoted ${\underline{\phi _1{\mathrm{{-mod}}}}}_{R}$ or ${\underline{\phi _1{\mathrm{{-mod}}}}}_{F'}$. The induced $\phi$-semi-linear map of $k$-vector spaces

$$\begin{equation*} \overline{\phi }_1:\operatorname {\mathcal{M}}_1/u\operatorname {\mathcal{M}}_1 \longrightarrow \operatorname {\mathcal{M}}/u\operatorname {\mathcal{M}} \end{equation*}$$

is bijective (because it is onto and $\# \operatorname {\mathcal{M}}_1/u\operatorname {\mathcal{M}}_1 = \# \operatorname {\mathcal{M}}_1[u] \leq \# \operatorname {\mathcal{M}}[u] = \# \operatorname {\mathcal{M}}/u\operatorname {\mathcal{M}}$). In particular, a map of Breuil modules

$$\begin{equation*} (\operatorname {\mathcal{M}},\operatorname {\mathcal{M}}_1,\phi _1) \longrightarrow (\operatorname {\mathcal{M}}',\operatorname {\mathcal{M}}'_1,\phi '_1) \end{equation*}$$

is an isomorphism if and only if the map $\operatorname {\mathcal{M}}\rightarrow \operatorname {\mathcal{M}}'$ on underlying $k[u]/u^{e\ell }$-modules is an isomorphism.

Lemma 5.1.1.

Suppose that

$$\begin{equation*} 0 \longrightarrow \operatorname {\mathcal{M}}' \longrightarrow \operatorname {\mathcal{M}}\longrightarrow \operatorname {\mathcal{M}}'' \longrightarrow 0 \end{equation*}$$

is a complex of Breuil modules. The following are equivalent.

(1)

The underlying sequence of $k[u]/u^{e\ell }$-modules is exact.

(2)

The underlying sequence of $k[u]/u^{e\ell }$-modules is exact as is the sequence$$\begin{equation*} 0 \longrightarrow \operatorname {\mathcal{M}}_1' \longrightarrow \operatorname {\mathcal{M}}_1 \longrightarrow \operatorname {\mathcal{M}}_1'' \longrightarrow 0. \end{equation*}$$

(3)

The complex of vector spaces$$\begin{equation*} 0 \longrightarrow \operatorname {\mathcal{M}}'/u \longrightarrow \operatorname {\mathcal{M}}/u \longrightarrow \operatorname {\mathcal{M}}''/u \longrightarrow 0 \end{equation*}$$

is exact.

Proof.

The second statement clearly implies the first. The first implies the third as Breuil modules are free over $k[u]/u^{e\ell }$. It remains to show that the third condition implies the second. Using Nakayama’s lemma and the freeness of Breuil modules we see that

$$\begin{equation*} 0 \longrightarrow \operatorname {\mathcal{M}}' \longrightarrow \operatorname {\mathcal{M}}\longrightarrow \operatorname {\mathcal{M}}'' \longrightarrow 0 \end{equation*}$$

is an exact sequence of $k[u]/u^{e\ell }$-modules. Using the bijectivity of $\overline{\phi }_1$, we see that the natural map

$$\begin{equation*} f_1:\operatorname {\mathcal{M}}_1 \longrightarrow \operatorname {\mathcal{M}}''_1 \end{equation*}$$

is surjective modulo $u$ and therefore is surjective. It remains to check that the inclusion of $k[u]/u^{e\ell }$-modules $\operatorname {\mathcal{M}}'_1 \subseteq \ker (f_1)$ is an equality. Since $f_1$ is compatible with $f:\operatorname {\mathcal{M}}\rightarrow \operatorname {\mathcal{M}}''$ via the inclusions $\operatorname {\mathcal{M}}_1 \subseteq \operatorname {\mathcal{M}}$, $\operatorname {\mathcal{M}}''_1 \subseteq \operatorname {\mathcal{M}}''$ and also via the maps $\phi _1$ and $\phi ''_1$, it is obvious that $\ker (f_1) \subseteq \ker (f) = \operatorname {\mathcal{M}}'$ and that $\phi _1(\ker (f_1)) \subseteq \operatorname {\mathcal{M}}'$. Since $\ker (f_1)$ contains $\operatorname {\mathcal{M}}'_1$, which in turn contains $u^e\operatorname {\mathcal{M}}'$, we see that $(\operatorname {\mathcal{M}}',\ker (f_1),\phi _1)$ is a Breuil module! Then $(\operatorname {\mathcal{M}}',\operatorname {\mathcal{M}}'_1,\phi '_1) \rightarrow (\operatorname {\mathcal{M}}',\ker (f_1),\phi _1)$ defined via the identity map on $\operatorname {\mathcal{M}}'$ is a map of Breuil modules which is an isomorphism on underlying $k[u]/u^{e\ell }$-modules, so it must be an isomorphism of Breuil modules. This forces $\ker (f_1) = \operatorname {\mathcal{M}}'_1$.

■

When the equivalent conditions of this lemma are met we call the sequence of Breuil modules

$$\begin{equation*} 0 \longrightarrow \operatorname {\mathcal{M}}' \longrightarrow \operatorname {\mathcal{M}}\longrightarrow \operatorname {\mathcal{M}}'' \longrightarrow 0 \end{equation*}$$

exact.

For any Breuil module $(\operatorname {\mathcal{M}},\operatorname {\mathcal{M}}_1,\phi _1)$, we define the Frobenius endomorphism $\phi :\operatorname {\mathcal{M}}\rightarrow \operatorname {\mathcal{M}}$ by

$$\begin{equation} \phi (m) = \frac{1}{c_{\pi }}\phi _1(u^e m), {\tag{5.1.2}} \end{equation}$$

where $c_\pi$ is defined as in (5.1.1). Note that this depends on our choice of uniformiser.

We let $N:W(k)[[u]] \rightarrow W(k)[[u]]$ denote the unique continuous $W(k)$-linear derivation satisfying $Nu=u$, i.e. $N=u\frac{d}{du}$. This operator “extends” to any Breuil module. More precisely, we have the following lemma.

Lemma 5.1.2.

Let $\mathcal{M}$ be an object of ${\underline{\phi _1{\mathrm{{-mod}}}}}_{R}$. There is a unique additive operator $N:{\mathcal{M}}\rightarrow {\mathcal{M}}$ (the monodromy operator) satisfying the following three conditions:

(1)

$N(sx)=N(s)x+sN(x)$, $s\in k[u]/u^{e\ell }$, $x\in {\mathcal{M}}$,

(2)

$N \circ \phi _1=\phi \circ N$ on ${\operatorname {\mathcal{M}}}_1$,

(3)

$N({\mathcal{M}})\subset u{\mathcal{M}}$.

Moreover, any morphism of Breuil modules ${\operatorname {\mathcal{M}}}\rightarrow {\operatorname {\mathcal{M}}}'$ automatically commutes with $N$.

Proof.

Let’s start with unicity. Recall we have an isomorphism $k[u]/u^{e\ell }\otimes _{k[u^\ell ]/u^{e\ell }}\phi _1({\mathcal{M}}_1)\buildrel \sim \over \rightarrow {\mathcal{M}}$ (Br2, 2.1.2.1). Suppose there are two operators $N$ and $N'$ satisfying (1), (2) and (3) above, so $\Delta =N-N'$ is $k[u]/u^{e\ell }$-linear and satisfies $\Delta \phi _1=\phi \Delta$ and $\Delta ({\mathcal{M}})\subset u{\mathcal{M}}$. Thus,

$$\begin{equation*} \Delta \phi _1({\mathcal{M}}_1)=\phi \Delta ({\mathcal{M}}_1)\subset \phi (u{\mathcal{M}})\subset u^\ell {\mathcal{M}}, \end{equation*}$$

so $\Delta ({\mathcal{M}})=\Delta (k[u]/u^{e\ell }\otimes _{k[u^\ell ]/u^{e\ell }}\phi _1({\mathcal{M}}_1))\subset u^\ell {\mathcal{M}}$. Iterating $\Delta \phi _1({\mathcal{M}}_1)\subset \phi \Delta ({\mathcal{M}})\subset u^{\ell ^2}{\mathcal{M}}$ so $\Delta ({\mathcal{M}})\subset u^{\ell ^2}{\mathcal{M}}$, and so on. As $u^{e\ell }=0$, we get $\Delta =0$. For the existence, let $N_0=N\otimes {\mathbf{1}}$ on

$$\begin{equation*} k[u]/u^{e\ell }\otimes _{k[u^\ell ]/u^{e\ell }}\phi _1({\mathcal{M}}_1)\simeq {\mathcal{M}}, \end{equation*}$$

and note $N_0$ satisfies $N_0(sx)=N(s)x+sN_0(x)$. Call a derivation of ${\operatorname {\mathcal{M}}}$ any additive operator satisfying this relation and define successive derivations of ${\operatorname {\mathcal{M}}}$ by the formula

$$\begin{equation*} N_{j+1}(s\otimes \phi _1(x))=N(s)\otimes \phi _1(x)+s\phi (N_j(x)), \end{equation*}$$

for $j \geq 0$. Note that $N_{j+1}$ is well defined by the following observations.

•

$N(u^\ell s)=u^\ell N(s)$ and $N_j(ux)=ux+uN_j(x)$ imply that $N_{j+1}(u^\ell s \otimes \phi _1(x))=N_{j+1}(s \otimes \phi _1(ux))$.

•

If $\phi _1(x)=0$, then $x \in u^e{\operatorname {\mathcal{M}}}$ (see (1) of Lemma 2.1.2.1 of Br1) and so $N_j(x) \in u^e{\operatorname {\mathcal{M}}}$ and $\phi (N_j(x))=0$.

As $N_0({\mathcal{M}})\subset u{\mathcal{M}}$, we have $(N_{j+1}-N_j)({\mathcal{M}})\subset u^{\ell ^{j+1}}{\mathcal{M}}$, so $N_j=N_{j+1}$ for $j \gg 0$. This $N_j$ satisfies (1), (2) and (3).

■

The reason for introducing Breuil modules (and putting the factor $c_\pi ^{-1}$ in the definition of $\phi$) is the following theorem.

Theorem 5.1.3.

(1)

Given the choice of uniformiser $\pi$ for $R$ there is a contravariant functor ${\operatorname {\mathcal{M}}}_\pi$ from finite flat $R$-group schemes which are killed by $\ell$ to ${\underline{\phi _1{\mathrm{{-mod}}}}}_{R}$ and a quasi-inverse functor ${\mathcal{G}}_\pi$.

(2)

If ${\mathcal{G}}$ is a finite flat $R$-group scheme killed by $\ell$, then ${\mathcal{G}}$ has rank $\ell ^r$ if and only if ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}})$ has rank $r$.

(3)

If ${\mathcal{G}}$ is a finite flat $R$-group scheme killed by $\ell$, then there is a canonical $k$-linear isomorphism$$\begin{equation*} D({\mathcal{G}}) \otimes _{k,\operatorname {Frob}_\ell } k \cong {\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}})/u{\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}). \end{equation*}$$

Under this identification, $\mathbf{F}\otimes \operatorname {Frob}_\ell$ corresponds to $\phi$ and $\mathbf{V}\otimes \operatorname {Frob}_\ell ^{-1}$ corresponds to the composite$$\begin{equation*} \mathbf{V}_{\operatorname {\mathcal{M}}}:\operatorname {\mathcal{M}}/u\operatorname {\mathcal{M}}\stackrel{\overline{\phi }_1^{-1}}{\longrightarrow } \operatorname {\mathcal{M}}_1/u\operatorname {\mathcal{M}}_1 \longrightarrow \operatorname {\mathcal{M}}/u\operatorname {\mathcal{M}}. \end{equation*}$$

(4)

If$$\begin{equation*} 0 \longrightarrow \mathcal{G}' \longrightarrow \mathcal{G}\longrightarrow \mathcal{G}'' \longrightarrow 0 \end{equation*}$$

is a diagram of finite flat group schemes over $R$ which are killed by $\ell$ and if$$\begin{equation*} 0 \longrightarrow \operatorname {\mathcal{M}}_\pi (\mathcal{G}'') \longrightarrow \operatorname {\mathcal{M}}_\pi (\mathcal{G}) \longrightarrow \operatorname {\mathcal{M}}_\pi (\mathcal{G}') \longrightarrow 0 \end{equation*}$$

is the corresponding diagram of Breuil modules, then the diagram of finite flat group schemes is a short exact sequence if and only if the diagram of Breuil modules is a short exact sequence.

Proof.

See §2.1.1, Proposition 2.1.2.2, Theorem 3.3.7, Theorem 4.2.1.6 and the proof of Theorem 3.3.5 of Br2. In 3.3.5 of Br2 it is shown that $\operatorname {\mathcal{M}}_\pi ({\mathcal{G}})/u\operatorname {\mathcal{M}}_\pi ({\mathcal{G}})$ can be $k$-linearly identified with the crystalline Dieudonné module of $\mathcal{G}\times k$. In 4.2.14 of BBM the crystalline Dieudonné module of $\mathcal{G}\times k$ is identified with $D({\mathcal{G}}) \otimes _{k,\operatorname {Frob}_\ell } k$. The equivalence of the two notions of exactness can be deduced from the compatibility of ${\operatorname {\mathcal{M}}}_\pi$ with Dieudonné theory, from Lemma 5.1.1, and from the fact that a complex of finite flat group schemes over $R$ is exact if and only if its special fibre is exact (see for example Proposition 1.1 of deJ).

■

5.2. Examples

For $0 \leq r \leq e$ an integer and for $a \in k^\times$, define a Breuil module $\operatorname {\mathcal{M}}(r,a)$ by

•

$\operatorname {\mathcal{M}}(r,a) = (k[u]/u^{e\ell })\mathbf{e}$,

•

$\operatorname {\mathcal{M}}(r,a)_1 = (k[u]/u^{e\ell }) u^r \mathbf{e}$,

•

$\phi _1(u^r\mathbf{e})=a\mathbf{e}$.

It is easy to check that $\phi _1$ is well defined (and uniquely determined by the given conditions). We will refer to $\mathbf{e}$ as the standard generator of ${\operatorname {\mathcal{M}}}(r,a)$ and write ${\mathcal{G}}(r,a)$ for ${\mathcal{G}}_\pi ({\operatorname {\mathcal{M}}}(r,a))$. The following lemma is easy to check.

Lemma 5.2.1.

(1)

Any Breuil module of rank $1$ over $k[u]/u^{e\ell }$ is isomorphic to some ${\operatorname {\mathcal{M}}}(r,a)$.

(2)

There is a non-zero morphism ${\operatorname {\mathcal{M}}}(r,a) \rightarrow {\operatorname {\mathcal{M}}}(r',a')$ if and only if $r' \geq r$, $r'\equiv r \bmod \ell -1$ and $a/a' \in (k^\times )^{\ell -1}$. All such morphisms are then of the form $\mathbf{e}\mapsto bu^{\ell (r'-r)/ (\ell -1)} \mathbf{e}'$, where $b^{\ell -1}=a/a'$.

(3)

The modules ${\operatorname {\mathcal{M}}}(r,a)$ and ${\operatorname {\mathcal{M}}}(r',a')$ are isomorphic if and only if $r=r'$ and $a/a' \in (k^\times )^{\ell -1}$, or equivalently if and only if there are non-zero morphisms ${\operatorname {\mathcal{M}}}(r,a)\rightarrow {\operatorname {\mathcal{M}}}(r',a')$ and ${\operatorname {\mathcal{M}}}(r',a')\rightarrow {\operatorname {\mathcal{M}}}(r,a)$.

(4)

If we order the ${\operatorname {\mathcal{M}}}(r,a)$ by setting ${\operatorname {\mathcal{M}}}(r,a) \geq {\operatorname {\mathcal{M}}}(r',a')$ if there is a non-zero morphism ${\operatorname {\mathcal{M}}}(r',a') \rightarrow {\operatorname {\mathcal{M}}}(r,a)$, then the set of isomorphism classes of ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}})$’s as ${\mathcal{G}}$ runs over models of a fixed finite flat $F'$-group scheme $G$ of order $\ell$ is well ordered.

(5)

On ${\operatorname {\mathcal{M}}}(r,a)$ we have $N\mathbf{e}=0$, so $N \circ \phi _1=0$.

(6)

${\mathcal{G}}(r,a)$ is étale (resp. multiplicative) if and only if $r=e$ (resp. $r=0$).

(7)

${\mathcal{G}}(0,1) \cong \mu _\ell$ and ${\mathcal{G}}(e,-G_\pi (0))\cong \mathbf{Z}/\ell \mathbf{Z}$.

(8)

The Cartier dual of ${\mathcal{G}}(r,-G_\pi (0))$ is ${\mathcal{G}}(e-r,1)$.

Proof.

The first three parts are easy computations. For the fourth part note that two finite flat group schemes ${\mathcal{G}}$ and ${\mathcal{G}}'$ of order $\ell$ over $R$ have isomorphic generic fibres if and only if there is a non-zero morphism ${\mathcal{G}}\rightarrow {\mathcal{G}}'$ or ${\mathcal{G}}' \rightarrow {\mathcal{G}}$. The fifth part is another easy computation and the sixth part follows on computing the Dieudonné module using Theorem 5.1.3.

By 3.1.2 of Br2 we see that the affine $R$-algebra of the group scheme attached to $\operatorname {\mathcal{M}}(r,a)$ is

$$\begin{equation*} R[X]/(X^\ell + \frac{\pi ^{e-r} \widetilde{a}}{G_{\pi }(\pi )}X), \end{equation*}$$

where $\widetilde{a}$ is a lift of $a$ to $W(k)$. This has constant generic fibre if and only if $-\pi ^{e-r}\widetilde{a}/G_{\pi }(\pi ) \in F'$ is an $(\ell -1)^{th}$ power. This occurs if and only if $r \equiv e \bmod \ell -1$ and $-a/G_{\pi }(0) \in k$ is an $(\ell -1)^{th}$ power. Thus $\operatorname {\mathcal{M}}(e,-G_{\pi }(0))$ corresponds to the étale group scheme $\mathbf{Z}/\ell \mathbf{Z}$ over $R$.

Next, we show that the group scheme $\mathcal{G}$ corresponding to the Breuil module $\operatorname {\mathcal{M}}(0,1)$ is isomorphic to $\mu _\ell$. By using the relation between Breuil modules and Dieudonné modules (see Theorem 5.1.3) we see that the Dieudonné module of the closed fibre of $\mathcal{G}$ is isomorphic to the Dieudonné module of the closed fibre of $\mu _\ell$. This forces ${\mathcal{G}}\stackrel{\sim }{\rightarrow }\mu _\ell$, since we may consider Cartier duals and observe that a finite flat $R$-group scheme ${\mathcal{G}}$ is étale if and only if its special fibre is étale, and then §18.5.15 of book IV$_4$ of EGA may be used.

This establishes the seventh part. The final part follows from parts four and seven.

■

Now suppose that $0 \leq r,s \leq e$ are integers and choose $a,b \in k^\times$ and $f \in u^{\max (0,r+s-e)}k[u]/u^{e\ell }$. We can define an extension class

$$\begin{equation*} (0) \longrightarrow {\operatorname {\mathcal{M}}}(s,b) \longrightarrow {\operatorname {\mathcal{M}}}(s,b;r,a;f) \longrightarrow {\operatorname {\mathcal{M}}}(r,a) \longrightarrow (0) \end{equation*}$$

in ${\underline{\phi _1{\mathrm{{-mod}}}}}_R$ by

•

$\operatorname {\mathcal{M}}(s,b;r,a;f) = (k[u]/u^{e\ell })\mathbf{e}\oplus (k[u]/u^{e\ell })\mathbf{e}'$,

•

$\operatorname {\mathcal{M}}(s,b;r,a;f)_1 = \langle u^s \mathbf{e}, u^r \mathbf{e}' + f\mathbf{e}\rangle$,

•

$\phi _1(u^s\mathbf{e})=b\mathbf{e}$ and $\phi _1(u^r \mathbf{e}' + f\mathbf{e})=a\mathbf{e}'$,

•

the standard generator of ${\operatorname {\mathcal{M}}}(s,b)$ maps to $\mathbf{e}$,

•

$\mathbf{e}$ maps to $0$ and $\mathbf{e}'$ maps to the standard generator in ${\operatorname {\mathcal{M}}}(r,a)$.

The following lemma is also easy to check.

Lemma 5.2.2.

(1)

Any extension of ${\operatorname {\mathcal{M}}}(r,a)$ by ${\operatorname {\mathcal{M}}}(s,b)$ in ${\underline{\phi _1{\mathrm{{-mod}}}}}_R$ is isomorphic to $\operatorname {\mathcal{M}}(s,b;r,a;f)$ for some $f \in u^{\max (0,r+s-e)}k[u]/u^{e\ell }$.

(2)

Two such extensions $\operatorname {\mathcal{M}}(s,b;r,a;f)$ and $\operatorname {\mathcal{M}}(s,b;r,a;f')$ are isomorphic as extension classes if and only if$$\begin{equation*} f'-f = u^sh-(b/a)u^rh^\ell \end{equation*}$$

for some $h \in k[u]/u^{e\ell }$, in which case one such isomorphism fixes $\mathbf{e}$ and sends $\mathbf{e}'$ to $\mathbf{e}'-(b/a)h^\ell \mathbf{e}$.

We remark that $f \in u^{\max (0,r+s-e)}k[u]/u^{e\ell }$ is required so that $\operatorname {\mathcal{M}}(s,b;r,a;f)_1 \supset u^e \operatorname {\mathcal{M}}(s,b;r,a;f)$. We will write ${\mathcal{G}}(s,b;r,a;f)$ for ${\mathcal{G}}_\pi ({\operatorname {\mathcal{M}}}(s,b;r,a;f))$.

We will also need some slight extensions of these results to allow for coefficients. To this end let $k'/\mathbf{F}_\ell$ be a finite extension linearly disjoint from $k$ and write $k'k$ for the field $k' \otimes _{\mathbf{F}_\ell } k$. For $0 \leq r \leq e$ an integer and for $a \in (k'k)^\times$, define a Breuil module, $\operatorname {\mathcal{M}}(k';r,a)$, with an action of $k'$ by

•

$\operatorname {\mathcal{M}}(k';r,a) = ((k'k)[u]/u^{e\ell })\mathbf{e}$,

•

$\operatorname {\mathcal{M}}(k';r,a)_1 = ((k'k)[u]/u^{e\ell }) u^r \mathbf{e}$,

•

$\phi _1(u^r\mathbf{e})=a\mathbf{e}$.

We will let $\phi$ denote the automorphism of $k'k[u]$, which is the identity on $k'$ and which raises elements of $k[u]$ to the $\ell ^{th}$ power. The following lemma is easy to check.

Lemma 5.2.3.

(1)

Any Breuil module with an action of $k'$ which is free of rank $[k':k]$ over $k[u]/u^{e\ell }$ is isomorphic to some ${\operatorname {\mathcal{M}}}(k';r,a)$.

(2)

There is a non-zero morphism ${\operatorname {\mathcal{M}}}(k';r,a) \rightarrow {\operatorname {\mathcal{M}}}(k';r',a')$ if and only if $r' \geq r$, $r'\equiv r \bmod \ell -1$ and $a/a' \in \phi (b)/b$ for some $b \in (k'k)^\times$. All such morphisms are then of the form $\mathbf{e}\mapsto b'u^{\ell (r'-r)/(\ell -1)} \mathbf{e}'$, where $b \in (k'k)^\times$ and $\phi (b')/b'= a/a'$.

(3)

The modules ${\operatorname {\mathcal{M}}}(k';r,a)$ and ${\operatorname {\mathcal{M}}}(k';r',a')$ are isomorphic if and only if $r=r'$ and $a/a' \in ((k'k)^\times )^{\phi -1}$.

(4)

On ${\operatorname {\mathcal{M}}}(k';r,a)$ we have $N\mathbf{e}=0$ and so $N \circ \phi _1=0$.

(5)

${\mathcal{G}}_\pi ({\operatorname {\mathcal{M}}}(k';r,a))$ is étale (resp. multiplicative) if and only if $r=e$ (resp. $r=0$).

Now choose $0 \leq r,s \leq e$ integers, $a,b \in (k'k)^\times$ and $f \in u^{\max (0,r+s-e)}(k'k)[u]/u^{e\ell }$. We define an extension class

$$\begin{equation*} (0) \longrightarrow {\operatorname {\mathcal{M}}}(k';s,b) \longrightarrow {\operatorname {\mathcal{M}}}(k';s,b;r,a;f) \longrightarrow {\operatorname {\mathcal{M}}}(k';r,a) \longrightarrow (0) \end{equation*}$$

in ${\underline{\phi _1{\mathrm{{-mod}}}}}_R$ with an action of $k'$ by

•

$\operatorname {\mathcal{M}}(k';s,b;r,a;f) = ((k'k)[u]/u^{e\ell })\mathbf{e}\oplus ((k'k)[u]/u^{e\ell })\mathbf{e}'$,

•

$\operatorname {\mathcal{M}}(k';s,b;r,a;f)_1 = \langle u^s \mathbf{e}, u^r \mathbf{e}' + f\mathbf{e}\rangle$,

•

$\phi _1(u^s\mathbf{e})=b\mathbf{e}$ and $\phi _1(u^r \mathbf{e}' + f\mathbf{e})=a\mathbf{e}'$,

•

the standard generator of ${\operatorname {\mathcal{M}}}(k';s,b)$ maps to $\mathbf{e}$,

•

$\mathbf{e}$ maps to $0$ and $\mathbf{e}'$ to the standard generators in ${\operatorname {\mathcal{M}}}(k';r,a)$.

Then the following lemma is easy to check.

Lemma 5.2.4.

(1)

Any extension of ${\operatorname {\mathcal{M}}}(k';r,a)$ by ${\operatorname {\mathcal{M}}}(k';s,b)$ in ${\underline{\phi _1{\mathrm{{-mod}}}}}_R$ with a compatible action of $k'$ is isomorphic to $\operatorname {\mathcal{M}}(k';s,b;r,a;f)$ for some $f \in u^{\max (0,r+s-e)}(k'k)[u]/u^{e\ell }$.

(2)

Two such extensions $\operatorname {\mathcal{M}}(k';s,b;r,a;f)$ and $\operatorname {\mathcal{M}}(k';s,b;r,a;f')$ are isomorphic (as extensions) if and only if$$\begin{equation*} f'-f = u^sh-(b/a)u^r\phi (h) \end{equation*}$$

for some $h \in (k'k)[u]/u^{e\ell }$, in which case one such isomorphism fixes $\mathbf{e}$ and sends $\mathbf{e}'$ to $\mathbf{e}'-(b/a)\phi (h)\mathbf{e}$.

We will write ${\mathcal{G}}(k';r,a;s,b;f)$ and ${\mathcal{G}}(k';r,a)$ for ${\mathcal{G}}_\pi ({\operatorname {\mathcal{M}}}(k';r,a;s,b;f))$ and ${\mathcal{G}}_\pi ({\operatorname {\mathcal{M}}}(k';r,a))$ respectively.

5.3. Relationship to syntomic sheaves

Let us first recall some of the notations of Br1 and Br2. Let ${\mathrm{Spf}}(R)_{\mathrm{syn}}$ be the small $\ell$-adic formal syntomic site over $R$, $S$ the $\ell$-adic completion of $W(k)[u,\frac{u^{ie}}{i!}]_{i\in {\mathbf{N}}}$, $S_n=S/\ell ^nS$, $E_n={\mathrm{Spec}}(S_n)$ and for any ${\mathfrak{X}}\in {\mathrm{Spf}}(R)_{\mathrm{syn}}$:

$$\begin{equation*} {\mathcal{O}}_{n,\pi }^{cris}({\mathfrak{X}})=H^0(({\mathfrak{X}}_n/E_n)_{cris},{\mathcal{O}}_{{\mathfrak{X}}_n/E_n}), \end{equation*}$$

where ${\mathfrak{X}}_n={\mathfrak{X}}\times _{R}R/\ell ^n$ is viewed over $E_n$ via the thickening $({\mathrm{Spec}}(R/\ell ^n)\hookrightarrow E_n, u\mapsto \pi )$. It turns out ${\mathcal{O}}_{n,\pi }^{cris}$ is the sheaf of $S_n$-modules on ${\mathrm{Spf}}(R)_{\mathrm{syn}}$ associated to the presheaf (cf. the proof of Lemma 2.3.2 in Br2):

$$\begin{equation} \begin{split} {\mathfrak{X}}&\mapsto \Big (W_n(k)[u]\otimes _{\phi ^n,W_n(k)}W_n(\Gamma ({\mathfrak{X}}_1,{\mathcal{O}}_{{\mathfrak{X}}_1}))\Big )^{DP}\\ &=\Big ((W_n(k)[u]/u^{e\ell ^n})\otimes _{\phi ^n,W_n(k)}W_n(\Gamma ({\mathfrak{X}}_1,{\mathcal{O}}_{{\mathfrak{X}}_1}))\Big )^{DP}.\end{split}{\tag{5.3.1}} \end{equation}$$

Here, the subscript “$\phi ^n$” means we twist by the $n^{th}$ power of the Frobenius when sending $W_n(k)$ to $W_n(k)[u]$ and the exponent “$DP$” means we take the divided power envelope with respect to the kernel of the canonical map:

$$\begin{equation*} \begin{matrix} W_n(k)[u]\otimes _{\phi ^n,W_n(k)}W_n(\Gamma ({\mathfrak{X}}_1,{\mathcal{O}}_{{\mathfrak{X}}_1}))&\longrightarrow &\Gamma ({\mathfrak{X}}_n,{\mathcal{O}}_{{\mathfrak{X}}_n})\\ s(u)\otimes (w_0,...,w_{n-1})& \longmapsto &s(\pi )(\hat{w}_0^{\ell ^n}+\ell \hat{w}_1^{\ell ^{n-1}}+...+\ell ^{n-1}\hat{w}_{n-1}^\ell ),\end{matrix} \end{equation*}$$

where $\hat{w}_i$ is a local lifting of $w_i$, these divided powers being required to be compatible with the usual divided powers $\gamma _i(\ell x)= \frac{\ell ^i}{i!}x^i$ (i.e. we take the divided power envelope relative to the usual divided power structure on the maximal ideal of $W_n(k)$). Note that the latter map induces a canonical surjection of sheaves of $S_n$-modules on ${\mathrm{Spf}}(R)_{\mathrm{syn}}$:

$$\begin{equation*} {\mathcal{O}}_{n,\pi }^{cris}\rightarrow {\mathcal{O}}_n, \end{equation*}$$

where ${\mathcal{O}}_n({\mathfrak{X}})=\Gamma ({\mathfrak{X}}_n,{\mathcal{O}}_{{\mathfrak{X}}_n})$. We denote by ${\mathcal{J}}_{n,\pi }^{cris}$ the kernel of this surjection. For any $n$, let $\phi :S_n\rightarrow S_n$ be the unique lifting of Frobenius such that $\phi (u)=u^\ell$ and $\phi (\frac{u^{ie}}{i!})=\frac{u^{ie\ell }}{i!}$. The sheaf ${\mathcal{O}}_{n,\pi }^{cris}$ is equipped with the crystalline Frobenius $\phi$, which is also induced by the map $s(u)\otimes (w_0,...,w_{n-1}) \mapsto \phi (s(u))\otimes (w_0^\ell ,...,w_{n-1}^\ell )$ on the above presheaf (5.3.1). (Here $\phi$ on $W_n(k)[u]$ is Frobenius on $W_n(k)$ and takes $u$ to $u^\ell$.) Since $\ell$ divides $\phi (x)-x^\ell$, we get $\phi ({\mathcal{J}}_{n,\pi }^{cris})\subset \ell {\mathcal{O}}_{n,\pi }^{cris}$ for all $n$, so we can define an $S_1$-linear $\phi _{1}=\frac{\phi }{\ell }|_{{\mathcal{J}}_{n,\pi }^{cris}}$ by the usual “flatness” trick (see §2.3 of Br2). Let $N:S_n\rightarrow S_n$ be the unique $W_n(k)$-linear derivation such that $N(u)=u$ and $N(\gamma _i(u^e))=eu^e\gamma _{i-1}(u^e)=ie\gamma _i(u^e)$. Finally define:

$$\begin{equation*} N:{\mathcal{O}}_{n,\pi }^{cris}\rightarrow {\mathcal{O}}_{n,\pi }^{cris} \end{equation*}$$

to be the unique $W_n(k)$-linear morphism of sheaves which on the presheaf (5.3.1) is given by $N(\gamma _i(\sum s\otimes w))=(\sum N(s)\otimes w)\gamma _{i-1}(\sum s\otimes w)$. Note that $N\circ \phi =\ell \phi \circ N$, so $N\circ \phi _1=\phi \circ N$ on ${\mathcal{J}}_{n,\pi }^{cris}$.

Let ${\mathcal{G}}$ be a finite flat group scheme over $R$, which is killed by $\ell$. Viewing ${\mathcal{G}}$ as a formal scheme over $R$, it is an object in ${\mathrm{Spf}}(R)_{\mathrm{syn}}$. Viewing it as a sheaf of groups on ${\mathrm{Spf}}(R)_{\mathrm{syn}}$, its associated Breuil module is defined as:

(1)

${\mathcal{M}}_{\pi }({\mathcal{G}})={\mathrm{Hom}}_{\mathrm{sheaves\ of\ groups}}({\mathcal{G}},{\mathcal{O}}_{1,\pi }^{cris})\otimes _{S_1}k[u]/u^{e\ell }$,

(2)

${\mathcal{M}}_{\pi }({\mathcal{G}})_1={\mathrm{image\ of\ Hom}}_{\mathrm{sheaves\ of\ groups}} ({\mathcal{G}},{\mathcal{J}}_{1,\pi }^{cris})\otimes _{S_1}k[u]/u^{e\ell }$ in ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}})$,

(3)

$\phi _1$ is induced by $\phi _1\otimes \phi$,

where the $S_1$-module structures are induced by the compatible $S_1$ actions on ${\mathcal{O}}_{1,\pi }^{cris}$ and ${\mathcal{J}}_{1,\pi }^{cris}$ (see §3.2 and §2.1.2.2 of Br2). Here $S_1\rightarrow k[u]/u^{e\ell }$ is the surjection that sends $u$ to $u$, $\gamma _i(u^e)$ to $\gamma _i(u^e)$ for $i<l$ and $\gamma _i(u^e)$ to $0$ for $i \geq l$.

We record for future reference the following straightforward observation.

Lemma 5.3.1.

If we denote by $\Delta$ (resp. ${\mathrm{pr}}_i$, $i\in \{1,2\}$) the coproduct (resp. the two projections)

$$\begin{equation*} {\mathcal{G}}\times _{{\mathrm{Spec}}(R)}{\mathcal{G}}\rightarrow {\mathcal{G}}, \end{equation*}$$

then for any sheaf of commutative groups $\mathcal{F}$ on ${\mathrm{Spf}}(R)_{\mathrm{syn}}$ we have:

$$\begin{equation*} {\mathrm{Hom}}_{\mathrm{sheaves\ of\ groups}}({\mathcal{G}},{\mathcal{F}})=\{x\in {\mathcal{F}}({\mathcal{G}})\ |\ (\Delta ^*-{\mathrm{pr}}_1^*-{\mathrm{pr}}_2^*)(x)=0\}. \end{equation*}$$

The operator $N$ on ${\mathcal{O}}_{1,\pi }^{cris}$ induces an operator $N$ on ${\mathrm{Hom}}_{\mathrm{sheaves\ of\ groups}}({\mathcal{G}},{\mathcal{O}}_{1,\pi }^{cris})$, hence on ${\mathcal{M}}_{\pi }({\mathcal{G}})$.

Lemma 5.3.2.

The above operator $N$ on ${\mathcal{M}}_{\pi }({\mathcal{G}})$ coincides with the operator $N$ defined in Lemma 5.1.2.

Proof.

By unicity in Lemma 5.1.2, we only have to prove that $N$ satisfies $N({\mathcal{M}}_{\pi }({\mathcal{G}})) \subset u{\mathcal{M}}_{\pi }({\mathcal{G}})$, since the other conditions are automatically satisfied. It’s enough to prove that $N(\phi _1(x))=(\phi \circ N)(x)\in u{\mathcal{M}}_{\pi }({\mathcal{G}})$ for any $x\in {\mathcal{M}}_{\pi }({\mathcal{G}})_1$. But $u^{e\ell -\ell }\phi \circ N=0$ on ${\mathcal{O}}_{1,\pi }^{cris}$ because it is so on $(k[u]\otimes \Gamma ({\mathfrak{X}}_1,{\mathcal{O}}_{{\mathfrak{X}}_1}))^{DP}$. Thus one also has $u^{e\ell -\ell }\phi \circ N=0$ on ${\mathrm{Hom}}_{\mathrm{groups}}({\mathcal{G}},{\mathcal{O}}_{1,\pi }^{cris})$, hence on ${\mathcal{M}}_{\pi }({\mathcal{G}})$. This implies $\phi \circ N({\mathcal{M}}_{\pi }({\mathcal{G}}))\subset u^\ell {\mathcal{M}}_{\pi }({\mathcal{G}})\subset u{\mathcal{M}}_{\pi }({\mathcal{G}})$ since ${\mathcal{M}}_{\pi }({\mathcal{G}})$ is free over $k[u]/u^{e\ell }$.

■

5.4. Base change

In this section we will examine the relationship of the functor ${\operatorname {\mathcal{M}}}_\pi$ with two instances of base change. First we consider unramified base change.

Let $k'$ be a perfect field of characteristic $\ell$ which is an extension of $k$ and $R'= R\otimes _{W(k)}W(k')$. Choose $\pi '=\pi \otimes 1$ as uniformiser in $R'$. If ${\mathfrak{X}}\in {\mathrm{Spf}}(R)_{\mathrm{syn}}$, let ${\mathfrak{X}}'= {\mathrm{Spf}}(R')\times _{{\mathrm{Spf}}(R)}{\mathfrak{X}}$ and define:

$$\begin{equation*} {\mathcal{O}}_{n,\pi }^{cris'}({\mathfrak{X}})={\mathcal{O}}_{n,\pi '}^{cris}( {\mathfrak{X}}')\quad {\mathrm{and}}\quad {\mathcal{J}}_{n,\pi }^{cris'}({\mathfrak{X}})={\mathcal{J}}_{n,\pi '}^{cris}({\mathfrak{X}}'). \end{equation*}$$

As in the proof of 2.3.2 of Br2, we have that ${\mathcal{O}}_{n,\pi }^{cris'}$ is the sheaf on ${\mathrm{Spf}}(R)_{\mathrm{syn}}$ associated to the presheaf:

$$\begin{equation*} \begin{array}{rcl} {\mathfrak{X}} &\mapsto &\Big (W_n(k')[u]\otimes _{\phi ^n,W_n(k')}W_n(\Gamma ( {\mathfrak{X}}'_1,{\mathcal{O}}_{{\mathfrak{X}}'_1}))\Big )^{DP} \\&=& \Big (W_n(k')[u]\otimes _{\phi ^n,W_n(k')}W_n(k'\otimes _{k}\Gamma ({\mathfrak{X}}_1, {\mathcal{O}}_{{\mathfrak{X}}_1}))\Big )^{DP}.\end{array} \end{equation*}$$

Define $S'_n$ as $S_n$ but with $k'$ instead of $k$. There is a canonical isomorphism of sheaves:

$$\begin{equation*} {\mathcal{O}}_{n,\pi }^{cris}\otimes _{S_n}S'_n={\mathcal{O}}_{n,\pi }^{cris}\otimes _{W_n(k)}W_n(k')\buildrel \sim \over \rightarrow {\mathcal{O}}_{n,\pi }^{cris'} \end{equation*}$$

coming from the obvious isomorphism:

$$\begin{multline*} (W_n(k')[u]/u^{e\ell ^n})\otimes _{\phi ^n,W_n(k)}W_n(\Gamma ({\mathfrak{X}}_1,{\mathcal{O}}_{{\mathfrak{X}}_1}))\\ \buildrel \sim \over \rightarrow (W_n(k')[u]/u^{e\ell ^n})\otimes _{\phi ^n,W_n(k')}W_n(k'\otimes _{k}\Gamma ( {\mathfrak{X}}_1,{\mathcal{O}}_{{\mathfrak{X}}_1})) \end{multline*}$$

and one easily sees it induces an isomorphism ${\mathcal{J}}_{n,\pi }^{cris}\otimes _{W_n(k)}W_n(k')\buildrel \sim \over \rightarrow {\mathcal{J}}_{n,\pi }^{cris'}$. Moreover, we have the following obvious lemma.

Lemma 5.4.1.

The diagram of sheaves on ${\mathrm{Spf}}(R)_{\mathrm{syn}}$:

$$\begin{equation*} \begin{matrix} {\mathcal{J}}_{n,\pi }^{cris}\otimes _{W_n(k)}W_n(k')&\buildrel \sim \over \rightarrow &{\mathcal{J}}_{n,\pi }^{cris'}\\ \downarrow {\scriptstyle \phi _{1}\otimes \phi } && {\scriptstyle \phi _1}\downarrow \\ {\mathcal{O}}_{n,\pi }^{cris}\otimes _{W_n(k)}W_n(k') &\buildrel \sim \over \rightarrow &{\mathcal{O}}_{n,\pi }^{cris'}\end{matrix} \end{equation*}$$

is commutative.

Using the identification from §5.3, Lemma 5.3.1 and Lemma 5.4.1 (for $n=1$), together with obvious functorialities, we obtain after tensoring by $k[u]/u^{e\ell }$ the following corollary.

Corollary 5.4.2.

Let ${\mathcal{G}}$ be a finite flat group scheme over $R$, which is killed by $\ell$. Let $k'/k$ be an extension of fields with $k'$ perfect and let $\pi '=\pi \otimes 1$, a uniformiser for $R'=R \otimes _{W(k)} W(k')$. Then there is a canonical isomorphism in the category ${\underline{\phi _1{\mathrm{{-mod}}}}}_R$

$$\begin{equation*} \Big ({\mathcal{M}}_{\pi }({\mathcal{G}})\otimes _{k}k', {\mathcal{M}}_{\pi }({\mathcal{G}})_1\otimes _{k}k',\phi _{1}\otimes \phi \Big )\buildrel \sim \over \rightarrow \Big ({\mathcal{M}}_{\pi '}({\mathcal{G}}'), {\mathcal{M}}_{\pi '}({\mathcal{G}}')_1,\phi _1\Big ) \end{equation*}$$

compatible with composites of such residue field extensions.

We will now turn to the case of base change by a continuous automorphism $g:R \stackrel{\sim }{\rightarrow }R$. For any $s=\sum w_iu^i \in W(k)[[u]]$, let ${^{(g)}s}=\sum g(w_i)u^i$ and ${^{(\phi )}s}=\sum \phi (w_i)u^i$, where $g$ and $\phi$ act on $W(k)$ through their action on $k$. Choose $H_{g}(u)\in W(k)[[u]]$ such that $g(\pi )=\pi H_g(\pi )$. Notice that $H_{g}(u)\in W(k)[[u]]^{\times }$. Define $\widehat{g}: W(k)[[u]]\buildrel \sim \over \rightarrow W(k)[[u]]$ by $\widehat{g} (\sum w_iu^i)=\sum g(w_i)u^iH_{g}(u)^i$.

Lemma 5.4.3.

There is a unique element ${_{g}t}(u)\in W(k)[[u]]$ such that, if ${_g\phi }$ is defined by ${_g\phi }(\sum w_iu^i)=\sum \phi (w_i)(u^\ell (1+\ell {_{g}t}(u)))^i$, one has $\widehat{g} \circ {_g\phi }=\phi \circ \widehat{g}$.

Proof.

One has to solve in $W(k)[[u]]$:

$$\begin{equation*} 1+\ell {_g^{(g)}t}(uH_{g}(u))=\Big (H_{g}(u)^{-1}\Big )^\ell {^{(\phi )}H_{g}}(u^\ell ) \end{equation*}$$

(where the two sides clearly belong to $1+\ell W(k)[[u]]$). As $H_{g}(u)\in W(k)[[u]]^{\times }$, there is a unique $K_{g}\in uW(k)[[u]]^{\times }$ such that $K_{g}(u)H_{g}(K_{g}(u))=u$, so we have

$$\begin{equation*} 1+\ell {_g^{(g)}t}(u)=\Big (H_{g}(K_{g}(u))^{-1}\Big )^\ell {^{(\phi )}H_{g}}(K_{g}(u)^\ell ). \end{equation*}$$ ■

For any object $\mathcal{M}$ of ${\underline{\phi _1{\mathrm{{-mod}}}}}_R$, define ${_g\phi }_1:{\mathcal{M}}_1\rightarrow {\mathcal{M}}$ by the following formula:

$$\begin{equation} {_g\phi }_1(x)=\phi _1(x)+{_gt}(u)N(\phi _1(x)) {\tag{5.4.1}} \end{equation}$$

where $N$ is as in Lemma 5.1.2.

For any ${\mathfrak{X}}\in {\mathrm{Spf}}(R)_{\mathrm{syn}}$, let ${^g\mathfrak{X}}={\mathrm{Spf}}(R)\times _{g^*,{\mathrm{Spf}}(R)}{\mathfrak{X}}$ and define:

$$\begin{equation*} {{\mathcal{O}}_{n,\pi }^{cris,(g)}}({\mathfrak{X}})={\mathcal{O}}_{n,\pi }^{cris}( {^g\mathfrak{X}})\quad {\mathrm{and}}\quad {{\mathcal{J}}_{n,\pi }^{cris,(g)}}({\mathfrak{X}})={\mathcal{J}}_{n,\pi }^{cris}({^g\mathfrak{X}}). \end{equation*}$$

Then ${\mathcal{O}}_{n,\pi }^{cris, (g)}$ is the sheaf on ${\mathrm{Spf}}(R)_{\mathrm{syn}}$ associated to the presheaf:

$$\begin{equation*} \begin{array}{rcl} {\mathfrak{X}}&\mapsto &\Big (W_n(k)[u]\otimes _{\phi ^n,W_n(k)}W_n(\Gamma ( {^g\mathfrak{X}}_1,{\mathcal{O}}_{{^g\mathfrak{X}}_1}))\Big )^{DP} \\&=& \Big (W_n(k)[u]\otimes _{\phi ^n,W_n(k)}W_n(R\otimes _{g,R}\Gamma ( {\mathfrak{X}}_1,{\mathcal{O}}_{{\mathfrak{X}}_1}))\Big )^{DP}. \end{array} \end{equation*}$$

Let $\widehat{g}: S_n\rightarrow S_n$ be the unique ring isomorphism such that

$$\begin{equation*} \widehat{g}\left(w_i\frac{u^{ei+j}}{i!}\right)=g(w_i)\frac{u^{ei+j}}{i!} H_{g}(u)^{ei+j} \end{equation*}$$

for $0 \leq j < e$, $i \geq 0$. There is a canonical isomorphism of sheaves:

$$\begin{equation*} {\mathcal{O}}_{n,\pi }^{cris}\otimes _{S_n,\hat{g}}S_n\buildrel \sim \over \longrightarrow {{\mathcal{O}}_{n,\pi }^{cris,(g)}} \end{equation*}$$

coming from the obvious $\widehat{g}$-semi-linear isomorphism:

$$\begin{multline*} (W_n(k)[u]/u^{e\ell ^n})\otimes _{\phi ^n,W_n(k)}W_n(\Gamma ( {\mathfrak{X}}_1,{\mathcal{O}}_{{\mathfrak{X}}_1}))\\ \buildrel \sim \over \longrightarrow (W_n(k)[u]/u^{e\ell ^n})\otimes _{\phi ^n,W_n(k)}W_n(R\otimes _{g,R}\Gamma ( {\mathfrak{X}}_1,{\mathcal{O}}_{{\mathfrak{X}}_1})) \end{multline*}$$

$$\begin{equation*} s\otimes (w_0,...,w_{n-1}) \longmapsto \widehat{g}(s)\otimes (1\otimes w_0,...,1\otimes w_{n-1}) \end{equation*}$$ and one easily sees it induces an isomorphism ${\mathcal{J}}_{n,\pi }^{cris}\otimes _{S_n,\hat{g}}S_n\buildrel \sim \over \rightarrow {{\mathcal{J}}_{n,\pi }^{cris,(g)}}$.

Define ${_g\phi }:S_n\rightarrow S_n$ as in Lemma 5.4.3 and define:

$$\begin{equation*} {_g\phi }:{\mathcal{O}}_{n,\pi }^{cris}\longrightarrow {\mathcal{O}}_{n,\pi }^{cris} \end{equation*}$$

to be the unique morphism of sheaves which is induced by ${_g\phi }(\gamma _i(\sum s\otimes w))=\gamma _i(\sum {_g\phi }(s)\otimes \phi (w))$ on the presheaf (5.3.1) (see §5.3 and note that this is well defined). Since ${_g\phi }({\mathcal{J}}_{n,\pi }^{cris})\subset \ell {\mathcal{O}}_{n,\pi }^{cris}$, we can define ${_g\phi }_1=\frac{_g\phi }{\ell }|_{{\mathcal{J}}_{n,\pi }^{cris}}$.

Lemma 5.4.4.

The diagram of sheaves on ${\mathrm{Spf}}(R)_{\mathrm{syn}}\!:$

$$\begin{equation*} \begin{matrix} {\mathcal{J}}_{n,\pi }^{cris}\otimes _{S_n,\hat{g}}S_n&\buildrel \sim \over \rightarrow &{\mathcal{J}}_{n,\pi }^{cris,(g)}\\ \downarrow {\scriptstyle {_g\phi }_1\otimes \phi } && {\scriptstyle \phi _1}\downarrow \\ {\mathcal{O}}_{n,\pi }^{cris}\otimes _{S_n,\hat{g}}S_n &\buildrel \sim \over \rightarrow &{\mathcal{O}}_{n,\pi }^{cris,(g)}\end{matrix} \end{equation*}$$

is commutative. Moreover we have on ${\mathcal{J}}_{n,\pi }^{cris}\!:$

$$\begin{equation*} {_g\phi }_1=\displaystyle {\sum _{i=0}^{\infty }}\Big ( \frac{\log (1+\ell {_gt}(u))}{\ell }\Big )^i\frac{N^i}{i!}\circ {\phi _1}, \end{equation*}$$

where $N$ is defined as in §5.3.

Proof.

By working modulo $\ell ^{n+1}$, i.e. with ${\mathcal{J}}_{n+1,\pi }^{cris}$ and ${_g\phi }$, and looking on the above presheaves, the proof is completely straightforward.

■

Let ${\mathcal{G}}$ be a finite flat group scheme over $R$ which is killed by $\ell$. Note that thanks to Lemma 5.3.2 and the formula for ${_g\phi }_1$ in Lemma 5.4.4, the operator ${\mathcal{M}}_{\pi }({\mathcal{G}})_1\rightarrow {\mathcal{M}}_{\pi }({\mathcal{G}})$ induced by the map ${_g\phi }_1: {\mathcal{J}}_{n,\pi }^{cris}\rightarrow {\mathcal{O}}_{n,\pi }^{cris}$ is precisely the operator denoted ${_g\phi }_1$ earlier in this section (see (5.4.1)). Using this, together with Lemma 5.3.1, Lemma 5.4.4 (for $n=1$) and obvious functorialities, we obtain, after tensoring by $k[u]/u^{e\ell }$, the following corollary.

Corollary 5.4.5.

Let $g:R \rightarrow R$ be a continuous automorphism.

(1)

Let ${\mathcal{G}}$ be a finite flat group scheme over $R$, which is killed by $\ell$. Then there is a canonical isomorphism in the category ${\underline{\phi _1{\mathrm{{-mod}}}}}_R\!:$$$\begin{multline*} \Big ({\mathcal{M}}_{\pi }({\mathcal{G}})\otimes _{k[u]/u^{e\ell }, \hat{g}}k[u]/ u^{e\ell },{\mathcal{M}}_{\pi }({\mathcal{G}})_1\otimes _{k[u]/u^{e\ell },\hat{g}}k[u]/u^{e\ell },{_g\phi }_1\otimes \phi \Big )\\ \buildrel \sim \over \longrightarrow \Big ({\mathcal{M}}_{\pi }({^g{\mathcal{G}}}), {\mathcal{M}}_{\pi }({^g{\mathcal{G}}})_1,\phi _1\Big ). \end{multline*}$$

(2)

If $f:{\mathcal{G}}\rightarrow {\mathcal{G}}'$ is a morphism of finite flat $R$-group schemes killed by $\ell$ and ${\mathcal{M}}_{\pi }(f)$ is the corresponding morphism in ${\underline{\phi _1{\mathrm{{-mod}}}}}_R$, then ${\mathcal{M}}_{\pi }(f)$ also commutes with the ${_g\phi }_1$ and there is a commutative diagram in ${\underline{\phi _1{\mathrm{{-mod}}}}}_R:$$$\begin{equation*} \begin{matrix} {\mathcal{M}}_{\pi }({\mathcal{G}}')\otimes _{k[u]/u^{e\ell },\hat{g}}(k[u]/u^{e\ell })&\buildrel {\mathcal{M}}_{\pi }(f)\otimes 1 \over \longrightarrow &{\mathcal{M}}_{\pi }({\mathcal{G}})\otimes _{k[u]/u^{e\ell },\hat{g}}(k[u]/u^{e\ell })\\ {\scriptstyle \wr }\downarrow && \downarrow {\scriptstyle \wr }\\ {\mathcal{M}}_{\pi }({^g{\mathcal{G}}'})&\buildrel {\mathcal{M}}_{\pi }({^gf})\over \longrightarrow &{\mathcal{M}}_{\pi }({^g{\mathcal{G}}})\end{matrix} \end{equation*}$$

(3)

If $g_1,g_2$ are two continuous automorphisms of $R$ and if we choose the unique $H_{g_2g_1} \in W(k)[[u]]$ such that $\widehat{g_2g_1}=\widehat{g}_2 \circ \widehat{g}_1$ on $W(k)[[u]]$, then on$$\begin{equation*} ({\mathcal{M}}_{\pi }({\mathcal{G}})\otimes _{k[u]/u^{e\ell },\hat{g}_1}k[u]/u^{e\ell })\otimes _{k[u]/u^{e\ell },\hat{g}_2}k[u]/u^{e\ell } \simeq {\mathcal{M}}_{\pi }({\mathcal{G}})\otimes _{k[u]/u^{e\ell },\widehat{g_2g_1}} k[u]/u^{e\ell }, \end{equation*}$$

one has ${_{g_2}({_{g_1}\phi }_1\otimes \phi )}\otimes \phi ={_{g_2g_1}\phi }_1\otimes \phi$.

Corollary 5.4.6.

Let ${\mathcal{G}}$ be a finite flat group scheme over $R$, which is killed by $\ell$. To give a morphism of schemes $[g]: {\mathcal{G}}\rightarrow {\mathcal{G}}$ such that the diagram of schemes

$$\begin{equation*} \begin{matrix} {\mathcal{G}}&\buildrel [g] \over \longrightarrow &{\mathcal{G}}\\ \downarrow &&\downarrow \\ \operatorname {Spec}(R)&\buildrel \operatorname {Spec}(g)\over \longrightarrow &\operatorname {Spec}(R) \end{matrix} \end{equation*}$$

is commutative and the induced morphism ${\mathcal{G}}\rightarrow \operatorname {Spec}(R)\times _{g,\operatorname {Spec}(R)}{\mathcal{G}}$ is a morphism of group schemes over $R$, is equivalent to giving an additive map $\widehat{g}:{\mathcal{M}}_{\pi }({\mathcal{G}})\rightarrow {\mathcal{M}}_{\pi }({\mathcal{G}})$ such that both of the following hold:

(1)

For all $s\in k[u]/u^{e\ell }$ and $x\in {\mathcal{M}}_{\pi }({\mathcal{G}})$, $\widehat{g}(sx)=\widehat{g}(s)\widehat{g}(x)$.

(2)

$\widehat{g}({\mathcal{M}}_{\pi }({\mathcal{G}})_1)\subset {\mathcal{M}}_{\pi }({\mathcal{G}})_1$ and $\phi _1 \circ \widehat{g} = \widehat{g} \circ \phi _1 +\widehat{g}({_gt}(u))\widehat{g}\circ N\circ \phi _1$ with ${_gt}$ as in Lemma 5.4.3 and $N$ as in Lemma 5.1.2.

Proof.

Note that the last condition is equivalent to $\phi _1 \circ \widehat{g} = \widehat{g} \circ {_g\phi }_{1}$. The first two conditions are equivalent to giving a morphism $\widehat{g}:{\mathcal{M}}_{\pi }({^g{\mathcal{G}}})\rightarrow {\mathcal{M}}_{\pi }({\mathcal{G}})$ in ${\underline{\phi _1{\mathrm{{-mod}}}}}_R$, which is equivalent to the last two by Corollary 5.4.5.

■

Finally we make some computations that concern the dependence of the above compatibilities on the choice of $H_g(u)$. Let $f(u)$ be an element of $(k[u]/u^{e\ell })_1=u^e(k[u]/u^{e\ell })$ and define, for any ${\mathcal{M}}$ in ${\underline{\phi _1{\mathrm{{-mod}}}}}_R$, the additive map ${\mathbf{1}}_f:\phi _1 ({\operatorname {\mathcal{M}}}_1) \rightarrow {\operatorname {\mathcal{M}}}$ via

$$\begin{equation*} {\mathbf{1}}_{f}=1+\Big (\sum _{i=1}^{\ell -1}\frac{(-1)^{i-1}}{i}f(u)^i\Big )N, \end{equation*}$$

where $N$ is as in Lemma 5.1.2. Using $k[u]/u^{e\ell }\otimes _{k[u^\ell ]/u^{e\ell }}\phi _1({\mathcal{M}}_1)\simeq {\mathcal{M}}$, we extend ${\mathbf{1}}_{f}$ to all of $\mathcal{M}$ by the formula:

$$\begin{equation*} {\mathbf{1}}_{f}(u^i x)=u^i(1+f(u))^i{\mathbf{1}}_{f} (x) \end{equation*}$$

for $x \in \phi _1({\operatorname {\mathcal{M}}}_1)$. If $x\in {\mathcal{M}}_1$, one checks that:

$$\begin{equation*} {\mathbf{1}}_{f}(\phi _1(u^ix))=u^{i\ell }{\mathbf{1}}_{f}(\phi _1(x))={\mathbf{1}}_{f}(u^{i\ell }\phi _1(x)) \end{equation*}$$

so ${\mathbf{1}}_{f}$ is well defined. Moreover, it is clear that ${\mathbf{1}}_{f}({\mathcal{M}}_1)\subset {\mathcal{M}}_1$. Let

$$\begin{equation*} {\mathbf{1}}_{f}:{\mathcal{O}}_{1,\pi }^{cris}\buildrel \sim \over \longrightarrow {\mathcal{O}}_{1,\pi }^{cris} \end{equation*}$$

be the unique isomorphism of sheaves coming from the semi-linear isomorphism of presheaves:

$$\begin{equation*} \begin{matrix} (k[u]/u^{e\ell })\otimes _{\phi ,k}\Gamma ({\mathfrak{X}}_1, {\mathcal{O}}_{{\mathfrak{X}}_1})&\buildrel \sim \over \longrightarrow & (k[u]/u^{e\ell })\otimes _{\phi ,k}(\Gamma ({\mathfrak{X}}_1,{\mathcal{O}}_{{\mathfrak{X}}_1}))\\ s(u)\otimes (w_0,...,w_{n-1}) & \longmapsto &s(u(1+f(u)))\otimes (w_0,...,w_{n-1})\end{matrix} \end{equation*}$$

(see 5.3.1).

Let $\mathcal{G}$ be a finite flat group scheme over $R$ killed by $\ell$ and recall that

$$\begin{equation*} {\mathcal{M}}_{\pi }({\mathcal{G}})={\mathrm{Hom}}_{\mathrm{sheaves\ of\ groups}} ({\mathcal{G}},{\mathcal{O}}_{1,\pi }^{cris})\otimes k[u]/u^{e\ell }. \end{equation*}$$

Lemma 5.4.7.

The operator ${\mathbf{1}}_{f}$ on ${\mathcal{M}}_{\pi }({\mathcal{G}})$ is induced by the operator ${\mathbf{1}}_{f}$ on ${\mathcal{O}}_{1,\pi }^{cris}$.

Proof.

One can check that the operator ${\mathbf{1}}_{f}$ on ${\mathcal{O}}_{1,\pi }^{cris}$ satisfies ${\mathbf{1}}_{f}\circ \phi _1=\phi _1+ \log (1+f)N\circ \phi _1$, where $N$ is defined as in §5.3 and $\log (1+f)$ is the usual expansion of $\log$ in $S_1$, which makes sense because of the assumption that $u^e|f$ and because of the divided powers $\gamma _i(u^e)=\frac{u^{ei}}{i!}$. After tensoring with $k[u]/u^{e\ell }$, we get ${\mathbf{1}}_{f}={\mathbf{1}}+(\sum _{i=1}^{\ell -1}\frac{(-1)^{i-1}}{i}f(u)^i)N$ on $\phi _1({\mathcal{M}}_{\pi }({\mathcal{G}})_1)$ which clearly implies the two ${\mathbf{1}}_f$’s are the same.

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Let $g=1$ and choose $H_{g}(u)=1+f(u)$ for some $f\in E_{\pi }(u)W(k)[[u]]$ (see the start of §5.1 for the definition of $E_\pi (u)$). Recall from Corollary 5.4.5 that we have a canonical isomorphism ${\mathcal{M}}_{\pi }({\mathcal{G}})\otimes _{k[u]/u^{e\ell },\widehat{g}}(k[u]/u^{e\ell })\buildrel \sim \over \rightarrow {\mathcal{M}}_{\pi }({^g{\mathcal{G}}})$.

Lemma 5.4.8.

The map ${\mathbf{1}}_{f}$ is the composite ${\mathcal{M}}_{\pi }({\mathcal{G}})\buildrel \sim \over \rightarrow {\mathcal{M}}_{\pi }({^g{\mathcal{G}}})\buildrel \sim \over \rightarrow {\mathcal{M}}_{\pi }({\mathcal{G}})$, where the first map is the one in Corollary 5.4.5 and the second comes from the obvious isomorphism ${\mathcal{G}}\buildrel \sim \over \rightarrow {^g{\mathcal{G}}}$. In other words, once $H_g(u)=1+f(u)$ has been chosen, ${\mathbf{1}}_{f}:{\mathcal{M}}_{\pi }({\mathcal{G}})\rightarrow {\mathcal{M}}_{\pi }({\mathcal{G}})$ is the map corresponding to the identity ${\mathbf{1}}_{\mathcal{G}}:{\mathcal{G}}\rightarrow {\mathcal{G}}$ under the equivalence of Corollary 5.4.6.

The proof is straightforward by looking at the usual presheaves and using Lemma 5.4.7. We remark that ${\mathbf{1}}_f$ is not necessarily the identity even though $1_{\mathcal{G}}$ is. However, with $f=0$, ${\mathbf{1}}_f$ is the identity.

5.5. Reformulation

In this section, we will reformulate Corollary 5.4.6.

Lemma 5.5.1.

There is a unique element ${t_g}(u)\in W(k)[[u]]$ such that if ${\phi _g}$ is defined by ${\phi _g}(\sum w_iu^i)=\sum \phi (w_i)(u^\ell (1+\ell {t_g}(u)))^i$, one has $\widehat{g} \circ {\phi }=\phi _g \circ \widehat{g}$.

Proof.

One has to solve in $W(k)[[u]]$:

$$\begin{equation*} u^\ell H_g(u)^\ell =u^\ell (1+\ell t_g(u)){^{(\phi )}H_g}(u^\ell (1+\ell t_g(u))). \end{equation*}$$

As $H_{g}(u)\in W(k)[[u]]^{\times }$, there is a unique $L_{g}\in uW(k)[[u]]^{\times }$ such that $L_{g}(uH_{g}(u))=u$. Applying $L_g$ to $u=K_g(u)H_g(K_g(u))$ (cf. the proof of Lemma 5.4.3), we get $L_g(u)=K_g(u)$. We must solve:

$$\begin{equation*} 1+\ell t_g(u)=\frac{{^{(\phi )}K_g}(u^\ell H_g(u)^\ell )}{u^\ell }. \end{equation*}$$ ■

Lemma 5.5.2.

There is a unique ${\lambda }_g(u)\in 1+uW(k)[[u]]$ such that if $N_g={\lambda }_g N$, then $N_g \circ {\widehat{g}} =\widehat{g}\circ N$. Similarly, there is a unique ${_g\lambda }(u)\in 1+uW(k)[[u]]$ such that if ${_gN}={_g\lambda } N$, then $\widehat{g} \circ {_gN}={N}\circ \widehat{g}$. Moreover, $N_g \circ {\phi _g}=\ell {\phi _g}\circ {N_g}$ and ${_gN} \circ {_g\phi }=\ell {_g\phi }\circ {_gN}$.

Proof.

Since $N$ is a derivation, so is $\lambda N$ for any $\lambda \in W(k)[[u]]$. One has to solve $\lambda _g(u)N(uH_{g}(u))=uH_g(u)$ and ${_g^{(g)}\lambda }(uH_g(u))=1+\frac{N(H_g(u))}{H_g(u)}$, which amounts to:

$$\begin{eqnarray*} \lambda _g(u)&=&\Big (1+\frac{N(H_g(u))}{H_g(u)}\Big )^{-1},\\ {_g^{(g)}\lambda }(u)&=&1+\frac{N(H_g)(K_{g}(u))}{H_g(K_{g}(u))}, \end{eqnarray*}$$

where $K_{g}$ is as in the proof of Lemma 5.4.3. The commutation relations with the Frobenius follow from $N\circ \phi =\ell \phi \circ N$, $\phi _g \circ \widehat{g}=\widehat{g} \circ \phi$, $N_g \circ \widehat{g} = \widehat{g} \circ N$, $\widehat{g} \circ {_g\phi }=\phi \circ \widehat{g}$, $\widehat{g} \circ {_gN}=N \circ \widehat{g}$ and the fact $\widehat{g}$ is bijective on $W(k)[[u]]$.

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We also denote by $_{g}N={_{g}\lambda } N$ and $N_{g}={\lambda _{g}} N$ the corresponding derivations on $k[u]/u^{e\ell }$. For any object $\mathcal{M}$ of ${\underline{\phi _1{\mathrm{{-mod}}}}}_R$, define $\phi _{1,g}:{\mathcal{M}}_1\rightarrow {\mathcal{M}}$ by the formula

$$\begin{equation*} \phi _{1,g}(x)=\phi _1(x)+t_g(u)N(\phi _1(x)), \end{equation*}$$

where $N$ is as in Lemma 5.1.2, and we recall that we defined ${_g\phi }_{1}$ in (5.4.1). One checks that $\phi _{1,g}(u^e)={_g\phi }_{1}(u^e)=\phi _1(u^e)=c_{\pi }$ (see (5.1.1)). Note that we also have $\phi _{1,g} \circ \widehat{g}=\widehat{g} \circ {_g\phi }_1$, $N_g \circ \widehat{g} = \widehat{g} \circ N$, $\widehat{g} \circ {_g\phi }_{1}=\phi _1 \circ \widehat{g}$, $\widehat{g} \circ {_gN}=N \circ \widehat{g}$ in $k[u]/u^{e\ell }$.

Lemma 5.5.3.

Let $\mathcal{M}$ be an object of ${\underline{\phi _1{\mathrm{{-mod}}}}}_R;$ then there is a unique operator ${N_g}:{\mathcal{M}}\rightarrow {\mathcal{M}}$ satisfying the following three conditions:

(1)

$N_g(sx)={N_g}(s)x+s {N_g}(x)$, $s\in k[u]/u^{e\ell }$, $x\in {\mathcal{M}}$,

(2)

$N_g {\phi _{1,g}}(x)={\phi _g} {N_g}(x)$, $x\in {\mathcal{M}}_1$, where $\phi _g(y)=\frac{1}{c_{\pi }}{\phi _{1,g}}(u^ey)$ if $y\in {\mathcal{M}}$,

(3)

$N_g({\mathcal{M}})\subset u{\mathcal{M}}$.

The same statement holds for $_gN$, $_g\phi$ and $_g\phi _1$.

Proof.

The proof is the same as for Lemma 5.1.2, using the fact we still have isomorphisms

$$\begin{equation*} k[u]/u^{e\ell } \otimes _{k[u^\ell ]/u^{e\ell }}{\phi _{1,g}}({\mathcal{M}}_1)\buildrel \sim \over \rightarrow {\mathcal{M}} \end{equation*}$$

(resp. with ${_g\phi }_{1}$ replacing $\phi _{1,g}$).

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Lemma 5.5.4.

For ${\operatorname {\mathcal{M}}}$ an object of ${\underline{\phi _1{\mathrm{{-mod}}}}}_R$, $N_g={\lambda }_g N$ and ${_gN}={_g\lambda }N$, where $N_{g}$, $_{g}N$ are as in Lemma 5.5.3, $\lambda _{g}$, $_{g}\lambda$ as in Lemma 5.5.2 and $N$ as in Lemma 5.1.2.

Proof.

By unicity of $N_g$, one has to check ${\lambda }_g N$ satisfies the three conditions of Lemma 5.5.3. The first and last are obvious. Note that $N\phi _1(u^ex)=\phi N(u^ex)=0$ so $\phi _{1,g}(u^ex)=\phi _1(u^ex)$, which implies $\phi = {\phi _g}$ on $\mathcal{M}$ ($\phi _g$ is as in Lemma 5.5.3). One computes:

$$\begin{eqnarray*} ({\lambda }_g(u) N) \circ {\phi _{1,g}}&=&{\lambda _g}(u) (1+N(t_g(u)))\phi \circ N,\\ \phi _g \circ ({\lambda _g(u)} N)&=&{^{(\phi )}\lambda _g}(u^\ell )\phi \circ N. \end{eqnarray*}$$

But the equality $N_g\circ \phi _g (u)=\ell \phi _g\circ N_g(u)$ in $W(k)[[u]]$ (from Lemma 5.5.2) yields

$$\begin{equation*} \lambda _g(u)(1+N(t_g(u)))-{^{(\phi )}\lambda _g}(u^\ell )\in \ell W(k)[[u]]. \end{equation*}$$

We thus get $({\lambda _g} N) \circ {\phi _{1,g}}={\phi _g} \circ ({\lambda _g} N)$, hence condition (2). For ${_gN}$, the proof is completely similar.

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Lemma 5.5.5.

Let $\mathcal{M}$ be an object of ${\underline{\phi _1{\mathrm{{-mod}}}}}_R$ and $\widehat{g}: {\mathcal{M}}\rightarrow {\mathcal{M}}$ an additive map such that for all $s\in k[u]/u^{e\ell }$ and $x\in {\mathcal{M}}$, $\widehat{g}(sx)=\widehat{g}(s)\widehat{g}(x)$ and $\widehat{g}({\mathcal{M}}_1)\subset {\mathcal{M}}_1$. If $\widehat{g} \circ \phi _1 = {\phi _{1,g}} \circ \widehat{g}$, then $\widehat{g} \circ N = {N_g} \circ \widehat{g}$. Similarly, if $\phi _1 \circ \widehat{g}=\widehat{g} \circ {_g\phi }_{1}$, then $N \circ \widehat{g}=\widehat{g} \circ {_gN}$.

Proof.

We prove the first case, the other one being the same. As in the proof of Lemma 5.1.2, we define $N_{g,0}$, $N_{g,1}$,..., with $N_g={N_{g,i}}$ for $i$ large enough, using $k[u]/u^{e\ell }\otimes _{k[u^\ell ]/u^{e\ell }}{\phi _{1,g}}({\mathcal{M}}_1)\buildrel \sim \over \rightarrow {\mathcal{M}}$. It is enough to show $\widehat{g} \circ N_i={N_{g,i}}\circ \widehat{g}$ for all $i$. Suppose $\widehat{g} \circ N_{i-1}={N_{g,i-1}}\circ \widehat{g}$ and let $s\in k[u]/u^{e\ell }$ and $x\in {\mathcal{M}}_1$. Then

$$\begin{eqnarray*} {N_{g,i}}\widehat{g}(s\phi _1(x))&=&{N_{g,i}}(\widehat{g}(s)\phi _{1,g}(\widehat{g}(x)))\\ &=&{N_g}(\widehat{g}(s))\phi _{1,g}(\widehat{g}(x))+\widehat{g}(s) {\phi _{1,g}}{N}_{g,i-1}(\widehat{g}(x))\\ &=&\widehat{g}(N(s)\phi _1(x))+\widehat{g}(s)\phi _{1,g}\widehat{g} (N_{i-1}(x))\\ &=&\widehat{g}(N(s)\phi _1(x))+\widehat{g}(s\phi _1(N_{i-1}(x)))\\ &=&\widehat{g} N_i(s\phi _1(x)), \end{eqnarray*}$$

so $\widehat{g} \circ N_i={N}_{g,i}\circ \widehat{g}$ by linearity. One easily checks by a similar computation that $N_{g,0}\circ \widehat{g}=\widehat{g} \circ N_{0}$, hence the result follows by induction.

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Lemma 5.5.6.

Let $\mathcal{M}$ be an object of ${\underline{\phi _1{\mathrm{{-mod}}}}}_R$ and $\widehat{g}: {\mathcal{M}}\rightarrow {\mathcal{M}}$ an additive map such that for all $s\in k[u]/u^{e\ell }$ and $x\in {\mathcal{M}}$, $\widehat{g}(sx)=\widehat{g}(s)\widehat{g}(x)$ and $\widehat{g}({\mathcal{M}}_1)\subset {\mathcal{M}}_1$. Then the following two conditions are equivalent:

(1)

$\phi _1 \circ \widehat{g} = \widehat{g} \circ \phi _1 +\widehat{g}({_gt}(u))\widehat{g}\circ N\circ \phi _1$,

(2)

and $\widehat{g} \circ \phi _1 = \phi _1\circ \widehat{g} +t_g(u)N\circ \phi _1\circ \widehat{g}$.

Proof.

One has to show $\phi _1 \circ \widehat{g}=\widehat{g} \circ {_g\phi }_{1}$ is equivalent to $\widehat{g} \circ \phi _1 = {\phi _{1,g}} \circ \widehat{g}$. We prove (1) $\Rightarrow$ (2), the other case being the same. On $\mathcal{M}$, we have $\widehat{g}\circ \phi =\phi \circ \widehat{g}$, because $\phi =\phi _g={_g\phi }$, as in the proof of Lemma 5.5.4. By Lemmas 5.5.4 and 5.5.5, we have $\widehat{g} \circ N=\widehat{g}({_g\lambda }^{-1})N\circ \widehat{g}$. Thus we get from (1), using $N\phi _1=\phi N$,

$$\begin{equation*} \widehat{g} \circ \phi _1=\phi _1 \circ \widehat{g}-\widehat{g}({_gt}(u))\widehat{g}({_g\lambda }(u)^{-1})^\ell N\circ \phi _1 \circ \widehat{g}. \end{equation*}$$

Playing the same game over $W_2(k)[[u]]$ with the relation $\phi \circ \widehat{g} =\widehat{g}\circ \phi + \widehat{g}({_gt}(u)) \widehat{g}\circ N\circ \phi$, which is easily checked to hold in $W_2(k)[[u]]$, we again end up with $\widehat{g} \circ \phi =\phi \circ \widehat{g} -\widehat{g}({_gt}(u))\widehat{g}({_g\lambda }(u)^{-1})^\ell N\circ \phi \circ \widehat{g}$ in $W_2(k)[[u]]$. But we also have in $W_2(k)[[u]]$ the equality:

$$\begin{equation*} \widehat{g} \circ \phi =\phi \circ \widehat{g} + t_g(u)N\circ \phi \circ \widehat{g}. \end{equation*}$$

Thus $-\widehat{g}({_gt}(u))\widehat{g}({_g\lambda }(u)^{-1})^\ell =t_g(u)$ in $k[u]/u^{e\ell }$, so relation (2) holds.

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We can now derive the variant of Corollary 5.4.6 which we will use.

Corollary 5.5.7.

Let ${\mathcal{G}}$ be a finite flat $R$-group scheme killed by $\ell$. Let $g:R\buildrel \sim \over \rightarrow R$ be a continuous automorphism, choose $H_g(u)\in W(k)[[u]]$ such that $g(\pi )=\pi H_g(\pi )$ and define $\widehat{g}:k[u]/u^{e\ell }\rightarrow k[u]/u^{e\ell }$ by $\widehat{g}(\sum w_iu^i)=\sum g(w_i)u^iH_g(u)^i$. To give a morphism of schemes $[g]: {\mathcal{G}}\rightarrow {\mathcal{G}}$ such that the diagram of schemes

$$\begin{equation*} \begin{matrix} {\mathcal{G}}&\buildrel [g] \over \longrightarrow &{\mathcal{G}}\\ \downarrow &&\downarrow \\ \operatorname {Spec}(R)&\buildrel \operatorname {Spec}(g)\over \longrightarrow &\operatorname {Spec}(R) \end{matrix} \end{equation*}$$

is commutative and the induced morphism ${\mathcal{G}}\rightarrow \operatorname {Spec}(R)\times _{g,\operatorname {Spec}(R)}{\mathcal{G}}$ is an morphism of group schemes over $R$, is equivalent to giving an additive map $\widehat{g}:{\mathcal{M}}_{\pi }({\mathcal{G}})\rightarrow {\mathcal{M}}_{\pi }({\mathcal{G}})$ such that both of the following hold:

(1)

For all $s\in k[u]/u^{e\ell }$ and $x\in {\mathcal{M}}_{\pi }({\mathcal{G}})$, $\widehat{g}(sx)=\widehat{g}(s)\widehat{g}(x)$.

(2)

$\widehat{g}({\mathcal{M}}_{\pi }({\mathcal{G}})_1)\subset {\mathcal{M}}_{\pi }({\mathcal{G}})_1$ and $\widehat{g} \circ \phi _1=(1+{t_g}(u)N)\circ \phi _1\circ \widehat{g}$, with ${t_g}$ as in Lemma 5.5.1 and $N$ as in Lemma 5.1.2.

Moreover, $[g]$ is an isomorphism if and only if $\widehat{g}$ is. Assume these are isomorphisms. Choose $H_{g^{-1}}$ such that $\widehat{g^{-1}}(u)= \widehat{g}^{-1}(u)$ on $W(k)[[u]]$, i.e. $H_{g^{-1}}(u)= \widehat{g}^{-1}(u)/ u$. Then the map $\widehat{g^{-1}}$ that corresponds to $[g]^{-1}$ is equal to $\widehat{g}^{-1}$. Also, if $g_1$, $g_2$ are two automorphisms of $R$ and if we choose $H_{g_1}, H_{g_2}$ as above, then $[g_1]\circ [g_2]$ corresponds to $\widehat{g}_2\circ \widehat{g}_1$ provided we choose $H_{g_2 g_1}$ such that $\widehat{g}_2(\widehat{g}_1(u))=uH_{g_2 g_1}(u)$.

Proof.

The equivalence is clear thanks to Corollary 5.4.6 and Lemma 5.5.6. The fact that $[g_1]\circ [g_2]$ corresponds to $\widehat{g}_2\circ \widehat{g}_1$ is automatic using Corollary 5.4.5 and the functor ${\mathcal{G}}\mapsto {\mathcal{M}}_{\pi }({\mathcal{G}})$. Applying this to $g_1=g$ and $g_2=g^{-1}$, we see that $1_{\mathcal{G}}= [g] \circ [g]^{-1}$ corresponds to $\widehat{g^{-1}} \circ \widehat{g}$. But by Lemma 5.4.8, $1_{\mathcal{G}}$ corresponds to ${\mathbf{1}}_f$ with $f$ defined by $(\widehat{g^{-1}} \circ \widehat{g})(u) = u(1+f)$ in $W(k)[[u]]$. We see that $f=0$ and that ${\mathbf{1}}_f$ is the identity on ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}})$. Thus $\widehat{g^{-1}}=\widehat{g}^{-1}$ on ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}})$.

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5.6. Descent data

Assume now that $R$ is endowed with a continuous left faithful action of a finite group $\Gamma$. Then $\Gamma$ becomes the Galois group of the fraction field $F'$ of $R$ over some subfield. For each $g \in \Gamma$, choose $H_{g}(u) \in W(k)[u]$ so that $g(\pi ) = \pi H_{g}(\pi )$, with the one condition that $H_{1}(u) = 1$. Recall from Lemma 5.5.1 that this uniquely determines elements $t_g(u)\in W(k)[[u]]$ such that

$$\begin{equation*} u^\ell H_g(u)^\ell =u^\ell (1+\ell t_g(u)){^{(\phi )}H_g}(u^\ell (1+\ell t_g(u))). \end{equation*}$$

Moreover, for any pair $g_1$, $g_2 \in \Gamma$, there is obviously a unique $f_{g_1,g_2}(u) \in E_{\pi }(u)W(k)[[u]]$ such that

$$\begin{equation*} \widehat{g}_1 \circ \widehat{g}_2(u) = \widehat{g_1 \circ g_2}(u(1+f_{g_1,g_2}(u))). \end{equation*}$$

If ${\operatorname {\mathcal{M}}}$ is an object of ${\underline{\phi _1{\mathrm{{-mod}}}}}_R$, then we will denote by ${\mathbf{1}}_{g_1,g_2}$ the unique $k$-linear map ${\operatorname {\mathcal{M}}}\rightarrow {\operatorname {\mathcal{M}}}$ such that for $x\in {\mathcal{M}}_1$ we have

•

${\mathbf{1}}_{g_1,g_2}(\phi _1(x))=\Big (1+(\sum _{i=1}^{\ell -1} \frac{(-1)^{i-1}}{i} f_{g_1,g_2}(u)^i)N\Big )(\phi _1(x))$, where $N$ is as in Lemma 5.1.2,

•

and ${\mathbf{1}}_{g_1,g_2}(u^i\phi _1(x))=u^i(1+f_{g_1,g_2}(u))^i {\mathbf{1}}_{g_1,g_2}(\phi _1(x)).$

(See §5.4 where we denoted ${\mathbf{1}}_{g_1,g_2}$ by ${\mathbf{1}}_{f_{g_1,g_2}}$.)

Suppose that ${\mathcal{G}}$ is a finite flat $R$-group scheme. Recall that by descent data on ${\mathcal{G}}$ for $\Gamma$ we mean isomorphisms of finite flat group schemes

$$\begin{equation*} [g]: {\mathcal{G}}\stackrel{\sim }{\rightarrow }{^g {\mathcal{G}}} \end{equation*}$$

for $g \in \Gamma$, such that

$$\begin{equation*} [gh]= ({^g[h]}) \circ [g] \end{equation*}$$

for all $g,h \in \Gamma$. Equivalently we may think of $[g]$ as a map of schemes ${\mathcal{G}}\rightarrow {\mathcal{G}}$ over $g^*:\operatorname {Spec}R \rightarrow \operatorname {Spec}R$ which induces an isomorphism of group schemes ${\mathcal{G}}\rightarrow {^g {\mathcal{G}}}$. In this picture the compatibility condition simply becomes

$$\begin{equation*} [gh]=[h][g]. \end{equation*}$$

Theorem 5.6.1.

Suppose that ${\mathcal{G}}$ is a finite flat $R$-group scheme killed by $\ell$. Fix $H_g(u)$ as above for all $g \in \Gamma$.

(1)

To give descent data on ${\mathcal{G}}$ relative to $\Gamma$ is equivalent to giving additive bijections $\widehat{g}:{\mathcal{M}}_{\pi }({\mathcal{G}}) \rightarrow {\mathcal{M}}_{\pi }({\mathcal{G}})$ for all $g \in \Gamma$ so that $\widehat{g}$ takes ${\mathcal{M}}_{\pi }({\mathcal{G}})_1$ into ${\mathcal{M}}_{\pi }({\mathcal{G}})_1$ and:

•

$\widehat{g}(w u^i m) = g(w) (uH_{g}(u))^i \widehat{g}(m)$ for $m \in {\mathcal{M}}_{\pi }({\mathcal{G}})$, $w \in k$,

•

$\widehat{g}\circ \phi _1 = (1 + t_{g}(u)N) \circ \phi _1 \circ \widehat{g}$ on ${\mathcal{M}}_{\pi }({\mathcal{G}})_1$,

•

$\widehat{1}_{\Gamma } = 1$ and $\widehat{g}_1 \circ \widehat{g}_2 = \widehat{g_1g_2} \circ {\mathbf{{1}}}_{g_1,g_2}$.

(2)

The above equivalence is functorial in ${\mathcal{G}}$ and is compatible with classical Dieudonné theory in the following sense: if the action $\{\widehat{g}\}_{g \in \Gamma }$ on ${\mathcal{M}}_{\pi }({\mathcal{G}})$ corresponds to descent data $\{[g]\}$ on ${\mathcal{G}}$, then the $g$-semi-linear map $D([g])$ induced on the contravariant Dieudonné module $D({\mathcal{G}})$ and the $g$-semi-linear map $\widehat{g} \bmod u$ induced on ${\mathcal{M}}_{\pi }({\mathcal{G}})/u{\mathcal{M}}_{\pi }({\mathcal{G}})$ are compatible via the isomorphism of Theorem 5.1.3.

Proof.

Part (1) is a consequence of Corollary 5.5.7, Lemma 5.4.8 and the choice $H_1=1$. The functoriality in (2) follows from Corollary 5.4.5, and the last statement there comes from ${^g({\mathcal{G}} \times k)}\cong {^g{\mathcal{G}}} \times k$, the functoriality of the isomorphism in Theorem 5.1.3 and the reduction modulo $u$ of Corollary 5.4.5.

■

Suppose that $\pi \in R^{\Gamma }$. Then we may take $H_g(u)=1$ for all $g \in \Gamma$. With this choice we see that $t_{g_1}=0$, $f_{g_1,g_2}=0$ and ${\mathbf{1}}_{g_1,g_2}=1$ for all $g_1,g_2 \in \Gamma$. In this case to give bijections $\widehat{g}: {\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}) \rightarrow {\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}})$ as in the lemma is equivalent to giving an $R$-semi-linear $\Gamma$-action on ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}})$ which commutes with $u$ and $\phi _1$ and preserves ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}})_1$. Thus $({\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}})^{\Gamma }, {\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}})_1^\Gamma , \phi _1)$ is a Breuil module over $R^{\Gamma }$ from which we can recover ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}})$ by tensoring with $W(k)$ over $W(k^{\Gamma })$. In other words, étale descent for group schemes translates in the obvious manner for Breuil modules if we choose $\pi$ to be $\Gamma$-invariant.

To build an action of $\Gamma$ on $\mathcal{G}$ using Theorem 5.6.1, the conditions $\widehat{g}_1 \circ \widehat{g}_2 = \widehat{g_1g_2} \circ {\mathbf{{1}}}_{g_1,g_2}$ are not very convenient to check in practice since there are too many of them. It is useful to have the following variant. Choose $d \in \mathbf{Z}_{>0}$ and a group surjection $\theta : \Gamma _d\rightarrow \Gamma$, where $\Gamma _d$ is the free group on $d$ generators $\gamma _1,...,\gamma _d$. The group $\Gamma _d$ still acts on $R$ (via its quotient $\Gamma$) and for each $i\in \{1,...,d\}$, choose elements $H_{\gamma _i}(u)\in W(k)[[u]]$ such that $\pi H_{\gamma _i}(\pi )=\gamma _i(\pi )$. This determines isomorphisms $\widehat{\gamma }_i$ on $W(k)[[u]]$ and $k[u]/u^{e\ell }$ and, by composition, isomorphisms $\widehat{\gamma }$ for all $\gamma \in \Gamma _d$. Note that if $\gamma \in \ker (\theta )$, then $H_{\gamma }(u)=u(1+f_{\gamma }(u))$ for some $f_{\gamma }\in E_{\pi }(u)W(k)[[u]]$. For such $\gamma$, denote by ${\mathbf{1}}_{\gamma }$ the unique $k$-vector space endomorphism of any object ${\operatorname {\mathcal{M}}}$ of ${\underline{\phi _1{\mathrm{{-mod}}}}}_R$ such that for $x\in {\mathcal{M}}_1$ we have

•

${\mathbf{1}}_{\gamma }=1+\Big (\sum _{i=1}^{\ell -1} \frac{(-1)^{i-1}}{i} f_{\gamma }(u)^i\Big ) N$ on the image of $\phi _1$,

•

and ${\mathbf{1}}_{\gamma }(u^i\phi _1(x))=u^i(1+f_{\gamma }(u))^i {\mathbf{1}}_{\gamma }(\phi _1(x)),$

where $N$ is as in Lemma 5.1.2. (See §5.4, where we denoted ${\mathbf{1}}_{\gamma }$ by ${\mathbf{1}}_{f_{\gamma }}$.) Let $\mathcal{R}$ be a subset of $\ker (\theta )$ such that $\ker (\theta )$ is the smallest normal subgroup of $\Gamma _d$ containing $\mathcal{R}$.

Corollary 5.6.2.

With the above notation, to give descent data on $\mathcal{G}$ for $\Gamma$ is equivalent to giving additive bijections $\widehat{\gamma _j}:{\mathcal{M}}_{\pi }({\mathcal{G}}) \rightarrow {\mathcal{M}}_{\pi }({\mathcal{G}})$ for $j\in \{1,...,d\}$ so that $\widehat{\gamma _j}$ takes ${\mathcal{M}}_{\pi }({\mathcal{G}})_1$ into ${\mathcal{M}}_{\pi }({\mathcal{G}})_1$ and:

•

$\widehat{\gamma _j}(w u^i m) = \gamma _j(w) (uH_{\gamma _j}(u))^i \widehat{\gamma _j}(m)$ for $m \in {\mathcal{M}}_{\pi }({\mathcal{G}})$, $w \in k$,

•

$\widehat{\gamma _j}\circ \phi _1 = (1 + t_{\gamma _j}(u)N) \circ \phi _1 \circ \widehat{\gamma _j}$ on ${\mathcal{M}}_{\pi }({\mathcal{G}})_1$,

•

if $\gamma = \gamma _{i_1}^{n_1} \cdot \dots \cdot \gamma _{i_m}^{n_m}\in {\mathcal{R}}$, where $i_j \in \{1,\dots ,d\}$, $n_j \in {\mathbf{Z}}$, and $i_j \ne i_{j+1}$ for $1 \le j < m$, and if we define $\widehat{\gamma } = \widehat{\gamma }_{i_1}^{n_1} \circ \dots \circ \widehat{\gamma }_{i_m}^{n_m}$, then $\widehat{\gamma }={\mathbf{1}}_{\gamma }$.

Proof.

Straightforward from Corollary 5.5.7 and Lemma 5.4.8.

■

We define a category ${\underline{\phi _1DD}}_{F'/(F')^\Gamma }$ of Breuil modules with descent data for $\Gamma$ in the obvious way. This category is additive but not necessarily abelian. We call a complex in ${\underline{\phi _1DD}}_{F'/(F')^\Gamma }$ exact if the underlying complex in ${\underline{\phi _1{\mathrm{{-mod}}}}}_R$ is exact. In the natural way, we extend ${\operatorname {\mathcal{M}}}_\pi$ to a functor from ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/(F')^\Gamma }$ to ${\underline{\phi _1DD}}_{F'/(F')^\Gamma }$.

5.7. More examples

In this section we will determine the possible descent data on a rank one Breuil module. Let $\Gamma$ be as in §5.6.

Lemma 5.7.1.

Suppose that ${\mathcal{G}}$ is a finite flat $R$-group scheme of order $\ell$ and that its generic fibre admits descent data over $(F')^\Gamma$. Then there is unique descent data on ${\mathcal{G}}$ over $(F')^\Gamma$ extending any choice of descent data on ${\mathcal{G}}\times F'$ over $(F')^\Gamma$. If ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}) \cong {\operatorname {\mathcal{M}}}(r,a)$ and if $\gamma \in \Gamma$ satisfies $\gamma (\pi )/\pi \equiv 1 \bmod (\pi )$, then

$$\begin{equation*} \widehat{\gamma }(\mathbf{e})=H_\gamma (u)^{-r\ell /(\ell -1)} \mathbf{e}, \end{equation*}$$

where $H_\gamma (u)^{-r\ell /(\ell -1)}$ denotes the unique $(\ell -1)^{th}$ root of $H_\gamma (u)^{-r\ell }$ in $k[u]/u^{e\ell }$ with constant term $1$.

We remark that since $\operatorname {Aut}({\operatorname {\mathcal{M}}}(r,a))=(\mathbf{Z}/\ell \mathbf{Z})^\times$ by consideration of the geometric generic fibre, the choice of isomorphism ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}) \cong {\operatorname {\mathcal{M}}}(r,a)$ does not matter.

Proof.

We first claim two such finite flat group schemes ${\mathcal{G}}$ and ${\mathcal{G}}'$ have isomorphic generic fibres if and only if there is a non-zero morphism ${\mathcal{G}}\rightarrow {\mathcal{G}}'$ or ${\mathcal{G}}' \rightarrow {\mathcal{G}}$. By Lemma 5.2.1 we see that if $G$ is a finite flat $F'$-group scheme, then the lattice of models for $G$ over $R$ is well ordered. Suppose all the integral models are ${\mathcal{G}}_1 < \cdots < {\mathcal{G}}_n$. For $\gamma \in \Gamma$, any isomorphism $[\gamma ]:G \stackrel{\sim }{\rightarrow }{^\gamma G}$ must then induce isomorphisms $[\gamma ]:{\mathcal{G}}_i \stackrel{\sim }{\rightarrow }{^\gamma {\mathcal{G}}_i}$ for all $i=1,...,n$. The first part of the lemma follows.

Let $\operatorname {\mathcal{M}}= \operatorname {\mathcal{M}}(r,a)$, so $\operatorname {\mathcal{M}}$ is a free $k[u]/u^{e\ell }$-module of rank 1 with the usual basis element $\mathbf{e}$. The submodule $\operatorname {\mathcal{M}}_1$ is spanned by $u^r \mathbf{e}$ and $\phi _1(u^r \mathbf{e}) = a \mathbf{e}$. From Theorem 5.2.1, we have $N \circ \phi _1 = 0$, which implies that

$$\begin{equation*} \widehat{\gamma } \circ \phi _1 = \phi _1 \circ \widehat{\gamma }. \end{equation*}$$

For $\gamma \in \Gamma _1$, $H_{\gamma }(0) \equiv 1 \bmod \ell$. Clearly

$$\begin{equation*} \widehat{\gamma }: c u^i \mathbf{e}\longmapsto c u^i H_{\gamma }(u)^i \widehat{\gamma }(\mathbf{e}) \end{equation*}$$

is a bijection if and only if $\widehat{\gamma }(\mathbf{e}) = \xi _{\gamma } \mathbf{e}$ for some unit $\xi _{\gamma } \in (k[u]/u^{e\ell })^{\times }$. Evaluating $\widehat{\gamma } \circ \phi '_1 = \phi '_1 \circ \widehat{\gamma }$ on the element $u^r\mathbf{e}\in \operatorname {\mathcal{M}}_1$, we get

$$\begin{equation*} \xi _{\gamma } = H_{\gamma }(u)^{r\ell }\xi _{\gamma }^\ell \end{equation*}$$

in $k[u]/u^{e\ell }$. Thus,

$$\begin{equation*} \xi _{\gamma } = \epsilon _{\gamma } H_{\gamma }^{-r\ell /(\ell -1)} \end{equation*}$$

for some unit $\epsilon _{\gamma } \in \mathbf{F}_\ell ^{\times }$.

Since Breuil module descent data always induces a $k$-linear action of the inertia group on the $k$-vector space $\operatorname {\mathcal{M}}/u\operatorname {\mathcal{M}}$ and in this case $\dim _{k} \operatorname {\mathcal{M}}/u\operatorname {\mathcal{M}}= 1$, the action of the element $\gamma$ of $\ell$-power order on $\operatorname {\mathcal{M}}/u\operatorname {\mathcal{M}}$ must be trivial. Thus $\epsilon _{\gamma } = 1$.

■

6. Some local fields

In order to apply the methods of §5, we need some more explicit information about the fields $F'$ introduced in §4. In this section we will collect this essentially elementary information. In each case we will give an explicit description of the Galois group $\operatorname {Gal}(F'/\mathbf{Q}_3)$. This is needed to carry out the delicate Breuil module calculations in subsequent sections. We will also specify a uniformiser $\pi$ of $F'$ and partially calculate the following polynomials and power series (depending on our choice of $\pi$).

•

$G(u)\in W(k_{F'})[u]$, a polynomial of degree at most $e(F'/\mathbf{Q}_3)-1$ such that $\pi$ has minimal polynomial $u^{e(F'/\mathbf{Q}_3)}-3G(u)$ over $\mathbf{Q}_3$.

•

$c_\pi \equiv -G(u)^3 \bmod (3,u^{3e(F'/\mathbf{Q}_3)})$.

•

For $\gamma \in \operatorname {Gal}(F'/\mathbf{Q}_3)$, the unique polynomial $H_\gamma (u)\in W(k_{F'})[u]$ of degree at most $e(F'/\mathbf{Q}_3)-1$ such that $\gamma (\pi )/\pi = H_\gamma (\pi )$.

•

In some cases power series $t_\gamma$ and $f_{\gamma ,\gamma '}$ as in §5.6.

6.1. The case of $F_1'$

Recall that $\tau _{1}$ corresponds to the order $3$ homomorphism

$$\begin{equation*} \mathbf{Z}_3^\times \longrightarrow \operatorname {GL}_2(\overline{\mathbf{Q}}_3) \end{equation*}$$

is determined by

$$\begin{equation*} \begin{array}{rcl} -1 & \longmapsto & 1 \\4 & \longmapsto & \zeta , \end{array} \end{equation*}$$

where $\det \zeta =1$ and $\zeta ^3=1$ but $\zeta \neq 1$. Recall also that $F_1'=F_1$ is any totally ramified cubic Galois extension of $\mathbf{Q}_3$. We may take $F_1'=F_1= \mathbf{Q}_3[\pi ]$, where $\pi$ is a root of $X^3-3X^2+3$. One may check that the other roots of $X^3-3X^2+3$ are $\pi ^2-2\pi$ and $3+\pi -\pi ^2$, so $\operatorname {Gal}(F_1'/\mathbf{Q}_3)$ is generated by one element $\gamma _3$, which sends $\pi$ to $\pi ^2-2\pi$ and satisfies $\gamma _3^3=1$. Also, $\pi$ is a uniformiser for $F_1'$, so

•

$G(u)=u^2-1$,

•

$c_\pi \equiv 1-u^6 \bmod (3,u^9)$,

•

$H_{\gamma _3}(u)=u-2$.

6.2. The case of $F_{-1}'$

Recall that $\tau _{-1}$ corresponds to the order $3$ homomorphism

$$\begin{equation*} \mathbf{Z}_3[\sqrt {-1}]^\times \longrightarrow GL_2(\overline{\mathbf{Q}}_3) \end{equation*}$$

determined by

$$\begin{equation*} \begin{array}{rcl} \sqrt [4]{-1} & \longmapsto & 1 \\4 & \longmapsto & 1 \\1+3\sqrt {-1} & \longmapsto & \zeta , \end{array} \end{equation*}$$

where $\det \zeta =1$ and $\zeta ^3=1$ but $\zeta \neq 1$. Recall also that $F_{-1}'/ \mathbf{Q}_3(\sqrt {-1})$ is the unique cubic extension such that $F_{-1}'/\mathbf{Q}_3$ is Galois but not abelian and that $F_{-1}$ is any cubic subfield. We may take $F_{-1}=\mathbf{Q}_3(\pi )$ and $F_{-1}'=F_{-1}(\sqrt {-1})$, where $\pi$ is a root of $X^3-3X^2+6$. The other roots of $X^3-3X^2+6$ are $(\sqrt {-1} \pi ^2 -\pi +3(1-\sqrt {-1}))/2$ and $(-\sqrt {-1} \pi ^2 -\pi +3(1+\sqrt {-1}))/2$. Thus, $\operatorname {Gal}(F_{-1}'/\mathbf{Q}_3)$ is generated by two elements $\gamma _2$ and $\gamma _3$ defined by

•

$\gamma _2(\pi )=\pi$,

•

$\gamma _2(\sqrt {-1})=-\sqrt {-1}$,

•

$\gamma _3(\pi )=(\sqrt {-1} \pi ^2 -\pi +3(1-\sqrt {-1}))/2$,

•

and $\gamma _3(\sqrt {-1})=\sqrt {-1}$.

We have $\gamma _2^2=\gamma _3^3=1$ and $\gamma _2\gamma _3=\gamma _3^2\gamma _2$, and $\pi$ is a uniformiser for $F_{-1}'$. Thus

•

$G(u)=u^2-2$,

•

$c_\pi \equiv -1-u^6 \bmod (3, u^9)$,

•

$H_{\gamma _2}(u)=1$,

•

$H_{\gamma _3}(u)=((\sqrt {-1}-1)u^2+(3-\sqrt {-1})u-2)/4$.

6.3. The case of $F_3'$

Recall that $\tau _3$ is the unique $3$-type such that $\tau _{3}|_{I_{\mathbf{Q}_3(\sqrt {3})}}$ corresponds to the order $6$ homomorphism

$$\begin{equation*} \mathbf{Z}_3[\sqrt {3}]^\times \longrightarrow GL_2(\overline{\mathbf{Q}}_3) \end{equation*}$$

determined by

$$\begin{equation*} \begin{array}{rcl} -1 & \longmapsto & -1 \\4 & \longmapsto & 1 \\1+\sqrt {3} & \longmapsto & \zeta , \end{array} \end{equation*}$$

where $\det \zeta =1$ and $\zeta ^3=1$ but $\zeta \neq 1$. Recall also that $F_3'$ is the degree $12$ abelian extension of $\mathbf{Q}_3(\sqrt {3})$ with norm subgroup in $\mathbf{Q}_3(\sqrt {3})^\times$ topologically generated by $3$, $4$ and $1+3\sqrt {3}$. We also let $\gamma _4^2$ denote the unique element of $I_{F_3'/\mathbf{Q}_3(\sqrt {3})}$ of order $3$ and we let $F_3$ denote the fixed field of some Frobenius lift of order $2$.

We claim that $F_3'= \mathbf{Q}_3(\sqrt {3})(\sqrt {-1},\alpha ,\beta )$, where $\alpha$ is a root of $X^3-3X+3$ and $\beta$ a root of $X^2-\sqrt {3}$. To verify this, set $F''=\mathbf{Q}_3(\sqrt {3})(\sqrt {-1},\alpha ,\beta )$. We must check that $F''/\mathbf{Q}_3(\sqrt {3})$ is abelian and that $N_{F''/\mathbf{Q}_3(\sqrt {3})}(F'')^\times$ contains $3$, $4$, and $1+3\sqrt {3}$. To see that $F''/\mathbf{Q}_3(\sqrt {3})$ is abelian, note that if $\alpha$ is one root of $X^3-3X+3$, then the other roots are $(2\sqrt {3}\alpha ^2-(-3\sqrt {3}+\sqrt {-5})\alpha -4\sqrt {3})/2 \sqrt {-5}$ and $(-2\sqrt {3}\alpha ^2-(3\sqrt {3}+\sqrt {-5})\alpha +4\sqrt {3})/2 \sqrt {-5}$ (where for definiteness we choose $\sqrt {-5} \in 1+3 \mathbf{Z}_3$). Note that $N_{F''/\mathbf{Q}_3(\sqrt {3})}(\alpha /\beta )=3$ and $N_{F''/\mathbf{Q}_3(\sqrt {3})} (1+\alpha )=5^4$.

Note that $\operatorname {Gal}(F_3'/\mathbf{Q}_3(\sqrt {3}))$ is generated by three commuting elements $\gamma _2$, $\gamma _4^2$ and $\gamma _3$ of respective orders $2$, $2$ and $3$. They may be defined by

•

$\gamma _2 \sqrt {-1} = -\sqrt {-1}$, $\gamma _2 \beta = \beta$ and $\gamma _2 \alpha =\alpha$;

•

$\gamma _4^2 \sqrt {-1} = \sqrt {-1}$, $\gamma _4^2 \beta = -\beta$ and $\gamma _4^2 \alpha =\alpha$;

•

$\gamma _3 \sqrt {-1} = \sqrt {-1}$, $\gamma _3 \beta = \beta$ and $\gamma _3 \alpha =(-2\sqrt {3}\alpha ^2-(3\sqrt {3}+\sqrt {-5})\alpha +4\sqrt {3})/2 \sqrt {-5}$.

Choose an element $\gamma \in I_{F_3'/\mathbf{Q}_3}-I_{F_3'/\mathbf{Q}_3(\sqrt {3})}$. Then $\gamma ^2 \in \langle \gamma _4^2, \gamma _3 \rangle$. As $\gamma \gamma _3 \gamma ^{-1} = \gamma _3^2$ we may alter our choice of $\gamma$ so that $\gamma ^2 \in \langle \gamma _4^2 \rangle$. As $\gamma \sqrt {3}=-\sqrt {3}$ we see that $\gamma \beta = \pm \sqrt {-1} \beta$, so $\gamma ^2=\gamma _4^2$. We will rename $\gamma$ as $\gamma _4$ and suppose it chosen so that $\gamma _4 \beta = \sqrt {-1}\beta$. Thus, $\operatorname {Gal}(F_3'/\mathbf{Q}_3)$ is generated by elements $\gamma _2$, $\gamma _3$ and $\gamma _4$ satisfying

•

$\gamma _2^2=\gamma _3^3=\gamma _4^4=1$,

•

$\gamma _2\gamma _3=\gamma _3 \gamma _2$,

•

$\gamma _4 \gamma _2 = \gamma _2 \gamma _4^{-1}$,

•

and $\gamma _4 \gamma _3=\gamma _3^2 \gamma _4$.

The element $\gamma _4^2$ is the unique element of $I_{F_3'/\mathbf{Q}_3(\sqrt {3})}$ of order $2$ and hence coincides with our previous definition. The element $\gamma _2$ is a Frobenius lift of order $2$ and so we may take $F_3$ to be its fixed field, i.e. $F_3=\mathbf{Q}_3(\pi )$, where $\pi =\alpha /\beta$ is a uniformiser for $F_3'$. (We are not asserting that $\gamma _2$ equals the element denoted $\widetilde{\gamma }_2$ in §4.) One can check that

$$\begin{equation*} \gamma _3(\pi )/\pi \equiv 1+\pi ^2 \bmod \pi ^4. \end{equation*}$$

Note also that $\langle \gamma _2, \gamma _4 \rangle$ projects isomorphically to the quotient of $\operatorname {Gal}(F_3'/\mathbf{Q}_3)$ by the wild inertia subgroup.

We conclude

•

$G(0)=1$,

•

$c_\pi \equiv -1 \bmod (3,u)$,

•

$H_{\gamma _2}(u)=1$,

•

$H_{\gamma _4}(u)=-\sqrt {-1}$,

•

$H_{\gamma _3}(u)\equiv 1+u^2 \bmod (3,u^4)$,

•

$t_g=f_{g,g'}=0$ for $g,g' \in \langle \gamma _2,\gamma _4 \rangle$.

6.4. The case of $F_{-3}'$

Recall that $\tau _{-3}$ is the unique $3$-type such that $\tau _{-3}|_{I_{\mathbf{Q}_3 (\sqrt {-3})}}$ corresponds to the order $6$ homomorphism

$$\begin{equation*} \mathbf{Z}_3[\sqrt {-3}]^\times \longrightarrow \operatorname {GL}_2(\overline{\mathbf{Q}}_3) \end{equation*}$$

determined by

$$\begin{equation*} \begin{array}{rcl} -1 & \longmapsto & -1 \\4 & \longmapsto & 1 \\1+3\sqrt {-3} & \longmapsto & 1 \\1+\sqrt {-3} & \longmapsto & \zeta , \end{array} \end{equation*}$$

where $\det \zeta =1$ and $\zeta ^3=1$ but $\zeta \neq 1$. Recall also that $F_{-3}'$ is the degree $12$ abelian extension of $\mathbf{Q}_3(\sqrt {-3})$ with norm subgroup in $Q_3(\sqrt {-3})^\times$ topologically generated by $-3$, $4$ and $1+3\sqrt {-3}$. We also let $\gamma _4^2$ denote the unique element of $I_{F_{-3}'/\mathbf{Q}_3(\sqrt {-3})}$ of order $3$ and we let $F_{-3}$ denote the fixed field of some Frobenius lift of order $2$.

We claim that $F_{-3}'= \mathbf{Q}_3(\sqrt {-3})(\sqrt {-1},\alpha ,\beta )$ where $\alpha$ is a root of $X^3-4$ and $\beta$ a root of $X^2+\sqrt {-3}$. To verify this, set $F''=\mathbf{Q}_3(\sqrt {-3})(\sqrt {-1},\alpha ,\beta )$. Then $F''/\mathbf{Q}_3(\sqrt {-3})$ is abelian and so we must check that $N_{F''/\mathbf{Q}_3(\sqrt {-3})}(F'')^\times$ contains $-3$, $4$ and $1+3\sqrt {-3}$. But note that we have the identities $N_{F''/\mathbf{Q}_3(\sqrt {-3})} ((\alpha -1)/\beta )=-3$, $N_{F''/\mathbf{Q}_3(\sqrt {-3})}(\alpha )=4^4$ and $N_{F''/\mathbf{Q}_3(\sqrt {-3})}(1-\beta )=(1+\sqrt {-3})^6$.

Note that $\operatorname {Gal}(F_3'/\mathbf{Q}_3(\sqrt {-3}))$ is generated by three commuting elements $\gamma _2$, $\gamma _4^2$ and $\gamma _3$ of respective orders $2$, $2$ and $3$. They may be defined by

•

$\gamma _2 \sqrt {-1} = -\sqrt {-1}$, $\gamma _2 \beta = \beta$ and $\gamma _2 \alpha =\alpha$;

•

$\gamma _4^2 \sqrt {-1} = \sqrt {-1}$, $\gamma _4^2 \beta = -\beta$ and $\gamma _4^2 \alpha =\alpha$;

•

$\gamma _3 \sqrt {-1} = \sqrt {-1}$, $\gamma _3 \beta = \beta$ and $\gamma _3 \alpha = (-1-\sqrt {-3})\alpha /2$.

Choose an element $\gamma \in I_{F_{-3}'/\mathbf{Q}_3}-I_{F_{-3}'/\mathbf{Q}_3(\sqrt {-3})}$, so $\gamma ^2 \in \langle \gamma _4^2, \gamma _3 \rangle$. As $\gamma \gamma _3 \gamma ^{-1} = \gamma _3^2$, we may alter our choice of $\gamma$ so that $\gamma ^2 \in \langle \gamma _4^2 \rangle$. As $\gamma \sqrt {-3}=-\sqrt {-3}$ we see that $\gamma \beta = \pm \sqrt {-1} \beta$, so $\gamma ^2=\gamma _4^2$. We will rename $\gamma$ as $\gamma _4$ and suppose it chosen so that $\gamma _4 \beta = \sqrt {-1}\beta$. Thus, $\operatorname {Gal}(F_3'/\mathbf{Q}_3)$ is generated by elements $\gamma _2$, $\gamma _3$ and $\gamma _4$ satisfying

•

$\gamma _2^2=\gamma _3^3=\gamma _4^4=1$,

•

$\gamma _2\gamma _3=\gamma _3 \gamma _2$,

•

$\gamma _4 \gamma _2 = \gamma _2 \gamma _4^{-1}$,

•

and $\gamma _4 \gamma _3=\gamma _3^2 \gamma _4$.

The element $\gamma _4^2$ is the unique element of $I_{F_{-3}'/\mathbf{Q}_3(\sqrt {-3})}$ of order $2$ and hence coincides with our previous definition. The element $\gamma _2$ is a Frobenius lift of order $2$ and so we may take $F_{-3}$ to be its fixed field, i.e. $F_{-3}=\mathbf{Q}_3(\pi )$, where $\pi =\alpha /\beta$ is a uniformiser for $F_{-3}'$. (We are not asserting that $\gamma _2$ equals the element denoted $\widetilde{\gamma }_2$ in §4.) One can check that

$$\begin{equation*} \gamma _3(\pi )/\pi \equiv 1+\pi ^2 \bmod \pi ^4. \end{equation*}$$

Note also that $\langle \gamma _2,\gamma _4 \rangle$ lifts tame inertia.

We conclude

•

$G(0)=-1$,

•

$c_\pi \equiv 1 \bmod (3, u)$,

•

$H_{\gamma _2}(u)=1$,

•

$H_{\gamma _4}(u)=-\sqrt {-1}$,

•

$H_{\gamma _3}(u)\equiv 1+u^2 \bmod (3,u^4)$,

•

$t_g=f_{g,g'}=0$ for $g,g' \in \langle \gamma _2,\gamma _4 \rangle$.

6.5. The case of $F_i'$

Here $i \in \mathbf{Z}/3\mathbf{Z}$ and we will let $\tilde{\imath }$ denote the unique lifting of $i$ to $\mathbf{Z}$ with $0 \leq \tilde{\imath }<3$. Recall that $\tau _i'$ is the unique extended $3$-type whose restrictions to $G_{\mathbf{Q}_3(\sqrt {-3})}$ correspond to the homomorphism

$$\begin{equation*} \mathbf{Q}_3(\sqrt {-3})^\times \longrightarrow \operatorname {GL}_2(\overline{\mathbf{Q}}_3) \end{equation*}$$

determined by

$$\begin{equation*} \begin{array}{rcl} \sqrt {-3} & \longmapsto & \zeta -\zeta ^{-1} \\-1 & \longmapsto & -1 \\4 & \longmapsto & 1 \\1+3\sqrt {-3} & \longmapsto & \zeta \\1+\sqrt {-3} & \longmapsto & \zeta ^i, \end{array} \end{equation*}$$

where $\det \zeta =1$ and $\zeta ^3=1$ but $\zeta \neq 1$. Recall also that $F_i'$ is the degree $12$ abelian extension of $\mathbf{Q}_3(\sqrt {-3})$ with norms the subgroup of $\mathbf{Q}_3(\sqrt {-3})^\times$ topologically generated by $-3$, $4$, $1+9\sqrt {-3}$ and $1+(1-3\tilde{\imath })\sqrt {-3}$. We let $\gamma _2$, $\gamma _3$ and $\gamma _4^2$ denote the elements of $\operatorname {Gal}(F_i'/\mathbf{Q}_3)$ which correspond respectively to $\sqrt {-3}$, $1-3\sqrt {-3}$ and $-1$.

We claim that $F_{i}'= \mathbf{Q}_3(\sqrt {-3})(\sqrt {-1},\alpha ,\beta )$, where $\alpha$ is a root of $X^3-3(1+3\tilde{\imath })$ and $\beta$ a root of $X^2+\sqrt {-3}$. To verify this, set $F''=\mathbf{Q}_3(\sqrt {-3})(\sqrt {-1},\alpha ,\beta )$, so $F''/\mathbf{Q}_3(\sqrt {-3})$ is abelian and we must check that $N_{F''/\mathbf{Q}_3(\sqrt {-3})}(F'')^\times$ contains $-3$, $4$, $1+9\sqrt {-3}$, and $1+(1-3\tilde{\imath })\sqrt {-3}$. But note that $N_{F''/\mathbf{Q}_3(\sqrt {-3})} (\alpha /\beta )=-3(1+3\tilde{\imath })^4$, $N_{F''/\mathbf{Q}_3(\sqrt {-3})}(1+\alpha )=(4+9\tilde{\imath })^4$ and

$$\begin{align*} N_{F''/\mathbf{Q}_3(\sqrt {-3})}(\beta (\sqrt {-3}-\alpha )/\alpha )&=(1+\sqrt {-3}+3\tilde{\imath })/ (1+3\tilde{\imath })^4\\ & \equiv 1+(1-3\tilde{\imath })\sqrt {-3} \bmod 9. \end{align*}$$

Note that $\gamma _4^2$ is an element of $I_{F_i'/\mathbf{Q}_3(\sqrt {-3})}$ of order $2$, $\gamma _2\neq \gamma _4^2$ but also has order $2$, and $\gamma _3$ is an element of $I_{F_i'/\mathbf{Q}_3(\sqrt {-3})}$ of order $3$. Thus,

•

$\gamma _4^2 \sqrt {-1} = \sqrt {-1}$, $\gamma _4^2 \beta = -\beta$ and $\gamma _4^2 \alpha =\alpha$;

•

$\gamma _2 \sqrt {-1} = -\sqrt {-1}$ and $\gamma _2 \alpha =\alpha$;

•

$\gamma _3 \sqrt {-1} = \sqrt {-1}$ and $\gamma _3 \beta = \beta$.

Moreover $\sqrt {-3}$ is a norm from $\mathbf{Q}_3(\sqrt {-3})(\alpha ,\beta )$, because $\alpha /\beta$ has norm $\sqrt {-3}(1+3\tilde{\imath })^2$, so

•

$\gamma _2(\beta )=\beta$.

The determination of $\gamma _3(\alpha )$ is more delicate. Let $\delta$ be a root of $X^3- (1+3\sqrt {-3})$, so $\delta =1+\sqrt {-3}\mu$, where $\mu$ is a root of $Y^3-\sqrt {-3}Y^2-Y+1$. Thus $\mathbf{Q}_3(\sqrt {-3})(\delta )/ \mathbf{Q}_3(\sqrt {-3})$ is unramified and

$$\begin{equation*} \operatorname {Frob}_3(\delta )/\delta \equiv (1+\sqrt {-3} \mu ^3)/(1+\sqrt {-3}\mu ) \equiv (-1+\sqrt {-3})/2 \bmod 3. \end{equation*}$$

The norms from $\mathbf{Q}_3(\sqrt {-3})(\delta )^\times$ to $\mathbf{Q}_3(\sqrt {-3})^\times$ are generated by $\mathbf{Z}_3[\sqrt {-3}]^\times$ and $3\sqrt {\!-3}$. The norms from $\mathbf{Q}_3(\sqrt {\!-3})(\alpha )^\times$ to $\mathbf{Q}_3(\sqrt {\!-3})^\times$ are generated by $1+9\mathbf{Z}_3[\sqrt {\!-3}]$, $1+(1-3\tilde{\imath })\sqrt {-3}$, $4$, $-1$ and $\sqrt {-3}$. The norms from $\mathbf{Q}_3(\sqrt {-3})(\alpha ,\delta )^\times$ to $\mathbf{Q}_3(\sqrt {-3})^\times$ are generated by $1+9\mathbf{Z}_3[\sqrt {-3}]$, $1+(1-3\tilde{\imath })\sqrt {-3}$, $4$, $-1$ and $3\sqrt {-3}$. Thus

$$\begin{equation*} \begin{array}{rcl} (\gamma _3,\operatorname {Frob}_3) & \in & \operatorname {Gal}(\mathbf{Q}_3(\sqrt {-3})(\alpha )/ \mathbf{Q}_3(\sqrt {-3})) \times \operatorname {Gal}(\mathbf{Q}_3(\sqrt {-3})(\delta )/\mathbf{Q}_3(\sqrt {-3})) \\& \cong & \operatorname {Gal}(\mathbf{Q}_3(\sqrt {-3})(\alpha ,\delta )/\mathbf{Q}_3(\sqrt {-3})) \end{array} \end{equation*}$$

corresponds to $\sqrt {-3}(1-3 \sqrt {-3}) \in \mathbf{Q}_3(\sqrt {-3})^\times$. As $\delta \alpha$ has norm to $\mathbf{Q}_3(\sqrt {-3})$ the product of $(\sqrt {-3}(1-3\sqrt {-3}))^2$ and $-(1+3\tilde{\imath })(1+3\sqrt {-3})/(1-3\sqrt {-3})^2$, we conclude that $(\gamma _3,\operatorname {Frob}_3)$ fixes $\delta \alpha$. Thus $\gamma _3(\alpha )/\alpha = \delta /\operatorname {Frob}_3(\delta ) = (-1-\sqrt {-3})/2$. In other words

•

$\gamma _3(\alpha )=(-1-\sqrt {-3})\alpha /2$.

Choose an element $\gamma \in I_{F_{i}'/\mathbf{Q}_3}-I_{F_{i}'/\mathbf{Q}_3(\sqrt {-3})}$. Then $\gamma ^2 \in \langle \gamma _4^2, \gamma _3 \rangle$. As $\gamma \gamma _3 \gamma ^{-1} = \gamma _3^2$ we may alter our choice of $\gamma$ so that $\gamma ^2 \in \langle \gamma _4^2 \rangle$. As $\gamma \sqrt {-3}=-\sqrt {-3}$ we see that $\gamma \beta = \pm \sqrt {-1} \beta$ and so $\gamma ^2=\gamma _4^2$. We will rename $\gamma$ as $\gamma _4$ and suppose it chosen so that $\gamma _4 \beta = \sqrt {-1}\beta$. Thus, $\operatorname {Gal}(F_i'/\mathbf{Q}_3)$ is generated by elements $\gamma _2$, $\gamma _3$ and $\gamma _4$ satisfying

•

$\gamma _2^2=\gamma _3^3=\gamma _4^4=1$,

•

$\gamma _2\gamma _3=\gamma _3 \gamma _2$,

•

$\gamma _4 \gamma _2 = \gamma _2 \gamma _4^{-1}$,

•

and $\gamma _4 \gamma _3=\gamma _3^2 \gamma _4$.

The element $\gamma _2$ is a Frobenius lift and it has fixed field $F_i=\mathbf{Q}_3(\pi )$, where $\pi =\alpha /\beta$ is a uniformiser for $F_i'$. One can check that

$$\begin{equation*} \gamma _3^{\pm 1}(\pi )/\pi =-(1 \mp (1+3\tilde{\imath })^{-2}\pi ^6)/2. \end{equation*}$$

We conclude

•

$G(u)=-(1+3\tilde{\imath })^4$,

•

$c_\pi \equiv 1 \bmod (3,u^{36})$,

•

$H_{\gamma _2}(u)=1$,

•

$H_{\gamma _4}(u)=-\sqrt {-1}$,

•

$H_{\gamma _3^{\pm 1}}(u)\equiv 1 \mp u^6\bmod 3$,

•

$t_{\gamma _3^{\pm 1}}(u)\equiv -1 \mp u^6 \bmod (3,u^{12})$,

•

$t_g=f_{g,g'}=0$ for $g,g' \in \langle \gamma _2,\gamma _4 \rangle$,

•

$f_{\gamma _3^{\pm 1},\gamma _3^{\pm 1}}(u)$, $f_{\gamma _3^{\pm 1},\gamma _3^{\mp 1}}(u) \equiv 0 \bmod (3,u^{12})$.

7. Proof of Theorem 4.4.1

In this section we will keep the notation of §4.4 and either §6.1 or §6.2 (depending on whether we are working with ${\operatorname {\mathcal{S}}}_1$ or ${\operatorname {\mathcal{S}}}_{-1}$). We will set $\delta =\pm 1$ in the case of ${\operatorname {\mathcal{S}}}_{\pm 1}$. We will write $F$ for $F_{\pm 1}$ and $F'$ for $F_{\pm 1}'$. If ${\mathcal{G}}$ (resp. ${\operatorname {\mathcal{M}}}$) is a finite flat ${\mathcal{O}}_F$-group scheme (resp. Breuil module over ${\mathcal{O}}_F$) we will write ${\mathcal{G}}'$ (resp. ${\operatorname {\mathcal{M}}}'$) for the unramified base change to ${\mathcal{O}}_{F'}$.

7.1. Rank one calculations

We recall from Lemma 5.2.1 that the only ${\mathcal{O}}_F$-models for $(\mathbf{Z}/3\mathbf{Z})_{/F}$ are ${\mathcal{G}}(3,\delta ) \cong (\mathbf{Z}/3\mathbf{Z})_{/{\mathcal{O}}_F}$ and ${\mathcal{G}}(1,\delta )$, and the only ${\mathcal{O}}_F$-models for $(\mu _3)_{/F}$ are ${\mathcal{G}}(0,1) \cong (\mu _3)_{/{\mathcal{O}}_F}$ and ${\mathcal{G}}(2,1)$. In each case, by Lemma 5.7.1, the base change to ${\mathcal{O}}_{F'}$ admits unique descent data over $\mathbf{Q}_3$ compatible with the canonical descent data on the generic fibre of $\mathbf{Z}/3\mathbf{Z}$ (resp. $\mu _3$) over $\mathbf{Q}_3$. We will refer to this descent data as the standard descent data on these finite flat group schemes.

7.2. Rank two calculations

Lemma 7.2.1.

The group of extensions of ${\operatorname {\mathcal{M}}}(2,1)$ by ${\operatorname {\mathcal{M}}}(1,\delta )$ over ${\mathcal{O}}_F$ is parametrised by $c \in \mathbf{F}_3$. The Breuil module ${\operatorname {\mathcal{M}}}(1,\delta ;2,1;c)$ corresponding to $c$ is free of rank two over $\mathbf{F}_3[u]/u^9$ with a basis $\{\mathbf{e}_1, \mathbf{e}_{\omega }\}$ such that

•

$\operatorname {\mathcal{M}}_1 = \langle u \mathbf{e}_1, u^2 \mathbf{e}_{\omega } + c\mathbf{e}_1 \rangle$,

•

$\phi _1(u\mathbf{e}_1) = \delta \mathbf{e}_1$, $\phi _1(u^2 \mathbf{e}_{\omega } + c\mathbf{e}_1) = \mathbf{e}_{\omega }$,

•

$N(\mathbf{e}_1) = 0$, $N(\mathbf{e}_{\omega }) = cu^6 \mathbf{e}_1$.

The standard descent data on ${\operatorname {\mathcal{M}}}(2,1)'$ and ${\operatorname {\mathcal{M}}}(1,\delta )'$ extends uniquely to descent data on ${\operatorname {\mathcal{M}}}(1,\delta ;2,1;c)'$. The corresponding representations $G_3 \rightarrow {\operatorname {GL}}_2(\mathbf{F}_3)$ are of the form

$$\begin{equation*} \left( \begin{array}{cc} {\omega } & {*} \\{0} & {1} \end{array} \right) \end{equation*}$$

and are peu ramifié. Any such peu-ramifié extension arises for a suitable choice of $c$.

Proof.

The classification of extensions of Breuil modules follows from Lemma 5.2.2. Next, we compute $N$ on ${\operatorname {\mathcal{M}}}={\operatorname {\mathcal{M}}}(1,\delta ;2,1;c)$. (We will not in fact need the result of this computation of $N$, but the calculation is given here as a representative sample of calculations needed later in more complicated settings.) By the last part of Lemma 5.2.1, $N(\mathbf{e}_1) = 0$ and $N(\mathbf{e}_{\omega }) = g \mathbf{e}_1$ for some $g \in \mathbf{F}_3[u]/u^{e\ell }$ divisible by $u$. In $\mathbf{F}_3[u]/u^9$ we compute

$$\begin{equation*} c_{\pi } = -\phi (G_{\pi }(u)) = -(u^2 - \delta )^3 = -u^6 + \delta , \end{equation*}$$

so

$$\begin{equation*} \frac{\delta u^6}{c_{\pi }} = u^6. \end{equation*}$$

Using the defining properties of $N$, we compute in $\mathbf{F}_3[u]/u^9$

$$\begin{eqnarray*} N(\mathbf{e}_{\omega }) &=&N\circ \phi _1(u^2 \mathbf{e}_{\omega } + c\mathbf{e}_1)\\ &=& \phi \circ N (u^2 \mathbf{e}_{\omega } + c\mathbf{e}_1)\\ &=& \phi (-u^2 \mathbf{e}_{\omega } + u^2 N(\mathbf{e}_{\omega }))\\ &=& \frac{\phi _1}{c_{\pi }}(-u^5 \mathbf{e}_{\omega } + u^5 N(\mathbf{e}_{\omega }))\\ &=& \frac{\phi _1}{c_{\pi }}(-u^3(u^2\mathbf{e}_{\omega } + c\mathbf{e}_1) + cu^3 \mathbf{e}_1 + u^5 N(\mathbf{e}_{\omega }))\\ &=& \frac{\phi _1}{c_{\pi }}(cu^3 \mathbf{e}_1) \end{eqnarray*}$$

since $u^5 N(\mathbf{e}_{\omega }) \in u^6 \operatorname {\mathcal{M}}= u^3 \cdot u^3 {\operatorname {\mathcal{M}}}\subseteq u^3 {\operatorname {\mathcal{M}}}_1$ and the Frobenius-semi-linear $\phi _1$ must kill $u^3 \operatorname {\mathcal{M}}_1$. Thus,

$$\begin{equation*} N(\mathbf{e}_{\omega }) = \frac{\phi _1}{c_{\pi }}(cu^2\cdot u\mathbf{e}_1) = \frac{cu^6}{c_{\pi }}\phi _1(u \mathbf{e}_1) = \frac{c\delta u^6}{c_{\pi }} \mathbf{e}_1 = cu^6 \mathbf{e}_1. \end{equation*}$$

To see existence and uniqueness of the descent data on ${\operatorname {\mathcal{M}}}(1,\delta ;2,1;c)'$ compatible with the standard descent data on ${\operatorname {\mathcal{M}}}(1,\delta )'$ and ${\operatorname {\mathcal{M}}}(2,1)'$ we will work on the side of finite flat group schemes. Because ${\mathcal{G}}(1,\delta ;2,1;c)'$ is the unique extension of ${\mathcal{G}}(1,\delta )'$ by ${\mathcal{G}}(2,1)'$ with generic fibre ${\mathcal{G}}(1,\delta ;2,1;c)' \times F'$ (by Lemma 4.1.2), uniqueness reduces to the corresponding questions on the generic fibre, which follows from the injectivity of

$$\begin{equation*} H^1(G_3,\omega ) \longrightarrow H^1(G_{F'}, \omega ) . \end{equation*}$$

For existence it suffices to exhibit a continuous representation $G_3 \rightarrow {\operatorname {GL}}_2(\mathbf{F}_3)$ of the form

$$\begin{equation*} \left( \begin{array}{cc} {\omega } & {*} \\{0} & {1} \end{array} \right) \end{equation*}$$

which is peu ramifié but not split, with restriction to $G_F$ corresponding to a local-local finite flat ${\mathcal{O}}_F$-group scheme ${\mathcal{G}}$. By Theorem 5.3.2 of Man we can find an elliptic curve $E_{/\mathbf{Q}_3}$ such that $E[3]$ furnishes the desired example. This also proves the final two assertions of the lemma.

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Lemma 7.2.2.

Suppose that $\widetilde{F}_1$ is a totally ramified abelian cubic extension of $\mathbf{Q}_3$ and suppose that ${\mathcal{G}}$ is a local-local finite flat ${\mathcal{O}}_{\widetilde{F}_1}$-group scheme killed by $3$ such that ${\mathcal{G}}\times \widetilde{F}_1$ is an extension of $\mathbf{Z}/3\mathbf{Z}$ by $\mu _3$. Then ${\mathcal{G}}\times _{{\mathcal{O}}_{\widetilde{F}_1}} \widetilde{F}_1 \cong G \times _{\mathbf{Q}_3} \widetilde{F}_1$ for some finite flat $\mathbf{Q}_3$-group scheme $G$.

Proof.

As in the proof of the last lemma we see that ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}) \cong {\operatorname {\mathcal{M}}}(1,1;2,1;c)$ for some $c \in \mathbf{F}_3$. As the only action of $\operatorname {Gal}(\widetilde{F}_1/\mathbf{Q}_3)$ on a one-dimensional $\mathbf{F}_3$-vector space is trivial, we see that each such $c$ gives a class in $H^1(G_{\widetilde{F}_1},\omega )$ which is invariant by $\operatorname {Gal}(\widetilde{F}_1/\mathbf{Q}_3)$. But

$$\begin{equation*} H^1(G_3,\omega ) \stackrel{\sim }{\longrightarrow }H^1(G_{\widetilde{F}_1},\omega )^{\operatorname {Gal}(\widetilde{F}_1/\mathbf{Q}_3)}, \end{equation*}$$

and so the lemma follows.

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Lemma 7.2.3.

The group of extensions of ${\operatorname {\mathcal{M}}}(1,\delta )$ by ${\operatorname {\mathcal{M}}}(2,1)$ over ${\mathcal{O}}_F$ is isomorphic to the group of linear polynomials $c+c'u$ in $\mathbf{F}_3[u]$. The Breuil module ${\operatorname {\mathcal{M}}}(2,1;1,\delta ;c+c'u)$ corresponding to $c+c'u$ is free of rank two over $\mathbf{F}_3[u]/u^9$ with a basis $\{ \mathbf{e}_{\omega }, \mathbf{e}_1\}$ such that

•

$\operatorname {\mathcal{M}}(2,1;1,\delta ;c+c'u)_1 = \langle u^2 \mathbf{e}_\omega , u \mathbf{e}_1 + (c+c'u)\mathbf{e}_\omega \rangle$,

•

$\phi _1(u^2\mathbf{e}_\omega ) = \mathbf{e}_\omega$, $\phi _1(u \mathbf{e}_1 + (c+c'u)\mathbf{e}_\omega ) = \delta \mathbf{e}_1$.

Each ${\operatorname {\mathcal{M}}}(2,1;1,\delta ;c+c'u)'$ admits unique descent data compatible with the standard descent data on ${\operatorname {\mathcal{M}}}(1,\delta )'$ and ${\operatorname {\mathcal{M}}}(2,1)'$. As $c,c'$ vary over $\mathbf{F}_3$ the corresponding descent to $\mathbf{Q}_3$ of the generic fibre of ${\mathcal{G}}_\pi ({\operatorname {\mathcal{M}}}(2,1;1,\delta ;c+c'u)')$ runs over all $9$ extensions of $\mu _3$ by $\mathbf{Z}/3\mathbf{Z}$. The corresponding representation of $G_3$ is peu ramifié if and only if $c=0$.

Proof.

The classification of extensions of Breuil modules follows from Lemma 5.2.2. The uniqueness of the descent data on ${\operatorname {\mathcal{M}}}(2,1;1,\delta ;c+c'u)'$ follows from Lemma 4.1.2 and the injectivity of $H^1(G_3,\omega ^{-1}) \rightarrow H^1(G_{F'}, \omega ^{-1})$ as in the proof of Lemma 7.2.1. Note that Frobenius vanishes on the Dieudonné module of ${\mathcal{G}}(2,1;1,\delta ;c+c'u)$ if and only if $c=0$. Thus the lemma will follow if for each $3$-torsion extension $G$ of $\mu _3$ by $\mathbf{Z}/3\mathbf{Z}$ over $\mathbf{Q}_3$ which is très ramifié, we can find a finite flat ${\mathcal{O}}_F$-group scheme ${\mathcal{G}}$ such that

•

the generic fibre of ${\mathcal{G}}$ is isomorphic to $G \times F$,

•

the closed fibre of ${\mathcal{G}}$ is local-local,

•

and Frobenius is not identically zero on $D({\mathcal{G}})$.

The splitting field of $G$ contains a cube root of $3v$ for some $v \equiv 1 \bmod 3$, where the three choices of $v \bmod 9$ correspond to the three choices of très ramifié ${\overline{\rho }}$. The calculations in §5.3 of Man give explicit additive reduction elliptic curves $E$ and $E'$ over $\mathbf{Q}_3$ with $E[3] \simeq E'[3] \simeq G$, where $E$ acquires good supersingular reduction over the non-Galois cubic ramified extension

$$\begin{equation*} \mathbf{Q}_3[X]/(X^3 - 3X + 2b), \end{equation*}$$

with $b^2 = 1 + 3v$, and $E'$ acquires good supersingular reduction over the abelian cubic ramified extension of $\mathbf{Q}_3$ with norm group generated by $3v \bmod (\mathbf{Q}_3^{\times })^3$. The appropriate ${\mathcal{G}}$ are provided by the $3$-torsion on the Néron models of $E$ or $E'$ over ${\mathcal{O}}_F$.

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Corollary 7.2.4.

Suppose that ${\mathcal{G}}$ is a finite flat ${\mathcal{O}}_F$-group scheme and that $\{ [g]\}$ is descent data on ${\mathcal{G}}'={\mathcal{G}}\times {\mathcal{O}}_{F'}$ such that $({\mathcal{G}}', \{[g]\})_{\mathbf{Q}_3}(\overline{\mathbf{Q}}_3)$ corresponds to ${\overline{\rho }}$. Then

$$\begin{equation*} {\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}) \cong {\operatorname {\mathcal{M}}}(2,1;1,\delta ;c+c'u) \end{equation*}$$

for some $c,c' \in \mathbf{F}_3$ with $c \neq 0$.

Proof.

From the connected-étale exact sequence and its dual we see that ${\mathcal{G}}\times \mathbf{F}_3$ must be local-local. The corollary now follows from Lemma 7.2.3 and the discussion of §7.1.

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Lemma 7.2.5.

The group of extensions of ${\operatorname {\mathcal{M}}}(1,\delta )$ by ${\operatorname {\mathcal{M}}}(1,\delta )$ over ${\mathcal{O}}_F$ is isomorphic to the group of linear polynomials $b+b'u$ in $\mathbf{F}_3[u]$. The Breuil module ${\operatorname {\mathcal{M}}}(1,\delta ;1,\delta ;b+b'u)$ corresponding to $b+b'u$ is free of rank two over $\mathbf{F}_3[u]/u^9$ with a basis $\{ \mathbf{e}, \mathbf{e}'\}$ such that

•

$\operatorname {\mathcal{M}}(1,\delta ;1,\delta ;b+b'u)_1 = \langle u \mathbf{e}, u \mathbf{e}' + (b+b'u)\mathbf{e}\rangle$,

•

$\phi _1(u\mathbf{e}) = \delta \mathbf{e}$, $\phi _1(u \mathbf{e}' + (b+b'u)\mathbf{e}) = \delta \mathbf{e}'$.

This extension splits over an unramified extension if and only if $b=0$. If $F'/\mathbf{Q}_3$ is non-abelian, then any descent data on ${\operatorname {\mathcal{M}}}(1,-1;1,-1;b+b'u)'$ compatible with the standard descent data on ${\operatorname {\mathcal{M}}}(1,-1)'$ satisfies

$$\begin{equation*} \widehat{\gamma }_2 \mathbf{e}=\mathbf{e}, \widehat{\gamma }_2 \mathbf{e}'=\mathbf{e}', \widehat{\gamma _3^{\pm 1}}(\mathbf{e}) = H_{\gamma _3^{\pm 1}}(u)^{3}\mathbf{e},\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}') = H_{\gamma _3^{\pm 1}}(u)^{3}\mathbf{e}' + h_{\gamma _3^{\pm 1}}(u)\mathbf{e}, \end{equation*}$$

where

$$\begin{equation*} h_{\gamma _3^{\pm 1}}(0) = -bH_{\gamma _3^{\pm 1}}'(0). \end{equation*}$$

Proof.

The classification of extensions of Breuil modules follows from Lemma 5.2.2. The computation of which of these split over an unramified extension follows from Lemma 5.2.2 and Corollary 5.4.2.

Now suppose that $F'/\mathbf{Q}_3$ is non-abelian. By Lemma 5.7.1, the only issue is to compute $h_{\gamma _3}(0)$. Since $H_{\gamma _3}(0) \equiv 1 \bmod 3$, by evaluating the congruence

$$\begin{equation*} \widehat{\gamma _3}\circ \phi '_1 \equiv \phi '_1 \circ \widehat{\gamma _3} \bmod u\operatorname {\mathcal{M}}(1,-1;1,-1;b+b'u)' \end{equation*}$$

on $u\mathbf{e}' + (b+b'u)\mathbf{e}$ and comparing constant terms of the coefficients of $\mathbf{e}$ on both sides we get

$$\begin{align*} h_{\gamma _3}(0)& = h_{\gamma _3}(0)^3 + b \left(\frac{1 - H_{\gamma _3}(u)}{u}\right)^3|_{u=0}\\ &= h_{\gamma _3}(0)^3 - bH_{\gamma _3}'(0)^3 = h_{\gamma _3}(0)^3 + bH_{\gamma _3}'(0) \end{align*}$$

in $\mathbf{F}_9$, where we have used the equality $H'_{\gamma _3}(0)^2 \equiv -1 \bmod 3$ (see §6.2).

In other words $h_{\gamma _3}(0)$ is a root of $T^3 - T + b H'_{\gamma _3}(0) = 0$. Since $H_{\gamma _3}'(0)^2 = -1$, we must have $h_{\gamma _3}(0) = -bH'_{\gamma _3}(0) + a$ for some $a \in \mathbf{F}_3$. Since $\gamma _2(H_{\gamma _3}(u)) = H_{\gamma _3^{-1}}(u)$ and $\gamma _2(h_{\gamma _3}(u)) = h_{\gamma _3^{-1}}(u)$ are forced by the identity $\gamma _2(\pi ) = \pi$, we see that $h_{\gamma _3^{-1}}(0)=-bH'_{\gamma _3^{-1}}(0) + a$ for the same $a \in \mathbf{F}_3$. The identity

$$\begin{equation*} \widehat{\gamma _3} \circ \widehat{\gamma _3^{-1}} \circ \phi '_1 \equiv \phi '_1 \bmod u\operatorname {\mathcal{M}}(1,-1;1,-1;b+b'u)' \end{equation*}$$

then implies $h_{\gamma _3}(0) + h_{\gamma _3^{-1}}(0) = 0$, so $a = 0$.

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Lemma 7.2.6.

The group of extensions of ${\operatorname {\mathcal{M}}}(2,1)$ by ${\operatorname {\mathcal{M}}}(2,1)$ over ${\mathcal{O}}_F$ is isomorphic to the group of quadratic polynomials vanishing at $0$, $(b+b'u)u$, in $\mathbf{F}_3[u]$. The Breuil module ${\operatorname {\mathcal{M}}}(2,1;2,1;(b+b'u)u)$ corresponding to $(b+b'u)u$ is free of rank two over $\mathbf{F}_3[u]/u^9$ with a basis $\{ \mathbf{e}, \mathbf{e}'\}$ such that

•

$\operatorname {\mathcal{M}}(2,1;2,1;(b+b'u)u)_1 = \langle u^2 \mathbf{e}, u^2 \mathbf{e}' + (b+b'u)u\mathbf{e}\rangle$,

•

$\phi _1(u^2\mathbf{e}) = \mathbf{e}$, $\phi _1(u^2 \mathbf{e}' + (b+b'u)u\mathbf{e}) = \mathbf{e}'$.

This extension splits over an unramified extension if and only if $b=0$. If $F'/\mathbf{Q}_3$ is non-abelian, then any descent data on ${\operatorname {\mathcal{M}}}(2,1;2,1;(b+b'u)u)'$ compatible with the standard descent data on ${\operatorname {\mathcal{M}}}(2,1)'$ satisfies

$$\begin{equation*} \widehat{\gamma _3}(\mathbf{e}) = H_{\gamma _3}(u)^{6}\mathbf{e},\,\,\, \widehat{\gamma _3}(\mathbf{e}') = H_{\gamma _3}(u)^{6}\mathbf{e}' + h_{\gamma _3}(u)\mathbf{e}, \end{equation*}$$

where

$$\begin{equation*} h_{\gamma _3}(0) = -bH_{\gamma _3}'(0). \end{equation*}$$

The sign in $h_{\gamma _3}(0) = -bH_{\gamma _3}'(0)$ will be very important in §7.4. The proof of this lemma is essentially the same as that of Lemma 7.2.5, but we repeat it anyway.

Proof.

The classification of extensions of Breuil modules follows from Lemma 5.2.2. The computation of which of these split over an unramified extension follows from Lemma 5.2.2 and Corollary 5.4.2.

Now suppose that $F'/\mathbf{Q}_3$ is non-abelian. By Lemma 5.7.1, the only issue is to compute $h_{\gamma _3}(0)$. Since $H_{\gamma _3}(0) \equiv 1 \bmod 3$, by evaluating the congruence

$$\begin{equation*} \widehat{\gamma _3}\circ \phi '_1 \equiv \phi '_1 \circ \widehat{\gamma _3} \bmod u\operatorname {\mathcal{M}}(2,1;2,1;(b+b'u)u)' \end{equation*}$$

on $u\mathbf{e}' + (b+b'u)\mathbf{e}$ and comparing constant terms of the coefficients of $\mathbf{e}$ on both sides we get

$$\begin{align*} h_{\gamma _3}(0)& = h_{\gamma _3}(0)^3 + b \left(\frac{1 - H_{\gamma _3}(u)}{u}\right)^3|_{u=0}\\ &= h_{\gamma _3}(0)^3 - bH_{\gamma _3}'(0)^3 = h_{\gamma _3}(0)^3 + bH_{\gamma _3}'(0) \end{align*}$$

in $\mathbf{F}_9$, where we have used the equality $H'_{\gamma _3}(0)^2 = -1$ (see §6.2).

In other words $h_{\gamma }(0)$ is a root of $T^3 - T + b H'_{\gamma _3}(0) = 0$. Since $H_{\gamma _3}'(0)^2 = -1$, we must have $h_{\gamma _3}(0) = -bH'_{\gamma _3}(0) + a$ for some $a \in \mathbf{F}_3$. Since $\gamma _2(H_{\gamma _3}(u)) = H_{\gamma _3^{-1}}(u)$ and $\gamma _2(h_{\gamma _3}(u)) = h_{\gamma _3^{-1}}(u)$ are forced by the identity $\gamma _2(\pi ) = \pi$, we see that $h_{\gamma _3^{-1}}(0)=-bH'_{\gamma _3^{-1}}(0) + a$ for the same $a \in \mathbf{F}_3$. The identity

$$\begin{equation*} \widehat{\gamma _3} \circ \widehat{\gamma _3^{-1}} \circ \phi '_1 \equiv \phi '_1 \bmod u\operatorname {\mathcal{M}}(2,1;2,1;b+b'u)' \end{equation*}$$

then implies $h_{\gamma _3}(0) + h_{\gamma _3^{-1}}(0) = 0$, so $a = 0$.

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7.3. Rank three calculations

Lemma 7.3.1.

Suppose that ${\mathcal{G}}$ is a finite flat group scheme over ${\mathcal{O}}_F$ which is killed by $3$. Suppose that there is a filtration by closed finite flat subgroup schemes ${\mathcal{G}}_1 \subset {\mathcal{G}}_2 \subset {\mathcal{G}}$ such that ${\mathcal{G}}_1 \cong {\mathcal{G}}(1,\delta )$, ${\mathcal{G}}_2/{\mathcal{G}}_1 \cong {\mathcal{G}}(2,1)$ and ${\mathcal{G}}/{\mathcal{G}}_2 \cong {\mathcal{G}}(1,\delta )$. Suppose finally that ${\mathcal{G}}_2 \times _{{\mathcal{O}}_F} F'$ descends to $\mathbf{Q}_3$ in such a way that it is a très ramifié extension of $\mu _3$ by $\mathbf{Z}/3\mathbf{Z}$. Then

$$\begin{equation*} {\mathcal{G}}/{\mathcal{G}}_1 \cong {\mathcal{G}}(2,1) \oplus {\mathcal{G}}(1,\delta ) \end{equation*}$$

compatibly with the extension class structure.

Proof.

Let ${\operatorname {\mathcal{M}}}= {\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}})$ and ${\operatorname {\mathcal{N}}}={\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}/{\mathcal{G}}_1)$. Using Lemmas 7.2.1 and 7.2.3 we see that we can write

•

$\operatorname {\mathcal{M}}= (\mathbf{F}_3[u]/u^{9}) \mathbf{e}_1 \oplus (\mathbf{F}_3[u]/u^{9})\mathbf{e}_{\omega } \oplus (\mathbf{F}_3[u]/u^{9}) \mathbf{e}'_1$,

•

$\operatorname {\mathcal{M}}_1 = \langle u\mathbf{e}_1, u^2 \mathbf{e}_{\omega } + b\mathbf{e}_1, u \mathbf{e}'_1 + (c+c'u)\mathbf{e}_{\omega } + f\mathbf{e}_1 \rangle$

for $b,c,c' \in \mathbf{F}_3$ with $c \ne 0$ and with $f \in \mathbf{F}_3[u]/u^{9}$. It suffices to show $b=0$. Since we must have $u^3 \operatorname {\mathcal{M}}\subseteq \operatorname {\mathcal{M}}_1$, we see that

$$\begin{align*} &(c+c'u)(u^2 \mathbf{e}_{\omega } + b\mathbf{e}_1) - u^2(u\mathbf{e}'_1 + (c+c'u)\mathbf{e}_{\omega } + f\mathbf{e}_1) + u^3\mathbf{e}'_1 \\ &\qquad = (bc+bc'u-u^2f)\mathbf{e}_1 \in \operatorname {\mathcal{M}}_1. \end{align*}$$

The Breuil module $\operatorname {\mathcal{N}}$ is spanned as a $\mathbf{F}_3[u]/u^{9}$-module by $\mathbf{e}_1$ and $\mathbf{e}_{\omega }$, so by Lemma 7.2.1 $u$ must divide $bc + bc'u - u^2f$. As $c \neq 0$ we must have $b=0$, as desired.

■

Combining this with Lemma 7.2.1 and the injectivity of $H^1(G_3, \omega ) \rightarrow H^1(G_{F'},\omega )$ we get the following corollary, which is also the first part of Theorem 4.7.3.

Corollary 7.3.2.

The natural map

$$\begin{equation*} \theta _0:\operatorname {Ext}^1_{{\operatorname {\mathcal{S}}}_{\pm 1}}({\overline{\rho }}, {\overline{\rho }}) \longrightarrow H^1(G_3,\omega ) \end{equation*}$$

is zero.

7.4. Conclusion of the proof of Theorem 4.4.1

Consider first the case of $F_1$. We still have to explain why

$$\begin{equation*} {\overline{\theta }}_\omega : H^1_{{\operatorname {\mathcal{S}}}_{1}}(G_3,\operatorname {ad}^0 {\overline{\rho }}) \longrightarrow H^1(I_3,\mathbf{F}_3) \end{equation*}$$

is zero. Suppose $x \in H^1_{{\operatorname {\mathcal{S}}}_{1}}(G_3,\operatorname {ad}^0 {\overline{\rho }})$ does not map to zero in $H^1(I_3,\mathbf{F}_3)$.

By our hypothesis on $x$ we may choose a totally ramified abelian cubic extension $\widetilde{F}_1/\mathbf{Q}_3$ such that $x$ restricts to zero under the natural map $H^1(G_3,\operatorname {ad}^0 {\overline{\rho }}) \rightarrow H^1(G_{\widetilde{F}_1},\mathbf{F}_3)$. Then the image of $x$ under the natural map $H^1(G_3,\operatorname {ad}^0 {\overline{\rho }}) \rightarrow H^1(G_{\widetilde{F}_1},{\overline{\rho }}\otimes \omega )$ is the image of some $\widetilde{x}\in H^1(G_{\widetilde{F}_1}, \omega )$ under the natural map $H^1(G_{\widetilde{F}_1}, \omega ) \rightarrow H^1(G_{\widetilde{F}_1},{\overline{\rho }}\otimes \omega )$. The element $\widetilde{x}$ parametrises a finite flat $\widetilde{F}_1$-group scheme $H$ which is an extension of $\mu _3$ by $\mathbf{Z}/3\mathbf{Z}$ and which is a subquotient of the restriction to $G_{\widetilde{F}_1}$ of the extension of ${\overline{\rho }}$ by itself parametrised by $x$. It follows that $H$ has a finite flat model ${\mathcal{H}}_{/{\mathcal{O}}_{\widetilde{F}_1}}$ (see Lemma 4.1.1) and the special fibre of ${\mathcal{H}}$ must be local-local (if $\widetilde{x}=0$, then the extension of ${\overline{\rho }}$ by itself parametrised by $x$ splits over $\widetilde{F}_1$ and this is clear, while if $\widetilde{x}\neq 0$ we would otherwise get a contradiction from the connected-étale sequence). By Lemma 7.2.2, we may therefore lift $\widetilde{x}$ to $H^1(G_3,\omega )$. Using the commutative diagram

$$\begin{equation*} \begin{matrix} H^1(G_3,\omega ) & \longrightarrow & H^1(G_3,{\overline{\rho }}\otimes \omega )\\ {\scriptstyle {\mathrm{{res}}}}\,\,\downarrow & &\downarrow \,\,{\scriptstyle {\mathrm{{res}}}}\\ H^1(G_{\widetilde{F}_1},\omega )& \longrightarrow & H^1(G_{\widetilde{F}_1},{\overline{\rho }}\otimes \omega ) \end{matrix} \end{equation*}$$

and noting that the right-hand vertical map is injective we conclude that

$$\begin{equation*} x \in H^1_{{\operatorname {\mathcal{S}}}_1}(G_3,\operatorname {ad}^0 {\overline{\rho }}) \subset H^1(G_3,{\overline{\rho }}\otimes \omega ) \end{equation*}$$

is in the image of $H^1(G_3,\omega ) \rightarrow H^1(G_3,{\overline{\rho }}\otimes \omega )$, a contradiction with the hypothesis that even the image of $x$ in $H^1(I_3,\mathbf{F}_3)$ is non-zero.

Now consider the case $F'=F_{-1}'$ which is non-abelian over $\mathbf{Q}_3$. We must show that

$$\begin{equation*} {\overline{\theta }}_\omega : \operatorname {Ext}^1_{{\operatorname {\mathcal{S}}}_{- 1}}({\overline{\rho }}, {\overline{\rho }}) \longrightarrow H^1(I_3,\mathbf{F}_3) \end{equation*}$$

is zero.

An element $x \in \operatorname {Ext}^1_{{\operatorname {\mathcal{S}}}_{- 1}}({\overline{\rho }}, {\overline{\rho }})$ gives rise to a finite flat ${\mathcal{O}}_{F_{-1}}$-group scheme ${\mathcal{G}}$ killed by $3$ and descent data $\{ [g]\}$ for $F_{-1}'/\mathbf{Q}_3$ on ${\mathcal{G}}'={\mathcal{G}}\times _{{\mathcal{O}}_{F_{-1}'}} F_{-1}'$, such that $({\mathcal{G}}', \{ [g] \})_{\mathbf{Q}_3}$ corresponds to the extension of ${\overline{\rho }}$ by itself classified by $x$. Let ${\operatorname {\mathcal{N}}}$ denote the Breuil module for ${\mathcal{G}}$ and let ${\operatorname {\mathcal{N}}}'={\operatorname {\mathcal{N}}}\otimes \mathbf{F}_9$. According to Lemmas 7.2.1, 7.2.3, 7.2.5, 7.2.6 and 7.3.1 we see that we can write

$$\begin{equation*} \operatorname {\mathcal{N}}= (\mathbf{F}_3[u]/u^{9})\mathbf{e}_{\omega } \oplus (\mathbf{F}_3[u]/u^{9})\mathbf{e}_1 \oplus (\mathbf{F}_3[u]/u^{9})\mathbf{e}'_{\omega } \oplus (\mathbf{F}_3[u]/u^{9})\mathbf{e}'_1 \end{equation*}$$

with

$$\begin{multline} \operatorname {\mathcal{N}}_1 = \langle u^2\mathbf{e}_{\omega }, u\mathbf{e}_1 + (c+c'u)\mathbf{e}_{\omega }, u^2\mathbf{e}'_{\omega } + (au + a'u^2)\mathbf{e}_{\omega }, \\ u\mathbf{e}'_1 + (c+c'u)\mathbf{e}'_{\omega }+ (b + b'u)\mathbf{e}_1 + h\mathbf{e}_{\omega } \rangle , {\tag{7.4.1}} \end{multline}$$

where $h \in \mathbf{F}_3[u]/u^{9}$ is some polynomial and where $a,a',b,b',c,c' \in \mathbf{F}_3$ with $c \ne 0$ (as ${\overline{\rho }}$ is très ramifié). By Lemma 7.2.6 what we must show is that $a=0$.

Note that $H_{{\gamma _3}}'(0)\neq 0$ in $\mathbf{F}_9$ by §6.2. By Lemmas 5.7.1 and 7.2.1, the action $\widehat{{\gamma _3}}$ is determined by

$$\begin{equation*} \widehat{{\gamma _3}}(\mathbf{e}_{\omega }) = H_{{\gamma _3}}(u)^6\mathbf{e}_{\omega },\,\,\, \widehat{{\gamma _3}}(\mathbf{e}_1) = H_{{\gamma _3}}(u)^3\mathbf{e}_1 + g_{{\gamma _3}}(u)\mathbf{e}'_{\omega }, \end{equation*}$$

$$\begin{equation*} \widehat{{\gamma _3}}(\mathbf{e}'_{\omega }) = H_{{\gamma _3}}(u)^6\mathbf{e}'_{\omega } + h_{{\gamma _3},\omega }(u)\mathbf{e}_{\omega }, \end{equation*}$$

$$\begin{equation*} \widehat{{\gamma _3}}(\mathbf{e}'_{\omega }) = H_{{\gamma _3}}(u)^3\mathbf{e}'_1 + g_{{\gamma _3}}(u)\mathbf{e}'_{\omega } + h_{{\gamma _3},{\mathbf{1}}}\mathbf{e}_1 + G_{{\gamma _3}}(u)\mathbf{e}_{\omega }, \end{equation*}$$

where $g_{{\gamma _3}}(u), G_{{\gamma _3}}(u) \in \mathbf{F}_9[u]/u^{9}$ and $h_{\gamma _3,\omega }$ and $h_{\gamma _3,{\mathbf{1}}}$ are as in Lemmas 7.2.6 and 7.2.5 respectively.

Due to the requirement $\widehat{{\gamma _3}}(\operatorname {\mathcal{N}}'_1) \subseteq \operatorname {\mathcal{N}}'_1$, we must have

$$\begin{equation*} \widehat{{\gamma _3}}(u\mathbf{e}'_1 + (c+c'u)\mathbf{e}'_{\omega } + (b+b'u)\mathbf{e}_1 + h(u)\mathbf{e}_{\omega }) \in \operatorname {\mathcal{N}}'_1, \end{equation*}$$

and this element is obviously equal to

$$\begin{equation*} \begin{array}{rr}(uH_{{\gamma _3}})(H_{{\gamma _3}}^3\mathbf{e}'_1 + g_{{\gamma _3}}\mathbf{e}'_{\omega } + h_{{\gamma _3},\mathbf{1}}\mathbf{e}_1 + G_{{\gamma _3}}\mathbf{e}_{\omega }) + (c+c'uH_{{\gamma _3}})(H_{{\gamma _3}}^6\mathbf{e}'_{\omega } + h_{{\gamma _3},\omega }\mathbf{e}_{\omega })\\+(b+b'uH_{{\gamma _3}})(H_{{\gamma _3}}^3\mathbf{e}_1 + g_{{\gamma _3}}\mathbf{e}_{\omega }) + h(uH_{{\gamma _3}})H_{{\gamma _3}}^6\mathbf{e}_{\omega }.\end{array} \end{equation*}$$

We now try to express this as a linear combination of the generators of $\operatorname {\mathcal{N}}'_1$ listed in (7.4.1), while working modulo $\langle u^3\operatorname {\mathcal{N}}', u^2\mathbf{e}_{\omega }\rangle \subseteq \operatorname {\mathcal{N}}'_1$. Using that $H_{{\gamma _3}}(0) = 1$ in $\mathbf{F}_9$ and $h(uH_{{\gamma _3}}) \equiv h(u) \bmod u^2$, we arrive at the expression

$$\begin{multline*} H_{{\gamma _3}}(u\mathbf{e}'_1 + (c+c'u)\mathbf{e}'_{\omega } + (b+b'u)\mathbf{e}_1 + h\mathbf{e}_{\omega })\\ + \left(\frac{c((1-H_{{\gamma _3}})/u) + g_{{\gamma _3}}}{u}\right) (u^2\mathbf{e}'_{\omega } + (au+a'u^2)\mathbf{e}_{\omega })\\ +\left(H_{{\gamma _3}}h_{{\gamma _3},{\mathbf{1}}} + b\left(\frac{1-H_{{\gamma _3}}}{u}\right)\right)(u\mathbf{e}_1 + (c+c'u)\mathbf{e}_{\omega }) + F_{{\gamma _3}}(u)\mathbf{e}_{\omega }, \end{multline*}$$

where

$$\begin{align*} F_{{\gamma _3}}(u) &= uH_{{\gamma _3}}G_{{\gamma _3}} + (c+c'uH_{{\gamma _3}})h_{{\gamma _3},\omega } + (b+b'uH_{{\gamma _3}})g_{{\gamma _3}} + h(u)(1 - H_{{\gamma _3}}) \\ &\quad - (a+a'u)(c(1-H_{{\gamma _3}})/u + g_{{\gamma _3}}) - (c+c'u)( H_{{\gamma _3}}h_{{\gamma _3},{\mathbf{1}}} + b((1-H_{{\gamma _3}})/u)) \end{align*}$$

in $\mathbf{F}_9[u]/u^{9}$. In particular, $c(1-H_{{\gamma _3}}(u))/u + g_{{\gamma _3}} \equiv 0 \bmod u$ and $F_{{\gamma _3}}(u) \equiv 0 \bmod u^2$. The condition $c((1-H_{{\gamma _3}})/u) + g_{{\gamma _3}} \equiv 0 \bmod u$ can be reformulated as

$$\begin{equation*} g_{{\gamma _3}}(0) = cH_{{\gamma _3}}'(0). \end{equation*}$$

Since $F_{{\gamma _3}}(u) \equiv 0 \bmod u^2$, we have to have $F_{{\gamma _3}}(0) = 0$. But a direct calculation using $g_{{\gamma _3}}(0) = cH_{{\gamma _3}}'(0)$ and the definition of $F_{{\gamma _3}}$ gives

$$\begin{align*} F_{{\gamma _3}}(0)& = 0 + ch_{{\gamma _3},\omega }(0) + bg_{{\gamma _3}}(0) +0 - 0 -c(h_{{\gamma _3},{\mathbf{1}}}(0) -bH_{{\gamma _3}}'(0))\\ & = c(h_{{\gamma _3},\omega }(0) - h_{{\gamma _3},{\mathbf{1}}}(0) -bH_{{\gamma _3}}'(0)), \end{align*}$$

so the non-vanishing of $c$ forces

$$\begin{equation*} h_{{\gamma _3},\omega }(0) - h_{{\gamma _3},{\mathbf{1}}}(0) = bH'_{{\gamma _3}}(0). \end{equation*}$$

Lemmas 7.2.6 and 7.2.5 give us the values

$$\begin{equation*} h_{{\gamma _3},\omega }(0) = -aH'_{{\gamma _3}}(0),\,\,\, h_{{\gamma _3},{\mathbf{1}}}(0) = -bH'_{{\gamma _3}}(0). \end{equation*}$$

Thus $(-a + b)H'_{{\gamma _3}}(0) = bH'_{{\gamma _3}}(0)$, and so $a = 0$. This completes the proof of Theorem 4.7.3 and hence of Theorem 4.4.1.

8. Proof of Theorem 4.5.1

In this section we will keep the notation of §4.5 and either §6.3 or §6.4 (depending on whether we are working with ${\operatorname {\mathcal{S}}}_3$ or ${\operatorname {\mathcal{S}}}_{-3}$). We will set $\delta =\pm 1$ in the case of ${\operatorname {\mathcal{S}}}_{\mp 3}$ (so that $c_\pi \equiv \delta \bmod (3,u)$). Note the signs. We will write $F$ for $F_{\pm 3}$, $F'$ for $F_{\pm 3}'$ and ${\mathcal{I}}$ for ${\mathcal{I}}_{\pm 3}$. If ${\mathcal{G}}$ (resp. ${\operatorname {\mathcal{M}}}$) is a finite flat ${\mathcal{O}}_F$-group scheme (resp. Breuil module over ${\mathcal{O}}_F$) we will write ${\mathcal{G}}'$ (resp. ${\operatorname {\mathcal{M}}}'$) for the base change to ${\mathcal{O}}_{F'}$.

8.1. Rank one calculations

We remark that with our choice of polynomials $H_g(u)$ in §6.3 and §6.4, any object ${\operatorname {\mathcal{M}}}$ in ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$ has an action of $\langle \gamma _2,\gamma _4 \rangle$ (via $\widehat{\gamma }_2$ and $\widehat{\gamma }_4$, the action of $\gamma _2$ being $\operatorname {Frob}_3$-semi-linear). Also, since $\gamma _3$ and $\gamma _2$ commute, $H_{\gamma _2}=1$ and $H_{\gamma _3^{\pm 1}}(u) \in \mathbf{Z}_3[u]$, we see that $\widehat{\gamma }_2$ must commute with $\widehat{\gamma _3^{\pm 1}}$ by Corollary 5.6.2.

We recall from Lemma 5.2.1 that the only models for $(\mathbf{Z}/3\mathbf{Z})_{/F}$ over ${\mathcal{O}}_{F}$ are ${\mathcal{G}}(r,\delta )$ for $r=0,2,4,6,8,10,12$ with ${\mathcal{G}}(12,\delta ) \cong (\mathbf{Z}/3\mathbf{Z})_{/{\mathcal{O}}_F}$, and the only models for $(\mu _3)_{/F}$ over ${\mathcal{O}}_{F}$ are ${\mathcal{G}}(r,1)$ for $r=0,2,4,6,8,10,12$ with ${\mathcal{G}}(0,1) \cong (\mu _3)_{/{\mathcal{O}}_F}$. In each case, the base change to ${\mathcal{O}}_{F'}$ admits unique descent data over $\mathbf{Q}_3$ such that descent of the generic fibre to $\mathbf{Q}_3$ is $\mathbf{Z}/3\mathbf{Z}$ (resp. $\mu _3$). (See Lemma 5.7.1.) We will write ${\mathcal{G}}_{r,1}'$ (resp. ${\mathcal{G}}_{r,\omega }'$) for the corresponding pair $({\mathcal{G}}(r,\delta ) \times {\mathcal{O}}_{F'}, \{ [g] \})$ (resp. $({\mathcal{G}}(r,1) \times {\mathcal{O}}_{F'}, \{ [g] \})$). We will also let ${\operatorname {\mathcal{M}}}_{r,1}'$ (resp. ${\operatorname {\mathcal{M}}}_{r,\omega }'$) denote the corresponding object of ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$. In particular, for $\chi = 1$ or $\omega$, the underlying $\mathbf{F}_9[u]/u^{36}$-module has the form $(\mathbf{F}_9[u]/u^{36})\mathbf{e}_\chi$ with $\mathbf{e}_\chi$ the standard generator, though we write $\mathbf{e}$ rather than $\mathbf{e}_\chi$ if $\chi$ is understood.

We have the following useful lemma.

Lemma 8.1.1.

Let $0 \le r \le e = 12$ be an even integer. The descent data on $\operatorname {\mathcal{M}}_{r,1}'$ is determined by

$$\begin{equation*} \widehat{\gamma _2}(\mathbf{e}) = \mathbf{e},\,\,\, \widehat{\gamma _4}(\mathbf{e}) = -(-\sqrt {-1})^{r/2}\mathbf{e},\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}) = H_{\gamma _3^{\pm 1}}(u)^{-3r/2}\mathbf{e}, \end{equation*}$$

and the descent data on $\operatorname {\mathcal{M}}_{r,\omega }'$ is determined by

$$\begin{equation*} \widehat{\gamma _2}(\mathbf{e}) = \mathbf{e},\,\,\, \widehat{\gamma _4}(\mathbf{e}) = (-\sqrt {-1})^{r/2}\mathbf{e},\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}) = H_{\gamma _3^{\pm 1}}(u)^{-3r/2}\mathbf{e}. \end{equation*}$$

In particular, $\gamma _4^2 = -1$ on $D({\mathcal{G}}_{r,1}')$ if and only if $\gamma _4^2 = -1$ on $D({\mathcal{G}}_{r,\omega }')$ if and only if $r=2$, $6$ or $10$.

Proof.

Certainly $\widehat{\gamma _2}(\mathbf{e}) = \mathbf{e}$. We have already seen in Lemma 5.7.1 that descent data must exist in each case, so our task is to compute the unique units $\xi _{{\gamma _4}}, \xi _{{\gamma _3}^{\pm 1}} \in (\mathbf{F}_9[u]/u^{36})^{\times }$ so that

$$\begin{equation*} \widehat{{\gamma _4}}(\mathbf{e}) = \xi _{{\gamma _4}}\mathbf{e},\,\,\,\, \widehat{{\gamma _3}^{\pm 1}}(\mathbf{e}) = \xi _{{\gamma _3}^{\pm 1}}\mathbf{e} \end{equation*}$$

corresponds to generic fibre descent data for the mod 3 cyclotomic or trivial character on $G_3$. The case of ${\gamma _3}^{\pm 1}$ follows from Lemma 5.7.1.

From the condition

$$\begin{equation*} \widehat{{\gamma _4}}\circ \phi '_1(u^r\mathbf{e}) = \phi '_1 \circ \widehat{{\gamma _4}}(u^r\mathbf{e}) \end{equation*}$$

we get $\xi _{{\gamma _4}}^2(u) = (-\sqrt {-1})^r$, so

$$\begin{equation*} \xi _{{\gamma _4}}(u) = \pm (-\sqrt {-1})^{r/2}. \end{equation*}$$

The non-zero morphisms ${\operatorname {\mathcal{M}}}_{r,1} \rightarrow {\operatorname {\mathcal{M}}}_{12,1}$ are given by $\mathbf{e}\mapsto \pm u^{3(12-r)/2} \mathbf{e}$ and the non-zero morphisms ${\operatorname {\mathcal{M}}}_{0,\omega } \rightarrow {\operatorname {\mathcal{M}}}_{r,\omega }$ are given by $\mathbf{e}\mapsto \pm u^{3r/2}\mathbf{e}$. Thus, it suffices to check that $\widehat{{\gamma _4}}\mathbf{e}= \mathbf{e}$ on ${\operatorname {\mathcal{M}}}_{12,1}'$ and $\widehat{{\gamma _4}}\mathbf{e}= \mathbf{e}$ on ${\operatorname {\mathcal{M}}}_{0,\omega }'$. In both cases we have shown that $\widehat{{\gamma _4}}\mathbf{e}= \pm \mathbf{e}$ and so we only need to check that $\gamma _4=1$ on $D({\mathcal{G}}_{12,1}')$ and $D({\mathcal{G}}_{0,\omega }')$. That is, we have to show that the ${\mathcal{O}}_{F'}$-group scheme maps $\mathbf{Z}/3\mathbf{Z}\rightarrow {^{\gamma _4} (\mathbf{Z}/3\mathbf{Z})}$ and $\mu _3 \rightarrow {^{\gamma _4} \mu _3}$ arising from the canonical generic fibre descent data induce the identity on the special fibres. This is easy.

■

Lemma 8.1.2.

Let ${\operatorname {\mathcal{M}}}$ be an object of ${\underline{\phi _1{\mathrm{{-mod}}}}}_{F}$ corresponding to a finite flat group scheme ${\mathcal{G}}$ and let $\{ [g] \}$ be descent data on ${\mathcal{G}}'= {\mathcal{G}}\times {\mathcal{O}}_{F'}$ relative to $\mathbf{Q}_3$. Assume that $(\mathcal{G}',\{\widehat{g}\})_{\mathbf{Q}_3}$ can be filtered so that each graded piece is isomorphic to $\mathbf{Z}/3\mathbf{Z}$ or $\mu _3$ and so that the corresponding filtration of $(\operatorname {\mathcal{M}}',\{\widehat{g}\})$ in ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$ has successive quotients of the form $\operatorname {\mathcal{M}}'_{r_j,\chi _j}$ with $r_j \in \{2,6,10\}$ and $\chi _j \in \{{\mathbf{1}}, \omega \}$. Then $\gamma _4^2 = -1$ on $\operatorname {\mathcal{M}}'/u\operatorname {\mathcal{M}}'$ and there exists a basis $\{\mathbf{e}_j\}$ of $\operatorname {\mathcal{M}}$ over $\mathbf{F}_3[u]/u^{36}$ so that for all $j$

•

$\mathbf{e}_j \in \phi _1(\operatorname {\mathcal{M}}_1)$,

•

$\mathbf{e}_j$ is an eigenvector of the $\mathbf{F}_9$-linear map $\widehat{\gamma _4}$ on $\operatorname {\mathcal{M}}'$,

•

$\mathbf{e}_j$ lies in the part of the filtration of $\operatorname {\mathcal{M}}'$ which surjects onto $\operatorname {\mathcal{M}}'_{r_j,\chi _j}$ and this surjection sends $\mathbf{e}_j$ onto the standard basis vector $\mathbf{e}$ of $\operatorname {\mathcal{M}}'_{r_j,\chi _j}$ over $\mathbf{F}_9[u]/u^{36}$.

Proof.

Since $\gamma _4^2$ acts linearly on $\operatorname {\mathcal{M}}'/u\operatorname {\mathcal{M}}'$ and $(\gamma _4^2)^2 = 1$, the action of $\gamma _4^2$ must be semi-simple. The eigenvalues of $\gamma _4^2$ are all equal to $-1$, so necessarily $\gamma _4^2 = -1$ on $\operatorname {\mathcal{M}}'/u\operatorname {\mathcal{M}}'$.

We now argue by induction on the number of Jordan-Hölder factors in the generic fibre, the case of length 1 being clear. Thus, we can assume we have a short exact sequence in ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$,

$$\begin{equation*} 0 \longrightarrow {\mathcal{N}}' \longrightarrow \operatorname {\mathcal{M}}' \longrightarrow \operatorname {\mathcal{M}}'_{r,\chi } \longrightarrow 0, \end{equation*}$$

so the lemma is known for $\mathcal{N}'$. We just have to find $\mathbf{e}_0 \in \phi _1(\operatorname {\mathcal{M}}_1)$ mapping onto the standard basis vector $\mathbf{e}$ in $\operatorname {\mathcal{M}}'_{r,\chi }$ such that $\mathbf{e}_0$ is an eigenvector of $\widehat{\gamma }_4$. Since $\phi '_1(\operatorname {\mathcal{M}}'_1) \rightarrow \phi '_1((\operatorname {\mathcal{M}}'_{r,\chi })_1)$ is a surjective map of $\mathbf{F}_9$-vector spaces which is compatible with the semi-simple $\mathbf{F}_9$-linear endomorphism $\widehat{\gamma }_4$ on each side, we can find $\mathbf{e}'_0 \in \phi '_1(\operatorname {\mathcal{M}}'_1)$ mapping onto $\mathbf{e}$ with $\mathbf{e}'_0$ an eigenvector of $\widehat{\gamma }_4$, say $\widehat{\gamma }_4(\mathbf{e}'_0) = (\sqrt {-1})^{\pm 1} \mathbf{e}'_0$. Since

$$\begin{equation*} \widehat{\gamma }_4\circ \widehat{\gamma }_2(\mathbf{e}'_0) = \widehat{{\gamma }}_2\circ \widehat{\gamma }_4^3(\mathbf{e}'_0) = \widehat{{\gamma }}_2(\sqrt {-1}^{\mp 1}\mathbf{e}'_0) = \sqrt {-1}^{\pm 1} \widehat{{\gamma }}_2(\mathbf{e}'_0), \end{equation*}$$

the element $\mathbf{e}_0 = (1/2)(\mathbf{e}'_0 + \widehat{\gamma }_2(\mathbf{e}'_0))$ maps to $\mathbf{e}$ and is an an eigenvector for $\widehat{\gamma }_4$. Also, $\mathbf{e}_0 \in \phi '_1(\operatorname {\mathcal{M}}'_1)$ is $\widehat{{\gamma }}_2$-invariant and $\widehat{{\gamma }}_2$ commutes with $\phi '_1$, so $\mathbf{e}_0 \in \phi _1(\operatorname {\mathcal{M}}_1)$.

■

8.2. Models for ${\overline{\rho }}$

Proposition 8.2.1.

There exists a unique object $({\mathcal{G}}',\{ [g]\})$ of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}$ such that $({\mathcal{G}}',\{ [g]\})_{\mathbf{Q}_3}$ corresponds to ${\overline{\rho }}$. If we set $({\operatorname {\mathcal{M}}}({\overline{\rho }})', \{ \widehat{g} \})={\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}', \{ [g] \})$, then $({\operatorname {\mathcal{M}}}({\overline{\rho }})', \{ \widehat{g} \})$ is an extension of ${\operatorname {\mathcal{M}}}_{2,\omega }'$ by ${\operatorname {\mathcal{M}}}_{10,1}'$ in ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$. Moreover Frobenius is not identically zero on $D({\mathcal{G}}')$.

Proof.

Let $({\mathcal{G}}',\{ [g]\})$ be an object of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}$ such that $({\mathcal{G}}',\{ [g]\})_{\mathbf{Q}_3}$ corresponds to ${\overline{\rho }}$, and set $({\operatorname {\mathcal{M}}}', \{ \widehat{g} \})={\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}', \{ [g] \})$. As in the discussion following Theorem 5.6.1, we have canonically $\operatorname {\mathcal{M}}' \simeq \mathbf{F}_9 \otimes _{\mathbf{F}_3} \operatorname {\mathcal{M}}$ for a Breuil module $\operatorname {\mathcal{M}}$ over ${\mathcal{O}}_F$, with $\widehat{\gamma }_2$ acting as $\gamma _2 \otimes 1$. By Lemma 8.1.1, there is a short exact sequence of Breuil modules over ${\mathcal{O}}_F$,

$$\begin{equation*} 0 \longrightarrow \operatorname {\mathcal{M}}(s,\delta ) \longrightarrow \operatorname {\mathcal{M}}\longrightarrow \operatorname {\mathcal{M}}(r,1) \longrightarrow 0, \end{equation*}$$

with $r, s \in \{2,6,10\}$ and this is compatible with descent data after base change to ${\mathcal{O}}_{F'}$ in the sense that we obtain an exact sequence

$$\begin{equation*} 0 \longrightarrow \operatorname {\mathcal{M}}_{s,1}' \longrightarrow \operatorname {\mathcal{M}}' \longrightarrow \operatorname {\mathcal{M}}_{r,\omega }' \longrightarrow 0, \end{equation*}$$

compatible with descent data. Because ${\overline{\rho }}$ is très ramifié, it follows that ${\overline{\rho }}|_{G_{F'}}$ is non-split, so the sequence

$$\begin{equation*} 0 \longrightarrow \operatorname {\mathcal{M}}(s,\delta ) \longrightarrow \operatorname {\mathcal{M}}\longrightarrow \operatorname {\mathcal{M}}(r,1) \longrightarrow 0 \end{equation*}$$

is non-split.

We first show that we must have $(r,s) = (2,10)$. Since ${\overline{\rho }}$ is self-dual, in order to prove $(r,s) = (2,10)$ we may use Cartier duality (and Lemma 5.2.1) in order to reduce to the case where $r + s \le e = 12$. We will first rule out cases with $r \ge s$ and then the case $(r,s) = (2,6)$.

By Lemmas 8.1.1 and 8.1.2, we can write

$$\begin{equation*} \operatorname {\mathcal{M}}= (\mathbf{F}_3[u]/u^{36})\mathbf{e}_1 \oplus (\mathbf{F}_3[u]/u^{36})\mathbf{e}'_{\omega }, \,\,\,\,\operatorname {\mathcal{M}}_1 = \langle u^s \mathbf{e}_1, u^r \mathbf{e}'_{\omega } + h\mathbf{e}_1 \rangle \end{equation*}$$

for some $h \in \mathbf{F}_3[u]/u^{36}$ so that

$$\begin{equation*} \phi _1(u^s\mathbf{e}_1) = \delta \mathbf{e}_1,\,\,\, \phi _1(u^r\mathbf{e}'_{\omega } + h\mathbf{e}_1) = \mathbf{e}'_{\omega } \end{equation*}$$

and

$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_1) = -(-\sqrt {-1})^{s/2}\mathbf{e}_1,\,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_{\omega }) = (-\sqrt {-1})^{r/2}\mathbf{e}'_{\omega }. \end{equation*}$$

Recall from Lemma 5.2.2 that the “parameter” $h$ gives an isomorphism of abstract groups

$$\begin{equation*} (\mathbf{F}_3[u]/u^{36})/\{u^s t - \delta u^r t^3|t\in \mathbf{F}_3[u]/u^{36}\} \simeq \operatorname {Ext}^1_{{\underline{\phi _1{\mathrm{{-mod}}}}}_F}(\operatorname {\mathcal{M}}(r,1),\operatorname {\mathcal{M}}(s,\delta )). \end{equation*}$$

It is easy to see that

$$\begin{equation*} \widehat{\gamma }_4(\operatorname {\mathcal{M}}'_1) \subseteq \operatorname {\mathcal{M}}'_1, \,\,\,\, \widehat{\gamma }_4\circ \phi '_1 = \phi '_1 \circ \widehat{\gamma }_4\,\,\,\,{\mathrm{{on}}}\, \,\,\operatorname {\mathcal{M}}'_1 \end{equation*}$$

if and only if $\widehat{\gamma }_4(u^r\mathbf{e}'_{\omega } + h\mathbf{e}_1) \in \operatorname {\mathcal{M}}'_1$ and $\widehat{\gamma }_4(\mathbf{e}'_{\omega }) = \phi '_1 \circ \widehat{\gamma }_4( u^r\mathbf{e}'_{\omega } + h\mathbf{e}_1)$, or equivalently

$$\begin{equation*} (\sqrt {-1})^{r/2}h(u) \equiv -(-\sqrt {-1})^{s/2}h(-\sqrt {-1}u) \bmod u^{12+s}. \end{equation*}$$

This says exactly that

$$\begin{equation} j \equiv 2 - (r+s)/2 \bmod 4 {\tag{8.2.1}} \end{equation}$$

for any $j < 12+s$ with a non-zero $u^j$ term appearing in $h$.

If $(r,s) = (6,2)$ this would force $h \equiv 0 \bmod u^2$, yet $\{u^2 t - \delta u^6 t^3 |t\in \mathbf{F}_3[u]/u^{36}\}$ contains all multiples of $u^2$, so

$$\begin{equation*} 0 \longrightarrow \operatorname {\mathcal{M}}(2,\delta ) \longrightarrow \operatorname {\mathcal{M}}\longrightarrow \operatorname {\mathcal{M}}(6,1) \longrightarrow 0 \end{equation*}$$

is split, a contradiction.

When $(r,s) = (10,2)$ or $(r,s) = (2,2)$ we see that $h \equiv h(0) \bmod u^4$, yet

$$\begin{equation*} u^4 (\mathbf{F}_3[u]/u^{36}) \subseteq \{u^s t - \delta u^r t^3|t\in \mathbf{F}_3[u]/u^{36}\}, \end{equation*}$$

so the choice of $\mathbf{e}'_{\omega }$ may be changed in order to arrange that

$$\begin{equation*} h \in \mathbf{F}_3 \end{equation*}$$

(though making this change of basis of $\operatorname {\mathcal{M}}$ may destroy the “diagonal” form of $\widehat{\gamma }_4$). Since

$$\begin{equation*} 0 \longrightarrow \operatorname {\mathcal{M}}(s,\delta ) \longrightarrow \operatorname {\mathcal{M}}\longrightarrow \operatorname {\mathcal{M}}(r,1) \longrightarrow 0 \end{equation*}$$

is non-split, necessarily $h \ne 0$, so by rescaling $\mathbf{e}'_{\omega }$ it can be assumed that $h = 1$. Then $\mathbf{V}_{\operatorname {\mathcal{M}}}(\mathbf{e}'_{\omega }) \equiv \mathbf{e}_1 \bmod u\operatorname {\mathcal{M}}$ (by Theorem 5.1.3) and

$$\begin{equation*} \phi (\mathbf{e}'_{\omega }) \equiv -(\delta /c_\pi )u^{3(12-r-s)}\mathbf{e}_1 \equiv -u^{3(12-r-s)}\mathbf{e}_1 \bmod u\operatorname {\mathcal{M}}. \end{equation*}$$

This forces $r+s = 12$. In particular, $(r,s) = (2,2)$ is ruled out.

For $(r,s) = (10,2)$, a splitting of the generic fibre ${\overline{\rho }}|_{F}$ is induced by the Breuil module map

$$\begin{equation*} \operatorname {\mathcal{M}}(0,1) \longrightarrow \operatorname {\mathcal{M}} \end{equation*}$$

defined by

$$\begin{equation*} \mathbf{e}\longmapsto u^{15}\mathbf{e}'_{\omega } + u^3 f\mathbf{e}_1 = u^5(u^{10}\mathbf{e}'_{\omega } + \mathbf{e}_1) + (uf-u^3)u^2\mathbf{e}_1, \end{equation*}$$

where $f \in \mathbf{F}_3[u]/u^{36}$ satisfies $f^3 - \delta f = u^6$ (i.e. $f = -\delta u^6 - u^{18}$, and a constant $c \in \mathbf{F}_3$ can even be added to this if $\delta = 1$). But ${\overline{\rho }}|_{G_{F'}}$ must be non-split, so this rules out $(r,s) = (10,2)$.

The remaining case with $r \ge s$ is $(r,s) = (6,6)$. In this case $\{u^s t - \delta u^r t^3|t\in \mathbf{F}_3[u]/u^{36}\}$ contains all multiples of $u^8$. But we have $j \equiv 0 \bmod 4$ for all $j < 12 + s = 18$ such that a non-zero $u^j$ term appears in $h$, so again (at the expense of possibly making the $\widehat{\gamma }_4$-action non-diagonal) we may assume

$$\begin{equation*} h = c + c'u^4 \end{equation*}$$

for some $c, c' \in \mathbf{F}_3$. Writing $\widehat{\gamma }_4(\mathbf{e}'_{\omega }) = (\sqrt {-1})\mathbf{e}'_{\omega } + h_{\gamma _4}(u)\mathbf{e}_1$, the commutativity of $\widehat{\gamma }_4$ and $\phi '_1$ amounts to $h_{\gamma _4} = -\delta h_{\gamma _4}^3$, so $h_{\gamma _4}(u) = b\sqrt {-\delta }$ for some $b \in \mathbf{F}_3$. The condition $\widehat{\gamma }_4^4(\mathbf{e}'_{\omega }) = \mathbf{e}'_{\omega }$ forces $b = 0$, so $\widehat{\gamma }_4$ still has diagonal action. This analysis shows that the map of $\mathbf{F}_3$-vector spaces

$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}(\operatorname {\mathcal{M}}'_{6,\omega },\operatorname {\mathcal{M}}'_{ 6,\mathbf{1}}) \longrightarrow \operatorname {Ext}^1_{{\underline{\phi _1{\mathrm{{-mod}}}}}_F}(\operatorname {\mathcal{M}}(6,1), \operatorname {\mathcal{M}}(6,\delta )) \end{equation*}$$

has at most a 2-dimensional image. If $c' + \delta c = 0$, then the Breuil module map

$$\begin{equation*} \mathbf{F}_9 \otimes \operatorname {\mathcal{M}}(0,1) \longrightarrow \operatorname {\mathcal{M}}' \end{equation*}$$

defined by

$$\begin{equation*} \mathbf{e}\longmapsto c \delta u^7 \mathbf{e}_1 + u^3(u^6 \mathbf{e}'_{\omega } + (c+c'u^4)\mathbf{e}_1) \end{equation*}$$

gives a splitting of the corresponding representation of $G_{F'}$. Thus the image of

$$\begin{equation} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}(\operatorname {\mathcal{M}}'_{6,\omega }, \operatorname {\mathcal{M}}'_{6,\mathbf{1}}) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_{F'}]}(1,\omega ) {\tag{8.2.2}} \end{equation}$$

is at most one dimensional and, because ${\overline{\rho }}|_{G_{F'}}$ is non-split, the pair $(c,c')$ corresponding to a model of ${\overline{\rho }}$ satisfies $c' + \delta c \ne 0$.

At this point, we treat the cases $\delta = \pm 1$ separately. Consider first the case $\delta =1$. We must have

$$\begin{equation*} \widehat{{\gamma }}_3(\mathbf{e}_1) = H_{{\gamma _3}}(u)^{-9}\mathbf{e}_1,\,\,\,\, \widehat{{\gamma }}_3(\mathbf{e}'_{\omega }) = H_{{\gamma _3}}(u)^{-9}\mathbf{e}'_{\omega } + h_{{\gamma _3}}(u)\mathbf{e}_1, \end{equation*}$$

where $h_{{\gamma _3}}(u) \in \mathbf{F}_9[u]/u^{36}$ lies in $\mathbf{F}_3[u]/u^{36}$ because $\widehat{{\gamma _3}}$ commutes with $\widehat{\gamma }_2$. Evaluating $\widehat{{\gamma }}_3\circ \phi '_1 \equiv \phi '_1 \circ \widehat{{\gamma }}_3 \bmod u\operatorname {\mathcal{M}}'$ on $u^6\mathbf{e}'_{\omega } + (c+c'u^4)\mathbf{e}_1 \in \operatorname {\mathcal{M}}'_1$ and using our knowledge of $H_{\gamma _3}(u) \bmod 3$, we arrive at

$$\begin{equation*} h_{{\gamma _3}}(0) = \delta (h_{{\gamma _3}}(0)^3 + (c + c')), \end{equation*}$$

which is impossible for $h_{{\gamma _3}}(0) \in \mathbf{F}_3$ with $\delta = 1$ because $c + c' = c + \delta c' \in \mathbf{F}_3^\times$.

Now let us turn to the case $\delta =-1$, still in the case $(r,s)=(6,6)$. In this case $\operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(1,\omega ) \rightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_{F'}]}(1,\omega )$ is injective and so by (8.2.2) we see that the image of

$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}(\operatorname {\mathcal{M}}'_{6,\omega },\operatorname {\mathcal{M}}'_{6,\mathbf{1}}) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(1,\omega ) \end{equation*}$$

is at most one dimensional. Thus to exclude the case $(r,s)=(6,6)$ and $\delta =-1$, it suffices to show that this image contains the peu ramifié line (as ${\overline{\rho }}$ is très ramifié). By Proposition 5.2.1 of Man, there is an elliptic curve $E'_{/\mathbf{Q}_3}$ which has supersingular reduction over $\mathbf{Q}_3(\sqrt {-1},\beta )$, with ${\overline{\rho }}_{E',3}$ a non-split, peu ramifié extension of $1$ by $\omega$. The representation ${\overline{\rho }}_{E',3}|_{F'}$ is non-split (again because $H^1(G_3,\omega ) \rightarrow H^1(G_{F'},\omega )$ is injective in the $\delta =-1$ case). Let $\operatorname {\mathcal{N}}'$ be the Breuil module corresponding to the 3-torsion on the Néron model of $E' \times _{\mathbf{Q}_3} F'$, so $\operatorname {\mathcal{N}}'$ admits descent data $\{\widehat{g}'\}$ via the universal property of Néron models. The filtration of ${\overline{\rho }}$ induces a short exact sequence in ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$

$$\begin{equation*} 0 \longrightarrow \operatorname {\mathcal{M}}'_{a,{\mathbf{1}}} \longrightarrow (\operatorname {\mathcal{N}}',\{\widehat{g}'\}) \longrightarrow \operatorname {\mathcal{M}}'_{b,\omega } \longrightarrow 0 \end{equation*}$$

for some even $a, b$ with $2 \le a, b \le 10$. The Néron model of $E'\times _{\mathbf{Q}_3} \mathbf{Q}_3(\sqrt {-1},\beta )$ has local-local 3-torsion, and the induced local-local integral models $\mathcal{G}_{\omega }$ and $\mathcal{G}_{\mathbf{1}}$ of the diagonal characters $\omega |_{\mathbf{Q}_3(\sqrt {-1},\beta )}$ and $1|_{\mathbf{Q}_3(\sqrt {-1},\beta )}$ must be the unique local-local models (uniqueness follows from Corollary 1.5.1 of Ra). Moreover, Corollary 1.5.1 of Ra makes it clear that base change to ${\mathcal{O}}_{F'}$ takes the order $3$ group schemes $\mathcal{G}_{\omega }$ and $\mathcal{G}_{\mathbf{1}}$ to the integral models that lie in the middle of the well-ordered sets of integral models of $\omega |_{F'}$ and $1|_{F'}$. It follows that $a = b = 6$, so the map

$$\begin{equation*} \operatorname {Ext}^1_{\underline{\phi _1DD}_{F'/\mathbf{Q}_3}}(\operatorname {\mathcal{M}}'_{6,\omega }, \operatorname {\mathcal{M}}'_{6,{\mathbf{1}}}) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(1,\omega ) \end{equation*}$$

indeed hits the peu ramifié line.

We next exclude the case $(r,s)=(2,6)$. As a first step, we check that there is at most one possibility for the underlying Breuil module $\operatorname {\mathcal{M}}$ (ignoring the extension class structure) if $(r,s) = (2,6)$. We can write

$$\begin{equation*} \operatorname {\mathcal{M}}= (\mathbf{F}_3[u]/u^{36})\mathbf{e}_1 \oplus (\mathbf{F}_3[u]/u^{36})\mathbf{e}'_{\omega },\,\,\,\, \operatorname {\mathcal{M}}_1 = \langle u^6\mathbf{e}_1, u^2\mathbf{e}'_{\omega } + h\mathbf{e}_1\rangle \end{equation*}$$

for some necessarily non-zero $h \in \mathbf{F}_3[u]/u^{36}$ with

$$\begin{equation*} \phi _1(u^6\mathbf{e}_1) = \delta \mathbf{e}_1,\,\,\,\, \phi _1(u^2\mathbf{e}'_{\omega } + h\mathbf{e}_1) = \mathbf{e}'_{\omega } \end{equation*}$$

and

$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_1) = -\sqrt {-1}\mathbf{e}_1,\,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_{\omega }) = -\sqrt {-1}\mathbf{e}'_{\omega }. \end{equation*}$$

The combined conditions $\widehat{\gamma }_4(\operatorname {\mathcal{M}}'_1) \subseteq \operatorname {\mathcal{M}}'_1$ and $\phi '_1 \circ \widehat{\gamma }_4 = \widehat{\gamma }_4 \circ \phi '_1$ on $\operatorname {\mathcal{M}}'_1$ are equivalent to

$$\begin{equation*} h(u) \equiv -h(-\sqrt {-1}u) \bmod u^{18}. \end{equation*}$$

Since $\{u^6t - \delta u^2t^3 |t\in \mathbf{F}_3[u]/u^{36}\}$ contains $u^6 - \delta u^2$ and all multiples of $u^9$, we may change $\mathbf{e}'_{\omega }$ (at the expense of possibly losing the diagonal form for $\widehat{\gamma }_4$) so that $h = cu^2$ for some $c \in \mathbf{F}_3$. Since $h$ is necessarily non-zero, we may rescale to get $h = u^2$, so there is indeed at most one possibility for the underlying Breuil module $\operatorname {\mathcal{M}}$ when $(r,s) = (2,6)$.

Again we treat the cases $\delta =\pm 1$ separately. Consider first the case $\delta =-1$. We have seen above that there is an extension $\mathcal{E}_{6,6} = (\operatorname {\mathcal{N}}',\{\widehat{g}'\})$ of $\operatorname {\mathcal{M}}'_{6,\omega }$ by $\operatorname {\mathcal{M}}'_{6,{\mathbf{1}}}$ in ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$ corresponding to a non-split, peu ramifié extension of $1$ by $\omega$. Pulling back $\mathcal{E}_{6,6}$ by a non-zero map

$$\begin{equation*} \operatorname {\mathcal{M}}'_{2,\omega } \longrightarrow \operatorname {\mathcal{M}}'_{6,\omega } \end{equation*}$$

in ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$ given by $\mathbf{e}\mapsto \pm u^6 \mathbf{e}$, we get an extension $\mathcal{E}_{2,6}$ of $\operatorname {\mathcal{M}}'_{2,\omega }$ by $\operatorname {\mathcal{M}}'_{6,{\mathbf{1}}}$ in ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$ corresponding to a non-split, peu ramifié extension of $1$ by $\omega$. The underlying Breuil module of $\mathcal{E}_{2,6}$ must be isomorphic to $\mathbf{F}_9 \otimes _{\mathbf{F}_3} \operatorname {\mathcal{M}}$ for our uniquely determined $\operatorname {\mathcal{M}}$ (with $h = u^2$). By the injectivity of $H^1(G_3,\omega ) \rightarrow H^1(G_{F'},\omega )$ in the $\delta =-1$ case, we conclude that $\mathbf{F}_9 \otimes _{\mathbf{F}_3} \operatorname {\mathcal{M}}$ cannot admit descent data giving rise to a très ramifié element in $\operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(1,\omega )$. This rules out the case $(r,s) = (2,6)$ and $\delta =-1$.

Now turn to the case $(r,s)=(2,6)$ and $\delta =1$. We will show that with the Breuil module ${\operatorname {\mathcal{M}}}$ constructed above (with $h=u^2$), the Breuil module ${\operatorname {\mathcal{M}}}'= \mathbf{F}_9 \otimes _{\mathbf{F}_3} \operatorname {\mathcal{M}}$ does not admit descent data relative to $F'/\mathbf{Q}_3$ (with $\widehat{\gamma }_2 = \gamma _2 \otimes 1$, without loss of generality). One checks that $N(\mathbf{e}_1) = N(\mathbf{e}'_{\omega }) = 0$, so

$$\begin{equation*} N \circ \phi _1 = 0. \end{equation*}$$

We must have

$$\begin{equation*} \widehat{\gamma }_3(\mathbf{e}_1) = H_{{\gamma _3}}(u)^{-9}\mathbf{e}_1,\,\,\,\, \widehat{\gamma }_3(\mathbf{e}'_{\omega }) = H_{{\gamma _3}}(u)^{-3}\mathbf{e}'_{\omega } + h_{{\gamma _3}}(u)\mathbf{e}_1 \end{equation*}$$

for some $h_{{\gamma _3}} \in \mathbf{F}_9[u]/u^{36}$. As usual, since $\widehat{\gamma }_3$ and $\widehat{\gamma }_2$ must commute, we have $h_{{\gamma _3}} \in \mathbf{F}_3[u]/u^{36}$. The condition $\widehat{\gamma }_3(\operatorname {\mathcal{M}}'_1) \subseteq \operatorname {\mathcal{M}}'_1$ is equivalent to

$$\begin{equation*} \widehat{\gamma }_3(u^2\mathbf{e}'_{\omega } + u^2\mathbf{e}_1) \in \operatorname {\mathcal{M}}'_1, \end{equation*}$$

which amounts to

$$\begin{equation*} h_{{\gamma _3}}(u) \equiv H_{{\gamma _3}}^{-3} - H_{{\gamma _3}}^{-9} \equiv 0 \bmod u^4, \end{equation*}$$

so

$$\begin{equation*} \widehat{\gamma }_3(u^2\mathbf{e}'_{\omega } + u^2\mathbf{e}_1) = H_{{\gamma _3}}(u)^{-1}(u^2\mathbf{e}'_{\omega } + u^2\mathbf{e}_1) + \left(\frac{h_{{\gamma _3}} - H_{{\gamma _3}}^{-3} + H_{{\gamma _3}}^{-9}}{u^4}\right)H_{{\gamma _3}}^2u^6\mathbf{e}_1. \end{equation*}$$

As $N \circ \phi _1 = 0$, we have

$$\begin{equation*} \widehat{\gamma }_3\circ \phi '_1 = \phi '_1 \circ \widehat{\gamma }_3 \end{equation*}$$

on $\operatorname {\mathcal{M}}'_1$. Evaluating this identity on $u^2 \mathbf{e}'_{\omega } + u^2\mathbf{e}_1 \in \operatorname {\mathcal{M}}_1$ gives

$$\begin{equation*} h_{{\gamma _3}} = H_{{\gamma _3}}^6\cdot \left(\frac{h_{{\gamma _3}} - H_{{\gamma _3}}^{-3} + H_{{\gamma _3}}^{-9}}{u^4}\right)^3, \end{equation*}$$

so $h_{{\gamma _3}}$ is a cube. Thus, $h_{{\gamma _3}} = u^6 g_{{\gamma _3}}$ for some $g_{{\gamma _3}} \in \mathbf{F}_3[u]/u^{36}$.

Since $H_{{\gamma _3}}^3 \equiv 1 + u^6 \bmod u^{12}$, we compute

$$\begin{equation*} H_{{\gamma _3}}^{-9} - H_{{\gamma _3}}^{-3} \equiv u^6 \bmod u^{12}, \end{equation*}$$

so

$$\begin{equation*} h_{{\gamma _3}} \equiv H_{{\gamma _3}}^6\cdot \left(\left(\frac{h_{{\gamma _3}}}{u^4}\right)^3 + u^6\right) \bmod u^7 \end{equation*}$$

and

$$\begin{equation*} g_{{\gamma _3}}(0) = g_{{\gamma _3}}(0)^3 + 1 \end{equation*}$$

in $\mathbf{F}_3$. This is absurd. This rules out all possibilities for $(r,s)$ aside from $(r,s) = (2,10)$. Uniqueness now follows from Corollary 4.1.5.

From Theorem 5.4.2 of Man and Proposition B.4.2 of CDT we see that there is an elliptic curve $E_{/\mathbf{Q}_3}$ such that $E[3](\overline{\mathbf{Q}}_3) \cong {\overline{\rho }}$ and $\rho _{E,3}$ has type $\tau _{\pm 3}$. Let ${\mathcal{E}}$ denote the Néron model of $E \times _{\mathbf{Q}_3} F'$ over ${\mathcal{O}}_{F'}$. By the Néron property of ${\mathcal{E}}_{/{\mathcal{O}}_{F'}}$ we see that ${\mathcal{E}}[3^\infty ]$ has descent data over $\mathbf{Q}_3$. As in §4.5 we see that ${\mathcal{I}}$ annihilates the Dieudonné module of ${\mathcal{E}}[3^\infty ] \times \mathbf{F}_9$. Thus ${\operatorname {\mathcal{M}}}({\overline{\rho }})' \cong {\operatorname {\mathcal{M}}}_\pi ({\mathcal{E}}[3])$ in ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$ and it follows that Frobenius is non-zero on $D({\mathcal{G}}')$.

■

8.3. Completion of the proof of Theorem 4.5.1

Lemma 8.3.1.

Let $({\mathcal{G}}',\{ [g]\})$ be the unique object of ${\mathcal{F}}{\operatorname {\mathcal{D}}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}$ such that $({\mathcal{G}}',\{ [g]\})_{\mathbf{Q}_3}$ corresponds to ${\overline{\rho }}$. Set $({\operatorname {\mathcal{M}}}({\overline{\rho }})', \{ \widehat{g} \})={\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}', \{ [g] \})$. The natural map of groups

$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}( (\operatorname {\mathcal{M}}({\overline{\rho }})',\{\widehat{g}\}), (\operatorname {\mathcal{M}}({\overline{\rho }})',\{\widehat{g}\})) \longrightarrow \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}(\operatorname {\mathcal{M}}'_{10,{\mathbf{1}}}, \operatorname {\mathcal{M}}'_{2,\omega }), \end{equation*}$$

using pushout by $(\operatorname {\mathcal{M}}({\overline{\rho }})',\{\widehat{g}\}) \rightarrow {\operatorname {\mathcal{M}}}_{2,\omega }'$ and pullback by ${\operatorname {\mathcal{M}}}_{10,{\mathbf{1}}}' \rightarrow (\operatorname {\mathcal{M}}({\overline{\rho }})',\{\widehat{g}\})$, is zero.

Proof.

Let $(\widetilde{\operatorname {\mathcal{M}}}',\{\widehat{g}\})$ represent a class in $\operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}( (\operatorname {\mathcal{M}}({\overline{\rho }})',\{\widehat{g}\}), (\operatorname {\mathcal{M}}({\overline{\rho }})',\{\widehat{g}\}))$ and let $(\operatorname {\mathcal{M}}',\{\widehat{g}\})$ be its image in $\operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}(\operatorname {\mathcal{M}}'_{10,{\mathbf{1}}}, \operatorname {\mathcal{M}}'_{2,\omega })$. By Lemma 5.2.2, $\operatorname {\mathcal{M}}' = \mathbf{F}_9 \otimes \operatorname {\mathcal{M}}$ with $\widehat{\gamma }_2 = \gamma _2 \otimes 1$ and

$$\begin{equation*} \operatorname {\mathcal{M}}= (\mathbf{F}_3[u]/u^{36})\mathbf{e}_{\omega }\oplus (\mathbf{F}_3[u]/u^{36})\mathbf{e}'_1,\,\,\,\, \operatorname {\mathcal{M}}_1 = \langle u^2\mathbf{e}_{\omega }, u^{10}\mathbf{e}'_1 + (c+c'u)\mathbf{e}_{\omega }\rangle , \end{equation*}$$

with $c, c' \in \mathbf{F}_3$. Also,

$$\begin{equation*} \phi _1(u^2\mathbf{e}_{\omega }) = \mathbf{e}_{\omega },\,\,\,\, \phi _1(u^{10}\mathbf{e}'_1 + (c+c'u)\mathbf{e}_{\omega }) = \delta \mathbf{e}'_1, \end{equation*}$$

and

$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_{\omega }) = -\sqrt {-1}\mathbf{e}_{\omega },\,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_1) = \sqrt {-1}\mathbf{e}'_1 + h_{\gamma _4}(u)\mathbf{e}_{\omega } \end{equation*}$$

for some $h_{\gamma _4} \in \mathbf{F}_9[u]/u^{36}$.

The properties $\widehat{\gamma }_4(\operatorname {\mathcal{M}}'_1) \subseteq \operatorname {\mathcal{M}}'_1$ and $\widehat{\gamma }_4\circ \phi '_1 = \phi '_1 \circ \widehat{\gamma }_4$ on $\operatorname {\mathcal{M}}'_1$ amount to

$$\begin{equation*} c' = 0,\,\,\,\, h_{\gamma _4} = -\delta h_{\gamma _4}^3 u^{24}, \end{equation*}$$

so $h_{\gamma _4} = 0$. If $c = 0$, then $N \circ \phi _1 = 0$, so $\widehat{\gamma }_3\circ \phi '_1 = \phi '_1\circ \widehat{\gamma }_3$ on $\operatorname {\mathcal{M}}'_1$. From this we readily see that $(\operatorname {\mathcal{M}}',\{\widehat{g}\})$ is split in ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$, as desired.

Now assume $c \ne 0$; we will deduce a contradiction. Consider the rank three Breuil module with descent data

$$\begin{equation*} (\operatorname {\mathcal{N}}, \{ \widehat{g} \})= (\widetilde{\operatorname {\mathcal{M}}}',\{\widehat{g}\})/ ({\operatorname {\mathcal{M}}}_{10,{\mathbf{1}}}', \{ \widehat{g} \}), \end{equation*}$$

where ${\operatorname {\mathcal{M}}}_{10,{\mathbf{1}}}' \hookrightarrow {\operatorname {\mathcal{M}}}({\overline{\rho }})' \hookrightarrow \widetilde{{\operatorname {\mathcal{M}}}}'$. Then $\operatorname {\mathcal{N}}$ has an ordered basis $\{\mathbf{e}_{\omega },\mathbf{e}'_1,\mathbf{e}'_{\omega }\}$ with respect to which

$$\begin{equation*} \operatorname {\mathcal{N}}_1 = \langle u^2\mathbf{e}_{\omega }, u^{10}\mathbf{e}'_1 + \mathbf{e}_{\omega }, u^2\mathbf{e}'_{\omega } + h\mathbf{e}'_1 + (b+b'u)\mathbf{e}_{\omega }\rangle \end{equation*}$$

for some $b,b' \in \mathbf{F}_9$ and $h=a+a'u^4+a''u^8\in \mathbf{F}_9[u]/(u^{36})$ defined modulo $\{ u^{10}t-\delta u^2 t^3 \}$ (see 8.2.1). Since our base field $F'$ has absolute ramification degree $12$, $\operatorname {\mathcal{N}}_1$ contains

$$\begin{equation*} u^{12}\mathbf{e}'_{\omega } = u^{10}(u^2\mathbf{e}'_{\omega } + h\mathbf{e}'_1 + (b+b'u)\mathbf{e}_{\omega }) - h(u^{10}\mathbf{e}'_1 + \mathbf{e}_{\omega }) + (h - u^{10}(b+b'u))\mathbf{e}_{\omega }. \end{equation*}$$

From the list of generators of ${\operatorname {\mathcal{N}}}_1$, it is not difficult to check that in the above expression for $u^{12} \mathbf{e}_\omega ' \in {\operatorname {\mathcal{N}}}_1$, $u^2$ must divide the coefficient of $\mathbf{e}_{\omega }$. Thus $a = 0$.

We must have $\operatorname {\mathcal{N}}/ \langle \mathbf{e}_\omega \rangle \cong {\operatorname {\mathcal{M}}}({\overline{\rho }})'$. Since $a = 0$, $\operatorname {\mathcal{M}}({\overline{\rho }})$ has basis $\{\mathbf{e}'_1,\mathbf{e}'_{\omega }\}$ and

$$\begin{equation*} \operatorname {\mathcal{M}}({\overline{\rho }})_1 = \langle u^{10}\mathbf{e}'_1, u^2\mathbf{e}'_{\omega }+(a'u^4+a''u^8)\mathbf{e}'_1 \rangle . \end{equation*}$$

Since $\phi _1$ for $\operatorname {\mathcal{M}}({\overline{\rho }})$ satisfies

$$\begin{equation*} \phi _1(u^{10}\mathbf{e}_1')=\delta \mathbf{e}_1', \,\,\, \phi _1(u^2\mathbf{e}_\omega '+(a'u^4+a'' u^8)\mathbf{e}_1')=\mathbf{e}_\omega ', \end{equation*}$$

it follows immediately that $\phi \equiv 0 \bmod u{\operatorname {\mathcal{M}}}({\overline{\rho }})$, which (using Theorem 5.1.3) contradicts Proposition 8.2.1.

■

Corollary 8.3.2.

The natural map

$$\begin{equation*} \theta _0: \operatorname {Ext}^1_{{\operatorname {\mathcal{S}}}_{\pm 3}}({\overline{\rho }},{\overline{\rho }}) \longrightarrow H^1(G_3,\omega ) \end{equation*}$$

is zero.

Theorem 4.7.4, and hence Theorem 4.4.1, now follow from the first case of the following lemma. We include the second case to simplify the proof.

Lemma 8.3.3.

The maps of groups

$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}( \operatorname {\mathcal{M}}'_{10,{\mathbf{1}}},\operatorname {\mathcal{M}}'_{10,{\mathbf{1}}}) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(1,1), \end{equation*}$$

$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}(\operatorname {\mathcal{M}}'_{2,\omega }, \operatorname {\mathcal{M}}'_{2,\omega }) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(\omega ,\omega ) \end{equation*}$$ have images inside the line of extension classes that split over an unramified extension of $\mathbf{Q}_3$.

Proof.

Since

$$\begin{equation*} H^1(G_3,\mathbf{Z}/3) \longrightarrow H^1(G_{F'},\mathbf{Z}/3) \end{equation*}$$

is injective and induces an isomorphism between the subgroups of unramified classes, it suffices to check that

$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}( \operatorname {\mathcal{M}}'_{10,{\mathbf{1}}},\operatorname {\mathcal{M}}'_{10,{\mathbf{1}}}) \longrightarrow \operatorname {Ext}^1_{{\underline{\phi _1{\mathrm{{-mod}}}}}_{F}}(1,1), \end{equation*}$$

$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}(\operatorname {\mathcal{M}}'_{2,\omega }, \operatorname {\mathcal{M}}'_{2,\omega }) \longrightarrow \operatorname {Ext}^1_{{\underline{\phi _1{\mathrm{{-mod}}}}}_F}(\omega ,\omega ) \end{equation*}$$ have images consisting of elements split over an unramified extension of $F$. By Cartier duality it suffices to consider only the second map.

Consider a representative $(\operatorname {\mathcal{M}}',\{\widehat{g}\})$ of an element in $\operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}(\operatorname {\mathcal{M}}'_{2,\omega }, \operatorname {\mathcal{M}}'_{2,\omega })$. Lemma 5.2.2 ensures that we can write

$$\begin{equation*} \operatorname {\mathcal{M}}= (\mathbf{F}_3[u]/u^{36})\mathbf{e}_{\omega } \oplus (\mathbf{F}_3[u]/u^{36})\mathbf{e}'_{\omega }, \,\,\,\, \operatorname {\mathcal{M}}_1 = \langle u^2\mathbf{e}_{\omega }, u^2\mathbf{e}'_{\omega } + h\mathbf{e}_{\omega }\rangle \end{equation*}$$

for some $h = c + c'u + c'' u^2$ with $c, c', c'' \in \mathbf{F}_3$ and

$$\begin{equation*} \phi _1(u^2\mathbf{e}_{\omega }) = \mathbf{e}_{\omega },\,\,\,\, \phi _1(u^2\mathbf{e}'_{\omega } + h\mathbf{e}_{\omega }) = \mathbf{e}'_{\omega }. \end{equation*}$$

We have

$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_{\omega }) = -\sqrt {-1}\mathbf{e}_{\omega },\,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_{\omega }) = -\sqrt {-1}\mathbf{e}'_{\omega } + h_{\gamma _4}(u)\mathbf{e}_{\omega } \end{equation*}$$

for some $h_{\gamma _4} \in \mathbf{F}_9[u]/u^{36}$, and the condition $\widehat{\gamma }_4(\operatorname {\mathcal{M}}'_1) \subseteq \operatorname {\mathcal{M}}'_1$ is equivalent to

$$\begin{equation*} h(u) \equiv -h(-\sqrt {-1}u) \bmod u^2, \end{equation*}$$

so $c = c' = 0$. The Breuil module extension class $\operatorname {\mathcal{M}}$ over ${\mathcal{O}}_F$ (ignoring descent data) therefore only depends on the parameter $c'' \in \mathbf{F}_3$. We then have a splitting $\overline{\mathbf{F}}_3 \otimes _{\mathbf{F}_3} \operatorname {\mathcal{M}}(2,1) \rightarrow \overline{\mathbf{F}}_3 \otimes _{\mathbf{F}_3} \operatorname {\mathcal{M}}$ determined by

$$\begin{equation*} \mathbf{e}\longmapsto a\mathbf{e}_{\omega } + \mathbf{e}'_{\omega }, \end{equation*}$$

where $a \in \overline{\mathbf{F}}_3$ satisfies $a^3 = a + c''$.

■

9. Proof of Theorems 4.6.1, 4.6.2 and 4.6.3

In this section we will keep the notation of §4.6 and §6.5. We will write $F$ for $F_i$, $F'$ for $F_i'$ and ${\mathcal{I}}$ for ${\mathcal{I}}_i$. If ${\mathcal{G}}$ (resp. ${\operatorname {\mathcal{M}}}$) is a finite flat ${\mathcal{O}}_F$-group scheme (resp. Breuil module over ${\mathcal{O}}_F$) we will write ${\mathcal{G}}'$ (resp. ${\operatorname {\mathcal{M}}}'$) for the base change to ${\mathcal{O}}_{F'}$.

9.1. Rank one calculations

We remark that with our choice of polynomials $H_g(u)$ in §6.5, any object ${\operatorname {\mathcal{M}}}$ in ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$ has an action of $\langle \gamma _2,\gamma _4 \rangle$ via $\widehat{\gamma }_2$ and $\widehat{\gamma }_4$. (The action of $\gamma _2$ is $\operatorname {Frob}_3$-semi-linear.) Since $\gamma _3$ and $\gamma _2$ commute and $H_{\gamma _3^{\pm 1}}(u) \in \mathbf{Z}_3[u]$ we see that $\widehat{\gamma }_2$ must commute with $\widehat{\gamma _3^{\pm 1}}$ (see Corollary 5.6.2).

By Lemma 5.2.1, the only models for $(\mathbf{Z}/3\mathbf{Z})_{/F}$ over ${\mathcal{O}}_{F}$ are ${\mathcal{G}}(r,1)$ for $r=0,2,4,6,8,10,12$ with ${\mathcal{G}}(12,1) \cong (\mathbf{Z}/3\mathbf{Z})_{/{\mathcal{O}}_F}$, and the only models for $(\mu _3)_{/F}$ over ${\mathcal{O}}_{F}$ are ${\mathcal{G}}(r,1)$ for $r=0,2,4,6,8,10,12$ with ${\mathcal{G}}(0,1) \cong (\mu _3)_{/{\mathcal{O}}_F}$. Lemma 5.7.1 ensures that the base changes to ${\mathcal{O}}_{F'}$ admit unique descent data over $\mathbf{Q}_3$ such that descent of the generic fibre to $\mathbf{Q}_3$ is $\mathbf{Z}/3\mathbf{Z}$ (resp. $\mu _3$). We will write ${\mathcal{G}}_{r,1}'$ (resp. ${\mathcal{G}}_{r,\omega }'$) for the corresponding pair $({\mathcal{G}}(r,1) \times _{{\mathcal{O}}_F} {\mathcal{O}}_{F'}, \{ [g] \})$ (resp. $({\mathcal{G}}(r,1) \times _{{\mathcal{O}}_F} {\mathcal{O}}_{F'}, \{ [g] \})$). We will also let ${\operatorname {\mathcal{M}}}_{r,1}'$ (resp. ${\operatorname {\mathcal{M}}}_{r,\omega }'$) denote the corresponding object of ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$.

We have the following useful lemmas, for which the proofs are identical to the proofs of Lemmas 8.1.1 and 8.1.2.

Lemma 9.1.1.

Let $0 \le r \le e = 12$ be an even integer. The descent data on $\operatorname {\mathcal{M}}_{r,1}$ is determined by

$$\begin{equation*} \widehat{\gamma _2}(\mathbf{e}) = \mathbf{e},\,\,\, \widehat{\gamma _4}(\mathbf{e}) = -(-\sqrt {-1})^{r/2}\mathbf{e},\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}) = H_{\gamma _3^{\pm 1}}(u)^{-3r/2}\mathbf{e}, \end{equation*}$$

and the descent data on $\operatorname {\mathcal{M}}_{r,\omega }$ is determined by

$$\begin{equation*} \widehat{\gamma _2}(\mathbf{e}) = \mathbf{e},\,\,\, \widehat{\gamma _4}(\mathbf{e}) = (-\sqrt {-1})^{r/2}\mathbf{e},\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}) = H_{\gamma _3^{\pm 1}}(u)^{-3r/2}\mathbf{e}. \end{equation*}$$

In particular, $\gamma _4^2 = -1$ on $D({\mathcal{G}}_{r,1})$ if and only if $\gamma _4^2 = -1$ on $D({\mathcal{G}}_{r,\omega })$ if and only if $r=2$, $6$ or $10$.

Lemma 9.1.2.

Let ${\operatorname {\mathcal{M}}}$ be an object of ${\underline{\phi _1{\mathrm{{-mod}}}}}_{F}$ corresponding to a finite flat group scheme ${\mathcal{G}}$ and let $\{ [g] \}$ be descent data on ${\mathcal{G}}'= {\mathcal{G}}\times _{{\mathcal{O}}_F} {\mathcal{O}}_{F'}$ over $\mathbf{Q}_3$. Assume that $(\mathcal{G}',\{\widehat{g}\})_{\mathbf{Q}_3}$ can be filtered so that each graded piece is isomorphic to $\mathbf{Z}/3\mathbf{Z}$ or $\mu _3$ and so that the corresponding filtration of $(\operatorname {\mathcal{M}}',\{\widehat{g}\})$ in ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$ has successive quotients of the form $\operatorname {\mathcal{M}}'_{r_j,\chi _j}$ with $r_j \in \{2,6,10\}$ and $\chi _j \in \{{\mathbf{1}}, \omega \}$. Then $\gamma _4^2 = -1$ on $\operatorname {\mathcal{M}}'/u\operatorname {\mathcal{M}}'$ and there exists a basis $\{\mathbf{e}_j\}$ of $\operatorname {\mathcal{M}}$ over $\mathbf{F}_3[u]/u^{36}$ so that for all $j$

•

$\mathbf{e}_j \in \phi _1(\operatorname {\mathcal{M}}_1)$,

•

$\mathbf{e}_j$ is an eigenvector of the $\mathbf{F}_9$-linear map $\widehat{\gamma _4}$ on $\operatorname {\mathcal{M}}'$,

•

$\mathbf{e}_j$ lies in the part of the filtration of $\operatorname {\mathcal{M}}'$ which surjects onto $\operatorname {\mathcal{M}}'_{r_j,\chi _j}$ and this surjection sends $\mathbf{e}_j$ onto the standard basis vector $\mathbf{e}$ of $\operatorname {\mathcal{M}}'_{r_j,\chi _j}$ over $\mathbf{F}_9[u]/u^{36}$.

9.2. Models for ${\overline{\rho }}$

Recall that we are assuming that ${\overline{\rho }}$ has the très ramifié form

$$\begin{equation*} \left( \begin{array}{cc} {\omega } & {*} \\{0} & {1} \end{array} \right), \end{equation*}$$

and is not split over $F'$. We will let ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3, {\mathcal{I}}}$ denote the full subcategory of ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$ consisting of objects ${\operatorname {\mathcal{M}}}'$ for which the ideal ${\mathcal{I}}$ acts trivially on $({\operatorname {\mathcal{M}}}'/u{\operatorname {\mathcal{M}}}') \otimes _{\mathbf{F}_9, \operatorname {Frob}_3} \mathbf{F}_9$.

Proposition 9.2.1.

Suppose that $({\operatorname {\mathcal{M}}}', \{ \widehat{g} \})$ is an object of ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}$ such that $({\operatorname {\mathcal{M}}}', \{ \widehat{g} \})_{\mathbf{Q}_3}$ is an extension of $\mathbf{Z}/3\mathbf{Z}$ by $\mu _3$. Then we have an exact sequence

$$\begin{equation*} (0) \longrightarrow {\operatorname {\mathcal{M}}}_{s,1}' \longrightarrow {\operatorname {\mathcal{M}}}' \longrightarrow {\operatorname {\mathcal{M}}}_{r,\omega }' \longrightarrow (0) \end{equation*}$$

with $(r,s)=(2,6)$, $(6,10)$, $(2,10)$ or $(6,6)$. Moreover we can write ${\operatorname {\mathcal{M}}}' = {\operatorname {\mathcal{M}}}\otimes _{\mathbf{F}_3} \mathbf{F}_9$ with $\widehat{\gamma }_2= 1 \otimes \operatorname {Frob}_3$, where ${\operatorname {\mathcal{M}}}$ has an $\mathbf{F}_3[u]/(u^{36})$-basis $\{ \mathbf{e}_1, \mathbf{e}'_\omega \}$ with $\mathbf{e}_1$ the standard basis element of ${\operatorname {\mathcal{M}}}(s,1)$ and $\mathbf{e}'_\omega$ mapping to the standard basis element of ${\operatorname {\mathcal{M}}}(r,1)$. More precisely we have the following exhaustive list of extension class possibilities, all of which are well defined. $(N$ denotes the monodromy operator described in Lemma 5.1.2.)

(1)

$(r,s) = (2,6)$: The natural map$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}}(\operatorname {\mathcal{M}}'_{2,\omega }, \operatorname {\mathcal{M}}'_{6,{\mathbf{1}}}) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(1,\omega ) \end{equation*}$$

is an isomorphism, with elements parametrised by pairs $(c,c_1) \in \mathbf{F}_3^2$ corresponding to$$\begin{equation*} \operatorname {\mathcal{M}}_1 = \langle u^6 \mathbf{e}_1, u^2 \mathbf{e}'_{\omega } + cu^2\mathbf{e}_1\rangle , \,\,\,\phi _1(u^6\mathbf{e}_1) = \mathbf{e}_1,\,\,\, \phi _1(u^2 \mathbf{e}'_{\omega } + cu^2\mathbf{e}_1) = \mathbf{e}'_{\omega } \end{equation*}$$

(so $N \circ \phi _1 = 0)$ with$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_1) = -\sqrt {-1}\mathbf{e}_1,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_{\omega }) = -\sqrt {-1}\mathbf{e}'_{\omega }, \end{equation*}$$

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_1) = \mathbf{e}_1,\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_{\omega }) = (1 \pm u^{18})(\mathbf{e}'_{\omega } \pm c_1u^6\mathbf{e}_1). \end{equation*}$$

The pairs with $c = 0$ are the ones which generically split over $F'$. In all cases $\phi \equiv 0 \bmod u{\operatorname {\mathcal{M}}}$.

(2)

$(r,s) = (6,10)$: The natural map$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}}(\operatorname {\mathcal{M}}'_{6,\omega }, \operatorname {\mathcal{M}}'_{10,\mathbf{1}}) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(1,\omega ) \end{equation*}$$

is an isomorphism, with elements parametrised by pairs $(c,c_1) \in \mathbf{F}_3^2$ corresponding to$$\begin{equation*} \operatorname {\mathcal{M}}_1 = \langle u^{10} \mathbf{e}_1, u^6 \mathbf{e}'_{\omega } + cu^6\mathbf{e}_1\rangle , \,\,\,\phi _1(u^{10}\mathbf{e}_1) = \mathbf{e}_1,\,\,\, \phi _1(u^6 \mathbf{e}'_{\omega } + cu^6\mathbf{e}_1) = \mathbf{e}'_{\omega } \end{equation*}$$

(so $N \circ \phi _1 = 0)$ with$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_1) = \sqrt {-1}\mathbf{e}_1,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_{\omega }) = \sqrt {-1}\mathbf{e}'_{\omega }, \end{equation*}$$

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_1) = (1 \mp u^{18})\mathbf{e}_1,\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_{\omega }) = \mathbf{e}'_{\omega } \pm c_1u^6\mathbf{e}_1. \end{equation*}$$

The pairs with $c = 0$ are the ones which generically split over $F'$. In all cases $\phi \equiv 0 \bmod u{\operatorname {\mathcal{M}}}$. These cases are Cartier dual to the $(2,6)$ cases above.

(3)

$(r,s) = (2,10)$: The natural map$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}}(\operatorname {\mathcal{M}}'_{2,\omega }, \operatorname {\mathcal{M}}'_{10,\mathbf{1}}) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(1,\omega ) \end{equation*}$$

is an isomorphism, with elements parametrised by pairs $(c,c_1) \in \mathbf{F}_3^2$ corresponding to$$\begin{equation*} \operatorname {\mathcal{M}}_1 = \langle u^{10} \mathbf{e}_1, u^2 \mathbf{e}'_{\omega } + cu^8\mathbf{e}_1\rangle , \,\,\,\phi _1(u^{10}\mathbf{e}_1) = \mathbf{e}_1,\,\,\, \phi _1(u^2 \mathbf{e}'_{\omega } + cu^8\mathbf{e}_1) = \mathbf{e}'_{\omega } \end{equation*}$$

(so $N \circ \phi _1 = 0)$ with$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_1) = \sqrt {-1}\mathbf{e}_1,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_{\omega }) = -\sqrt {-1}\mathbf{e}'_{\omega }, \end{equation*}$$

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_1) = (1\mp u^{18})\mathbf{e}_1,\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_{\omega }) = (1 \pm u^{18})(\mathbf{e}'_{\omega } \pm c_1 u^{12}\mathbf{e}_1). \end{equation*}$$

The pairs with $c = 0$ are the ones which generically split over $F'$. In all cases $\phi \equiv 0 \bmod u{\operatorname {\mathcal{M}}}$.

(4)

$(r,s) = (6,6)$: The natural map$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}}(\operatorname {\mathcal{M}}'_{6,\omega }, \operatorname {\mathcal{M}}'_{6,\mathbf{1}}) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(1,\omega ) \end{equation*}$$

is an isomorphism, with elements parametrised by pairs $(c,c') \in \mathbf{F}_3^2$ corresponding to$$\begin{equation*} \operatorname {\mathcal{M}}_1 = \langle u^6 \mathbf{e}_1, u^6 \mathbf{e}'_{\omega } + (c+c'u^4)\mathbf{e}_1\rangle , \,\,\,\phi _1(u^6\mathbf{e}_1) = \mathbf{e}_1,\,\,\, \phi _1(u^6 \mathbf{e}'_{\omega } + (c+c'u^4)\mathbf{e}_1) = \mathbf{e}'_{\omega } \end{equation*}$$

(it is easily checked that $N(\mathbf{e}_1) = 0$ and $N(\mathbf{e}'_{\omega }) = c' u^{30}\mathbf{e}_1)$ and$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_1) = -\sqrt {-1}\mathbf{e}_1,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_{\omega }) = \sqrt {-1}\mathbf{e}'_{\omega }, \end{equation*}$$

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_1) = \mathbf{e}_1,\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_{\omega }) = \mathbf{e}'_{\omega } +(\pm c \mp c' u^{12} - c'u^{30})\mathbf{e}_1. \end{equation*}$$

In particular, $\phi \equiv 0 \bmod u{\operatorname {\mathcal{M}}}$ if and only if $c=0$.

In the first three cases, the peu ramifié condition on a class in $\operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(1,\omega )$ is equivalent to the vanishing of $c_1$. In the fourth case it is equivalent to the vanishing of $c$.

Proof.

By Lemma 9.1.1 we have an exact sequence

$$\begin{equation*} (0) \longrightarrow {\operatorname {\mathcal{M}}}_{s,1}' \longrightarrow {\operatorname {\mathcal{M}}}' \longrightarrow {\operatorname {\mathcal{M}}}_{r,\omega }' \longrightarrow (0) \end{equation*}$$

with $r,s \in \{ 2,6,10 \}$. As usual

$$\begin{equation*} {\operatorname {\mathcal{M}}}_1=\langle u^s\mathbf{e}_1, u^r\mathbf{e}_\omega '+h\mathbf{e}_1 \rangle . \end{equation*}$$

In the cases $(r,s)=(2,2)$ and $(6,2)$ as in the proof of Proposition 8.2.1 we may take $h=0$. We will show that in the case $(r,s)=(10,2)$ we also have $h=0$. Following the proof of Proposition 8.2.1 we may suppose that $h \in \mathbf{F}_3$. Without loss of generality we can take $h=1$ and look for a contradiction. Again following the proof of Proposition 8.2.1 and using

$$\begin{equation*} {\operatorname {\mathcal{M}}}_1=\langle u^2 \mathbf{e}_1, u^{10}\mathbf{e}_\omega '+\mathbf{e}_1 \rangle \end{equation*}$$

we find that $\phi \mathbf{e}_\omega ' \equiv -\mathbf{e}_1 \bmod u {\operatorname {\mathcal{M}}}$. Also

$$\begin{equation*} \begin{array}{rcl} \widehat{\gamma _3^{\pm 1}} \mathbf{e}_1 &=& (1 \pm u^{18})\mathbf{e}_1, \\\widehat{\gamma _3^{\pm 1}} \mathbf{e}_\omega ' &=&(1 \mp u^{18})\mathbf{e}_\omega '+h_{\pm 1}(u) \mathbf{e}_1 \end{array} \end{equation*}$$

for some $h_{\pm 1}(u) \in \mathbf{F}_9[u]/(u^{36})$, which must actually lie in $\mathbf{F}_3[u]/(u^{36})$ (using, as usual, the fact that $\widehat{\gamma }_2$ and $\widehat{\gamma }_3$ commute). Thus

$$\begin{equation*} \begin{array}{rcl} -\mathbf{e}_1 & \equiv & \phi \mathbf{e}_\omega ' \\& \equiv & (\widehat{\gamma }_3 \widehat{\gamma }_2-\widehat{\gamma _3^{-1}} \widehat{\gamma }_2)(\mathbf{e}_\omega ') \\& \equiv & (\widehat{\gamma }_3 - \widehat{\gamma _3^{-1}})(\mathbf{e}_\omega ') \\& \equiv & (h_1(0)-h_{-1}(0))\mathbf{e}_1 \bmod u {\operatorname {\mathcal{M}}}'. \end{array} \end{equation*}$$

The inverse linear maps $\widehat{\gamma _3^{\pm 1}}$ on ${\operatorname {\mathcal{M}}}'/u{\operatorname {\mathcal{M}}}'$ have matrices

$$\begin{equation*} \left( \begin{array}{cc} 1& h_{\pm 1}(0) \\0&1 \end{array}\right) \end{equation*}$$

with respect to the basis $\{ \mathbf{e}_1, \mathbf{e}_\omega ' \}$, so that $h_{-1}(0)=- h_1(0)$. Thus $h_1(0)=1$. On the other hand evaluating $\widehat{\gamma }_3 \phi _1' \equiv \phi _1' \widehat{\gamma }_3 \bmod u {\operatorname {\mathcal{M}}}'$ on $u^{10}\mathbf{e}_\omega '+ \mathbf{e}_1$ and comparing coefficients of $\mathbf{e}_1$ gives $h_1(0)=0$, a contradiction.

Thus if any case $(r,2)$ arises, the underlying Breuil module must be a split extension

$$\begin{equation*} \begin{array}{ll} {\operatorname {\mathcal{M}}}=(\mathbf{F}_3[u]/(u^{36})) \mathbf{e}_1 \oplus (\mathbf{F}_3[u]/(u^{36})) \mathbf{e}_\omega ', & {\operatorname {\mathcal{M}}}_1 = \langle u^2 \mathbf{e}_1, u^r \mathbf{e}_\omega ' \rangle , \\\phi _1(u^2\mathbf{e}_1)=\mathbf{e}_1, & \phi _1(u^r \mathbf{e}_\omega ')=\mathbf{e}_\omega ' \end{array} \end{equation*}$$

(so $N \circ \phi _1=0$), with

$$\begin{equation*} \widehat{\gamma }_2 \mathbf{e}_1 = \mathbf{e}_1, \ \ \widehat{\gamma }_2 \mathbf{e}_\omega '=\mathbf{e}_\omega ', \end{equation*}$$

$$\begin{equation*} \widehat{\gamma }_4 \mathbf{e}_1 = \sqrt {-1}\mathbf{e}_1, \ \ \widehat{\gamma }_4 \mathbf{e}_\omega '=(-\sqrt {-1})^{r/2}\mathbf{e}_\omega '. \end{equation*}$$ We also have

$$\begin{equation*} \begin{array}{ll} \widehat{\gamma _3^{\pm 1}} \mathbf{e}_1=H_{\gamma _3^{\pm 1}} (u)^{-3} \mathbf{e}_1 & \widehat{\gamma _3^{\pm 1}} \mathbf{e}_\omega '=H_{\gamma _3^{\pm 1}} (u)^{-3r/2} \mathbf{e}_\omega '+ h_{\pm 1}(u) \mathbf{e}_1 \end{array} \end{equation*}$$

for some $h_{\pm 1} \in \mathbf{F}_3[u]/(u^{36})$. Since $N \circ \phi _1=0$, we have $\widehat{\gamma _3^{\pm 1}} \phi _1'=\phi _1' \widehat{\gamma _3^{\pm 1}}$ on ${\operatorname {\mathcal{M}}}_1'$. Evaluating this on $u^r\mathbf{e}_\omega '$ and comparing coefficients of $\mathbf{e}_1$ gives $h_{\pm 1}(u)=u^{3(r-2)}h_{\pm 1}(u)^3 H_{\gamma _3^{\pm 1}}(u)^{3r}$. This forces $h_{\pm 1}(u)=0$ if $r \neq 2$. If $r=2$ it forces $h_{\pm 1}(u)= c_{\pm 1}(1 \pm u^{18})$ for some $c_{\pm 1} \in \mathbf{F}_3$. We will show $c_{-1}= c_1=0$. Indeed, evaluating the congruence

$$\begin{equation*} \phi \equiv (\widehat{\gamma }_3 \widehat{\gamma }_2-\widehat{\gamma _3^{-1}} \widehat{\gamma }_2) \bmod u {\operatorname {\mathcal{M}}}' \end{equation*}$$

on $\mathbf{e}_\omega '$ gives

$$\begin{equation*} 0 = \phi (\mathbf{e}_\omega ') \equiv (c_1-c_{-1}) \mathbf{e}_1 \bmod u {\operatorname {\mathcal{M}}}' \end{equation*}$$

so that $c_{-1}=c_1$. On the other hand the congruence

$$\begin{equation*} \widehat{\gamma }_3 \widehat{\gamma }_4\equiv \widehat{\gamma }_4 \widehat{\gamma _3^{-1}} \bmod u {\operatorname {\mathcal{M}}}' \end{equation*}$$

gives

$$\begin{equation*} \left( \begin{array}{cc} {1} & {c_1} \\{0} & {1} \end{array} \right) \left( \begin{array}{cc} {\sqrt {-1}} & {0} \\{0} & {-\sqrt {-1}} \end{array} \right) = \left( \begin{array}{cc} {\sqrt {-1}} & {0} \\{0} & {-\sqrt {-1}} \end{array} \right) \left( \begin{array}{cc} {1} & {c_{-1}} \\{0} & {1} \end{array} \right) \end{equation*}$$

in $M_2(\mathbf{F}_3)$, so $c_{-1}=-c_1$. Thus $c_{-1}=c_1=0$ and $h_{\pm 1}=0$ for $r=2$ as well. Thus for $r=2$, $6$ and $10$ the Breuil module with descent data ${\operatorname {\mathcal{M}}}'$ is split, so ${\overline{\rho }}$ is split, a contradiction.

This rules out the possibilities $(2,2)$, $(6,2)$ and $(10,2)$. Using Cartier duality we can also rule out $(10,10)$ and $(10,6)$. We are left with the four possible pairs $(r,s)$ as asserted in the proposition and must determine which possibilities arise in each case.

Next consider the case $(r,s) = (2,6)$. Using the same analysis as in the $(r,s) = (2,6)$ case in Proposition 8.2.1, we find that the possibilities for the Breuil module $\operatorname {\mathcal{M}}$ are the ones in the statement of the proposition (and $N \circ \phi _1 = 0$ is easy to check), though we only know that

$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_1) = -\sqrt {-1} \mathbf{e}_1,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_{\omega }) = -\sqrt {-1}\mathbf{e}'_{\omega } + h_{\gamma _4}(u)\mathbf{e}_1 \end{equation*}$$

for some $h_{\gamma _4}(u) \in \mathbf{F}_9[u]/u^{36}$. The conditions

$$\begin{equation*} \widehat{\gamma }_4(\operatorname {\mathcal{M}}'_1) \subseteq \operatorname {\mathcal{M}}'_1,\,\,\,\, \widehat{\gamma }_4\circ \phi '_1 = \phi '_1\circ \widehat{\gamma }_4 \,\,\,{\mathrm{{on}}}\,\,\,\operatorname {\mathcal{M}}'_1 \end{equation*}$$

are equivalent to

$$\begin{equation*} h_{\gamma _4} \equiv 0 \bmod u^4,\,\,\, h_{\gamma _4} = -\left(\frac{h_{\gamma _4}}{u^4}\right)^3. \end{equation*}$$

The solutions to this are $h_{\gamma _4} = a\sqrt {-1}u^6$ for $a \in \mathbf{F}_3$. Replacing $\mathbf{e}'_{\omega }$ by $\mathbf{e}'_{\omega } + au^6\mathbf{e}_1$ preserves our standardized form but makes $h_{\gamma _4}=0$:

$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_1) = -\sqrt {-1} \mathbf{e}_1,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_{\omega }) = -\sqrt {-1}\mathbf{e}'_{\omega } . \end{equation*}$$

The wild descent data must have the form

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_1) = \mathbf{e}_1,\,\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_{\omega }) = (1 \pm u^{18})\mathbf{e}'_{\omega } + h_{\pm 1}(u)\mathbf{e}_1 \end{equation*}$$

for some $h_{\pm 1} \in \mathbf{F}_9[u]/u^{36}$. The conditions

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\operatorname {\mathcal{M}}'_1) \subseteq \operatorname {\mathcal{M}}'_1,\,\,\,\, \widehat{\gamma _3^{\pm 1}}\circ \phi '_1 = \phi '_1 \circ \widehat{\gamma _3^{\pm 1}}\,\,\,{\mathrm{{on}}}\,\,\,\operatorname {\mathcal{M}}'_1 \end{equation*}$$

(recall $N \circ \phi _1 = 0$) are equivalent to

$$\begin{equation*} h_{\pm 1} \equiv 0 \bmod u^4,\,\,\, h_{\pm 1} = (1 \pm u^{18})\left(\frac{h_{\pm 1}}{u^4}\right)^3, \end{equation*}$$

whose solutions are

$$\begin{equation*} h_{\pm 1} = c_{\pm 1} u^6(1 \pm u^{18}) \end{equation*}$$

for some $c_{\pm 1} \in \mathbf{F}_3$. Since $N \circ \phi _1 = 0$, we have

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}} \circ \widehat{\gamma _3^{\mp 1}} \circ \phi '_1 = \phi '_1, \end{equation*}$$

so

$$\begin{equation*} c_{-1} = -c_1. \end{equation*}$$

Using Lemma 5.2.2 and Corollary 5.6.2, we see that all of these possibilities are well defined. We also see that ${\mathcal{I}}$ annihilates ${\operatorname {\mathcal{M}}}/u{\operatorname {\mathcal{M}}}\otimes \mathbf{F}_9$. It is straightforward to check that generic splitting over $F'$ (which is equivalent to generic splitting over $F$) is equivalent to $c = 0$, and that such splitting is compatible with descent data (i.e. descends to $\mathbf{Q}_3$) if and only if $c = c_1 = 0$. For dimension reasons, the map on $\operatorname {Ext}^1$’s is therefore an isomorphism.

Now consider the case $(r,s) = (2,10)$. Here we have

$$\begin{equation*} \operatorname {\mathcal{M}}_1 = \langle u^{10} \mathbf{e}_1, u^2\mathbf{e}'_{\omega } + h\mathbf{e}_1\rangle \end{equation*}$$

for some $h \in \mathbf{F}_3[u]/u^{36}$, with

$$\begin{equation*} \phi _1(u^{10}\mathbf{e}_1) = \mathbf{e}_1,\,\,\, \phi _1(u^2\mathbf{e}'_{\omega } + h\mathbf{e}_1) = \mathbf{e}'_{\omega } \end{equation*}$$

and

$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_1) = \sqrt {-1}\mathbf{e}_1,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_{\omega }) = -\sqrt {-1}\mathbf{e}'_{\omega }. \end{equation*}$$

In order that

$$\begin{equation*} \widehat{\gamma }_4(\operatorname {\mathcal{M}}'_1) \subseteq \operatorname {\mathcal{M}}'_1,\,\,\,\, \widehat{\gamma }_4\circ \phi '_1 = \phi '_1 \circ \widehat{\gamma }_4 \,\,\,{\mathrm{{on}}}\,\,\,\operatorname {\mathcal{M}}'_1, \end{equation*}$$

it is necessary and sufficient that

$$\begin{equation*} h \equiv h(-\sqrt {-1}u) \bmod u^{22}. \end{equation*}$$

But $\{u^{10}t - u^2t^3|t\in \mathbf{F}_3[u]/u^{36}\}$ is spanned by $u^{13} - u^{11}$, $u^{12} - u^8$, $u^{11} - u^5$, $u^{10} - u^2$, and all multiples of $u^{15}$, so we may suppose

$$\begin{equation} h = c'' + c' u^4 + c u^8, {\tag{9.2.1}} \end{equation}$$

for some $c'', c', c \in \mathbf{F}_3$, at the expense of possibly losing the diagonal form of $\widehat{\gamma }_4$.

The monodromy operator satisfies

$$\begin{equation*} N(\mathbf{e}_1) = 0,\,\,\,\, N(\mathbf{e}'_{\omega }) = (c'' u^6- c' u^{18} + c'' u^{30})\mathbf{e}_1. \end{equation*}$$

Since the wild descent data must take the form

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_1) = \ (1 \mp u^{18})\mathbf{e}_1,\,\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_{\omega }) = (1 \pm u^{18})\mathbf{e}'_{\omega } + h_{\pm 1}\mathbf{e}_1 \end{equation*}$$

for some $h_{\pm 1} \in \mathbf{F}_9[u]/u^{36}$, we compute

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(u^2\mathbf{e}'_{\omega } + h\mathbf{e}_1) = H_{\gamma _3^{\pm 1}}^2(u)(1 \pm u^{18})(u^2\mathbf{e}'_{\omega }+ h\mathbf{e}_1) + f_{\pm 1}(u)\mathbf{e}_1, \end{equation*}$$

where

$$\begin{equation} f_{\pm 1}(u) = -h H_{\gamma _3^{\pm 1}}^2(u)(1 \pm u^{18}) + u^2 H_{\gamma _3^{\pm 1}}^2 h_{\pm 1} + (1 \mp u^{18})h(uH_{\gamma _3^{\pm 1}}). {\tag{9.2.2}} \end{equation}$$

Thus, in order that $\widehat{\gamma _3^{\pm 1}}(\operatorname {\mathcal{M}}'_1) \subseteq \operatorname {\mathcal{M}}'_1$, it is necessary and sufficient that $f_{\pm 1}$ satisfies

$$\begin{equation*} f_{\pm 1} \equiv 0 \bmod u^{10}. \end{equation*}$$

Using (9.2.1) and $H_{\gamma _3^{\pm 1}} \equiv 1 \mp u^6 \bmod 3$, this amounts to

$$\begin{equation} h_{\pm 1} \equiv \pm c'' u^4 \bmod u^8. {\tag{9.2.3}} \end{equation}$$

However, $N\circ \phi _1(\operatorname {\mathcal{M}}_1) \subseteq u^6\operatorname {\mathcal{M}}$, so

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}} \circ \phi '_1 \equiv \phi '_1 \circ \widehat{\gamma _3^{\pm 1}} \bmod u^6 \operatorname {\mathcal{M}}' \end{equation*}$$

when evaluated on $\operatorname {\mathcal{M}}'_1$. This gives

$$\begin{equation*} h_{\pm 1} \equiv \left(\frac{f_{\pm 1}}{u^{10}}\right)^3 \bmod u^6. \end{equation*}$$

Since $h_{\pm 1}$ is a cube modulo $u^6$, by (9.2.3) we must have $c''=0$, and so $N\circ \phi _1 \equiv 0 \bmod u^{18}\operatorname {\mathcal{M}}$. Thus, $\widehat{\gamma _3^{\pm 1}}$ and $\phi '_1$ commute modulo $u^{18}\operatorname {\mathcal{M}}'$ when evaluated on $\operatorname {\mathcal{M}}'_1$, so we get

$$\begin{equation} h_{\pm 1} \equiv \left(\frac{f_{\pm 1}}{u^{10}}\right)^3 \bmod u^{18}, {\tag{9.2.4}} \end{equation}$$

and $h_{\pm 1}$ is a cube modulo $u^{18}$.

On the other hand, with $c'' = 0$, we see from (9.2.3) that

$$\begin{equation*} h_{\pm 1} \equiv 0 \bmod u^8. \end{equation*}$$

Because $h_{\pm 1}$ is a cube modulo $u^{18}$, we get the slight improvement

$$\begin{equation*} h_{\pm 1} \equiv 0 \bmod u^9. \end{equation*}$$

Combining this with the vanishing of $c''$, we deduce from (9.2.2) that $f_{\pm 1} \equiv \pm c' u^{10} \bmod u^{11}$, so by (9.2.4)

$$\begin{equation*} h_{\pm 1} \equiv \pm c' \bmod u. \end{equation*}$$

This forces $c'=0$, so $N \circ \phi _1 = 0$. Thus, $\widehat{\gamma _3^{\pm 1}}$ and $\phi '_1$ commute on $\operatorname {\mathcal{M}}'_1$, so

$$\begin{equation*} h_{\pm 1} = \left(\frac{f_{\pm 1}}{u^{10}}\right)^3 \end{equation*}$$

in $\mathbf{F}_9[u]/u^{36}$. Using $h=cu^8$ this becomes (via (9.2.2))

$$\begin{equation*} h_{\pm 1} = (1 \pm u^{18})\left(\frac{h_{\pm 1}}{u^8}\right)^3, \end{equation*}$$

so

$$\begin{equation*} h_{\pm 1} = c_{\pm 1}u^{12}(1 \pm u^{18}) \end{equation*}$$

for some $c_{\pm 1} \in \mathbf{F}_3$. As before, we get $c_{-1} = -c_1$.

Now we “diagonalise” $\widehat{\gamma }_4$. Since we have

$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_1) = \sqrt {-1}\mathbf{e}_1,\,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_{\omega }) = -\sqrt {-1}\mathbf{e}'_{\omega } + h_{\gamma _4}(u)\mathbf{e}_1 \end{equation*}$$

for some $h_{\gamma _4} \in \mathbf{F}_9[u]/u^{36}$, the conditions

$$\begin{equation*} \widehat{\gamma }_4(\operatorname {\mathcal{M}}'_1) \subseteq \operatorname {\mathcal{M}}'_1,\,\,\,\, \widehat{\gamma }_4 \circ \phi '_1 = \phi '_1 \circ \widehat{\gamma }_4 \,\,\,{\mathrm{{on}}}\,\,\,\operatorname {\mathcal{M}}'_1 \end{equation*}$$

are equivalent to

$$\begin{equation*} h_{\gamma _4} \equiv 0 \bmod u^8,\,\,\,\, h_{\gamma _4} = -\left(\frac{h_{\gamma _4}}{u^8}\right)^3, \end{equation*}$$

which is to say

$$\begin{equation*} h_{\gamma _4} = a\sqrt {-1}u^{12} \end{equation*}$$

for some $a \in \mathbf{F}_3$. Replacing $\mathbf{e}'_{\omega }$ by $\mathbf{e}'_{\omega } + au^{12}\mathbf{e}_1$ then puts us in a setting with $a = 0$. Thus all extensions have the form asserted in the proposition. It is easy to check that in each case ${\mathcal{I}}$ annihilates $({\operatorname {\mathcal{M}}}/u{\operatorname {\mathcal{M}}}) \otimes \mathbf{F}_9$.

Pushout by the non-zero map $\operatorname {\mathcal{M}}'_{6,{\mathbf{1}}} \rightarrow \operatorname {\mathcal{M}}'_{10,{\mathbf{1}}}$ in ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$ induced by $\mathbf{e}\mapsto u^6\mathbf{e}$ takes our (2,6) examples to our (2,10) examples (compatibly with the labelling of parameters $c, c_1$ as in the statement of the proposition). Thus all $9$ possibilities for $(c,c_1)$ do occur and we get an isomorphism of $\operatorname {Ext}^1$’s as asserted. Moreover, generic splitting over $F'$ (which is equivalent to generic splitting over $F$) is equivalent to $c = 0$, and such splitting is compatible with descent data (i.e. descends to $\mathbf{Q}_3$) if and only if $c = c_1 = 0$.

Using Cartier duality and the case $(r,s) = (2,6)$, we see that in the case $(r,s) = (6,10)$ the map of $\operatorname {Ext}^1$’s is an isomorphism. It is easy to check that the objects in our asserted list of $9$ possibilities for $(r,s) = (6,10)$ are well defined and that pullback by the non-zero map $\operatorname {\mathcal{M}}'_{2,\omega } \rightarrow \operatorname {\mathcal{M}}'_{6,\omega }$ induced by $\mathbf{e}\mapsto u^6\mathbf{e}$ takes these to our $(2,10)$ examples (compatibly with the labelling of parameters $c, c_1$).

Finally, we turn to the case $(r,s) = (6,6)$. Choosing a basis with respect to which $\widehat{\gamma }_4$ has a diagonal action, the conditions

$$\begin{equation} \widehat{\gamma }_4(\operatorname {\mathcal{M}}'_1) \subseteq \operatorname {\mathcal{M}}'_1,\,\,\,\, \widehat{\gamma }_4\circ \phi '_1 = \phi '_1\circ \widehat{\gamma }_4\,\,\, {\mathrm{{on}}}\,\,\,\operatorname {\mathcal{M}}'_1 {\tag{9.2.5}} \end{equation}$$

are equivalent to

$$\begin{equation*} h(u) \equiv h(-\sqrt {-1}u) \bmod u^{18}. \end{equation*}$$

Since $\{u^6t - u^6 t^3|t\in \mathbf{F}_3[u]/u^{36}\}$ consists of multiples of $u^7$, we may change $\mathbf{e}'_{\omega }$ to get

$$\begin{equation*} h = c + c'u^4 \end{equation*}$$

for some $c, c' \in \mathbf{F}_3$, with

$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_1) = -\sqrt {-1}\mathbf{e}_1,\,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_{\omega }) = \sqrt {-1}\mathbf{e}'_{\omega } + h_{\gamma _4}(u)\mathbf{e}_1 \end{equation*}$$

for some $h_{\gamma _4} \in \mathbf{F}_9[u]/u^{36}$. Feeding this into (9.2.5) we get $h_{\gamma _4} = -h_{\gamma _4}^3$, so $h_{\gamma _4} = \sqrt {-1}a$ for some $a \in \mathbf{F}_3$. Replacing $\mathbf{e}'_{\omega }$ by $\mathbf{e}'_{\omega } - a\mathbf{e}_1$ returns us to the setting with “diagonal” $\widehat{\gamma }_4$-action and preserves the standardizations we have made so far.

It is easy to compute $N(\mathbf{e}'_{\omega }) = c'u^{30}\mathbf{e}_1$ (and we know $N(\mathbf{e}_1) = 0$). The “wild” descent data is

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_1) = \mathbf{e}_1,\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_{\omega }) = \mathbf{e}'_{\omega } + h_{\pm 1}\mathbf{e}_1 \end{equation*}$$

for some $h_{\pm 1} \in \mathbf{F}_9[u]/u^{36}$. Using the congruence for $t_{\gamma _3^{\pm 1}}$ in §6.5, the identity

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}\circ \phi '_1 = (1 + t_{\gamma _3^{\pm 1}}\cdot N)\circ \phi '_1 \circ \widehat{\gamma _3^{\pm 1}} \end{equation*}$$

on $\operatorname {\mathcal{M}}'_1$ amounts to the condition

$$\begin{equation*} h_{\pm 1} = h_{\pm 1}^3 \mp c' u^{12} - c'u^{30}, \end{equation*}$$

whose solutions are

$$\begin{equation*} h_{\pm 1} = c_{\pm 1} \mp c' u^{12} -c'u^{30} \end{equation*}$$

for some $c_{\pm 1} \in \mathbf{F}_3$. The identity

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}\circ \widehat{\gamma _3^{\mp 1}} \equiv 1 \bmod u\operatorname {\mathcal{M}}' \end{equation*}$$

implies $c_{-1} = -c_1$. Thus

$$\begin{equation*} -\mathbf{V}_{\operatorname {\mathcal{M}}'}(\mathbf{e}'_{\omega }) \equiv \phi (\mathbf{e}'_{\omega }) \equiv -c\mathbf{e}_1 \bmod u\operatorname {\mathcal{M}}', \,\,\,(\widehat{\gamma }_3\circ \widehat{\gamma }_2 - \widehat{\gamma _3^{-1}} \widehat{\gamma }_2)(\mathbf{e}'_{\omega }) \equiv -c_1 \mathbf{e}_1 \bmod u\operatorname {\mathcal{M}}'. \end{equation*}$$

Thus ${\mathcal{I}}$ annihilates $({\operatorname {\mathcal{M}}}/u{\operatorname {\mathcal{M}}}) \otimes _{ \mathbf{F}_3} \mathbf{F}_9$ if and only if $c_1=c$.

By Lemma 5.2.2 and Corollary 5.6.2, it is easy to see that all of these objects are well defined. The kernel of

$$\begin{equation} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}}( \operatorname {\mathcal{M}}'_{6,\omega },\operatorname {\mathcal{M}}'_{6,{\mathbf{1}}}) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_F]}(1,\omega ) {\tag{9.2.6}} \end{equation}$$

consists of pairs $(c,-c)$, where generic splittings are induced by any of the (non-zero) Breuil module maps

$$\begin{equation*} \operatorname {\mathcal{M}}(0,1) \longrightarrow \operatorname {\mathcal{M}} \end{equation*}$$

defined by

$$\begin{equation*} \mathbf{e}\longmapsto u^9 \mathbf{e}'_{\omega } + (\tilde{c}u^9 + cu^3)\mathbf{e}_1 = u(c+u^2 \tilde{c})u^6\mathbf{e}_1 + u^3(u^6\mathbf{e}'_{\omega } + (c-c u^4)\mathbf{e}_1) \end{equation*}$$

with $\tilde{c} \in \mathbf{F}_3$. Thus, the pairs $(c,c')$ corresponding to the ${\overline{\rho }}$ which are split over $F$ (or equivalently, split over $F'$) are exactly those for which $c + c' = 0$. The map

$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}}( \operatorname {\mathcal{M}}'_{6,\omega },\operatorname {\mathcal{M}}'_{6,{\mathbf{1}}}) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(1,\omega ) \end{equation*}$$

is therefore injective, because the splitting given above respects descent data if and only if $\tilde{c}=c_1=0$.

It remains to establish which of the given extensions of Breuil modules correspond to peu ramifié extensions of $\mathbf{Z}/3\mathbf{Z}$ by $\mu _3$ over $\mathbf{Q}_3$. We noted above that the maps among the $\operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}$’s in the (2,6), (6,10), (2,10) cases induced by pushout/pullback along $\mathbf{e}\mapsto u^6\mathbf{e}$ are compatible with the parametrisation by pairs $(c,c_1)$. With a little more care, one checks that the maps

$$\begin{multline*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}}(\operatorname {\mathcal{M}}'_{2,\omega }, \operatorname {\mathcal{M}}'_{6,\mathbf{1}}) \longleftarrow \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}}(\operatorname {\mathcal{M}}'_{6,\omega }, \operatorname {\mathcal{M}}'_{6,\mathbf{1}})\\ \longrightarrow \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}}(\operatorname {\mathcal{M}}'_{6,\omega }, \operatorname {\mathcal{M}}'_{10,\mathbf{1}}) \end{multline*}$$

induced by $\mathbf{e}\mapsto u^6\mathbf{e}$ send the pair $(c,c')$ in the middle to the pair $(c+c',c)$ on either end (to construct the necessary commutative diagrams of short exact sequences in the two cases, use the maps

$$\begin{equation*} (\mathbf{e}'_{\omega },\mathbf{e}_1) \longmapsto (u^6\mathbf{e}'_{\omega }-c'\mathbf{e}_1,\mathbf{e}_1),\,\,\, (\mathbf{e}'_{\omega },\mathbf{e}_1) \longmapsto (\mathbf{e}'_{\omega } + c'\mathbf{e}_1,u^6\mathbf{e}_1) \end{equation*}$$

respectively). This reduces us to checking the $(6,6)$ case.

By Corollary 2.3.2, the two très ramifié extensions, ${\overline{\rho }}_1$ and ${\overline{\rho }}_2$, of $1$ by $\omega$ which are non-split over $F$ arise from elliptic curves, $E_1$ and $E_2$, over $\mathbf{Q}_3$ for which $\rho _{E_j,3}$ is potentially Barsotti-Tate with extended type $\tau _i'$ (see §6.5). Let ${\mathcal{G}}_j$ denote the $3$-torsion in the Néron model of $E_j$ over ${\mathcal{O}}_F$. From the universal property of Néron models we see that ${\mathcal{G}}_j'={\mathcal{G}}_j \times _{{\mathcal{O}}_F} {\mathcal{O}}_{F'}$ inherits descent data $\{ [g]\}$ over $\mathbf{Q}_3$. By the same argument used at the end of §4.6 we see that $({\mathcal{G}}_j',\{ [g]\})$ is an object of ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}$. Moreover we see that $\mathbf{F}\neq 0$ on $D({\mathcal{G}}_j)$. Since all non-$(6,6)$ cases above have $\phi \equiv 0 \bmod u {\operatorname {\mathcal{M}}}$, by the parts of Proposition 9.2.1 which we have already proved we see that ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}_j',\{ [g]\})$ is an extension of ${\operatorname {\mathcal{M}}}_{6,\omega }'$ by ${\operatorname {\mathcal{M}}}_{6,1}'$ and correspond to a pair $(c,c')$ with $c \neq 0$ (since $\mathbf{F}\neq 0$) and $c+c' \neq 0$ (by our analysis of (9.2.6), since ${\overline{\rho }}_i$ is non-split over $F$). Hence ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}_1',\{ [g]\})$ and ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}_2',\{ [g]\})$ must correspond in some order to the lines $c'=0$ and $c=c'$ in $\mathbf{F}_3^2$.

As a non-split peu ramifié extension of $1$ by $\omega$ remains non-split over $F'$, we see that the peu ramifié line in

$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}}(\operatorname {\mathcal{M}}'_{6,\omega }, \operatorname {\mathcal{M}}'_{6,\mathbf{1}}) \cong \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(1,\omega ) \end{equation*}$$

cannot correspond to $c+c'=0$. By the above analysis it cannot correspond to $c'=0$ or $c-c'=0$. Thus it must correspond to the remaining line $c=0$.

■

The properties of $\phi$ in the cases listed in Proposition 9.2.1 make it clear that the $(6,6)$ case there is “different”. We will see further manifestations of this difference later.

9.3. Further rank two calculations

Lemma 9.3.1.

For $(r,s)=(2,6)$, $(6,10)$ and $(2,10)$ we have

$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}}({\operatorname {\mathcal{M}}}_{s,1}',{\operatorname {\mathcal{M}}}_{r,\omega }')=(0). \end{equation*}$$

Proof.

The (6,10) case follows from the (2,6) case by Cartier duality. Thus, we assume $r = 2$, $s \in \{6, 10\}$. Let $(\operatorname {\mathcal{M}}',\{\widehat{g}\})$ be such an extension. By Lemma 8.1.2, $(\operatorname {\mathcal{M}}',\{\widehat{g}\})$ arises from a Breuil module over ${\mathcal{O}}_F$ of the form

$$\begin{equation*} \operatorname {\mathcal{M}}= (\mathbf{F}_3[u]/u^{36})\mathbf{e}_{\omega } \oplus (\mathbf{F}_3[u]/u^{36})\mathbf{e}'_1,\,\,\,\, \operatorname {\mathcal{M}}_1 = \langle u^2\mathbf{e}_{\omega }, u^s\mathbf{e}'_1 + h\mathbf{e}_{\omega }\rangle \end{equation*}$$

with

$$\begin{equation*} \phi _1(u^2\mathbf{e}_{\omega }) = \mathbf{e}_{\omega },\,\,\,\, \phi _1(u^s\mathbf{e}'_1 + h\mathbf{e}_{\omega }) = \mathbf{e}'_1 \end{equation*}$$

and

$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_{\omega }) = -\sqrt {-1}\mathbf{e}_{\omega },\,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_1) = -(-\sqrt {-1})^{s/2}\mathbf{e}'_1, \end{equation*}$$

where $h \in \mathbf{F}_3[u]/u^{36}$.

The combined conditions

$$\begin{equation*} \widehat{\gamma }_4(\operatorname {\mathcal{M}}'_1) \subseteq \operatorname {\mathcal{M}}'_1,\,\,\,\, \widehat{\gamma }_4\circ \phi '_1 = \phi '_1 \circ \widehat{\gamma }_4\,\,\, {\mathrm{{on}}}\,\,\,\operatorname {\mathcal{M}}'_1 \end{equation*}$$

are equivalent to

$$\begin{equation*} (\sqrt {-1})^{s/2}h(u) \equiv (\sqrt {-1})h(-\sqrt {-1}u) \bmod u^{14}. \end{equation*}$$

Treating the cases $s = 6$ and $s = 10$ separately, we conclude from Lemma 5.2.2 that we may change $\mathbf{e}'_1$ so that $h = 0$ when $s = 6$ and $h \in \mathbf{F}_3$ when $s = 10$. As a result of this change, we only have

$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_{\omega }) = -\sqrt {-1}\mathbf{e}_{\omega },\,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_1) = -(-\sqrt {-1})^{s/2}\mathbf{e}'_1 + h_{\gamma _4}(u) \mathbf{e}_{\omega }. \end{equation*}$$

However, with $h \in \mathbf{F}_3$ when $s = 10$ and $h = 0$ when $s = 6$, the condition

$$\begin{equation*} \widehat{\gamma }_4\circ \phi '_1 = \phi '_1 \circ \widehat{\gamma }_4\,\,\, {\mathrm{{on}}}\,\,\,\operatorname {\mathcal{M}}'_1 \end{equation*}$$

forces $h_{\gamma _4} = -u^{3(s-2)}h_{\gamma _4}^3$, so that in fact $h_{\gamma _4} = 0$ after all.

When $h = 0$, so $\operatorname {\mathcal{M}}$ is split in ${\underline{\phi _1{\mathrm{{-mod}}}}}_F$ (compatibly with $\widehat{\gamma }_4$ on $\operatorname {\mathcal{M}}'$), and it is easy to check (using $N=0$) that the “wild” descent data $\widehat{\gamma _3^{\pm 1}}$ must also be diagonal, so we have the desired splitting in ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$.

It remains to consider the case $(r,s) = (2,10)$ with $h = c \in \mathbf{F}_3$. It is easy to compute

$$\begin{equation*} N(\mathbf{e}_{\omega }) = 0,\,\,\, N(\mathbf{e}'_1) = -cu^{30}\mathbf{e}_{\omega }. \end{equation*}$$

The wild descent data must have the form

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_{\omega }) = (1 \pm u^{18})\mathbf{e}_{ \omega },\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_1) = (1 \mp u^{18})\mathbf{e}'_1 + h_{\pm 1}\mathbf{e}_{\omega } \end{equation*}$$

with $h_{\pm 1} \in \mathbf{F}_9[u]/u^{36}$.

It is straightforward to check that $\widehat{\gamma _3^{\pm 1}}(\operatorname {\mathcal{M}}'_1) \subseteq \operatorname {\mathcal{M}}'_1$, and then the condition

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}\circ \phi '_1 = (1 + t_{\gamma _3^{\pm 1}}\cdot N)\circ \phi '_1 \circ \widehat{\gamma _3^{\pm 1}} \end{equation*}$$

on $\operatorname {\mathcal{M}}'_1$ gives

$$\begin{equation*} h_{\pm 1} = \pm c u^{12} + h_{\pm 1}^3 u^{24} (1\mp u^{18}) + cu^{30}. \end{equation*}$$

The unique solution to this is

$$\begin{equation*} h_{\pm 1} = c(\pm u^{12} + u^{30}). \end{equation*}$$

Thus $\widehat{\gamma }_3\widehat{\gamma }_2-\widehat{\gamma _3^{-1}} \widehat{\gamma }_2 \equiv 0 \bmod u$, while $\phi (\mathbf{e}_1') \equiv -c\mathbf{e}_\omega \bmod u$. This forces $c=0$. With $c=0$ we obviously have only the split extension class.

■

Lemma 9.3.2.

The natural map

$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}}(\operatorname {\mathcal{M}}'_{6,{\mathbf{1}}}, \operatorname {\mathcal{M}}'_{6,\omega }) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(\omega ,1) \end{equation*}$$

is an isomorphism, with elements parametrised by pairs $(c,c') \in \mathbf{F}_3^2$ corresponding to

$$\begin{equation*} \operatorname {\mathcal{M}}= (\mathbf{F}_3[u]/u^{36})\mathbf{e}_{\omega } \oplus (\mathbf{F}_3[u]/u^{36})\mathbf{e}'_1,\,\,\, \operatorname {\mathcal{M}}_1 = \langle u^6\mathbf{e}_{\omega }, u^6\mathbf{e}'_1 + (c + c'u^4)\mathbf{e}_{\omega } \rangle , \end{equation*}$$

where

$$\begin{equation*} \phi _1(u^6\mathbf{e}_{\omega }) = \mathbf{e}_{\omega }, \,\,\, \phi _1(u^6\mathbf{e}'_1 + (c + c'u^4)\mathbf{e}_{\omega }) = \mathbf{e}'_1,\,\,\, N(\mathbf{e}_{\omega }) = 0,\,\,\, N(\mathbf{e}'_1) = c' u^{30}\mathbf{e}_{\omega } \end{equation*}$$

and the descent data is

$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_{\omega }) = \sqrt {-1}\mathbf{e}_{\omega },\,\,\, \widehat{\gamma }_4(\mathbf{e}'_1) = -\sqrt {-1}\mathbf{e}'_1, \end{equation*}$$

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_{\omega }) = \mathbf{e}_{\omega },\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_1) = \mathbf{e}'_1 + (\pm c \mp c' u^{12} - c' u^{30})\mathbf{e}_{\omega }. \end{equation*}$$

Proof.

The proof is identical to the proof of the case $(r,s)=(6,6)$ in Proposition 9.2.1, except $\sqrt {-1}$ is everywhere replaced by $-\sqrt {-1}$ and when we study splitting we give ${\operatorname {\mathcal{M}}}(0,1)$ the descent data for the trivial mod $3$ character (which amounts to using $\widehat{\gamma }_4(\mathbf{e})=-\mathbf{e}$ rather than $\widehat{\gamma }_4(\mathbf{e})=\mathbf{e}$).

■

Lemma 9.3.3.

For $r \in \{2,10\}$, the maps

$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}(\operatorname {\mathcal{M}}'_{r,\omega }, \operatorname {\mathcal{M}}'_{r,\omega }) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(\omega ,\omega ) \end{equation*}$$

and

$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}(\operatorname {\mathcal{M}}'_{r,{{\mathbf{1}}}}, \operatorname {\mathcal{M}}'_{r,{{\mathbf{1}}}}) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(1,1) \end{equation*}$$

are injective and have image consisting of the $1$-dimensional space of classes which split over an unramified extension of $\mathbf{Q}_3$.

Proof.

The cases $r=10$ follow from the cases $r=2$ using Cartier duality. Thus we suppose $r=2$. We treat only the case of ${\operatorname {\mathcal{M}}}_{2,\omega }'$, the case ${\operatorname {\mathcal{M}}}_{2,1}'$ being exactly the same except that $-\sqrt {-1}$ replaces $\sqrt {-1}$ everywhere.

Let $(\operatorname {\mathcal{M}}',\{\widehat{g}\})$ represent an element in $\operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}(\operatorname {\mathcal{M}}'_{2,\omega }, \operatorname {\mathcal{M}}'_{2,\omega })$. Lemma 8.1.2 ensures the existence of an ordered $\mathbf{F}_3[u]/u^{36}$-basis $\mathbf{e}_{\omega }$, $\mathbf{e}'_{\omega }$ of $\operatorname {\mathcal{M}}$ such that

$$\begin{equation*} \operatorname {\mathcal{M}}_1 = \langle u^2 \mathbf{e}_{\omega }, u^2 \mathbf{e}'_{\omega } + h\mathbf{e}_{\omega }\rangle ,\,\,\,\phi _1(u^2\mathbf{e}_{\omega }) = \mathbf{e}_{\omega },\,\,\, \phi _1(u^2 \mathbf{e}'_{\omega } + h\mathbf{e}_{\omega }) = \mathbf{e}'_{\omega } \end{equation*}$$

with

$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_{\omega }) = -\sqrt {-1}\mathbf{e}_{\omega },\,\,\, \widehat{\gamma }_4(\mathbf{e}'_{\omega }) = -\sqrt {-1}\mathbf{e}'_{\omega }. \end{equation*}$$

Carrying out the usual calculation,

$$\begin{equation} \widehat{\gamma }_4(\operatorname {\mathcal{M}}'_1) \subseteq \operatorname {\mathcal{M}}'_1,\,\,\, \phi '_1\circ \widehat{\gamma }_4 = \widehat{\gamma }_4\circ \phi '_1 \,\,\,{\mathrm{{on}}}\,\,\,\operatorname {\mathcal{M}}'_1 {\tag{9.3.1}} \end{equation}$$

if and only if

$$\begin{equation*} h \equiv -h(-\sqrt {-1}u) \bmod u^{14}. \end{equation*}$$

Combining this with Lemma 5.2.2, we may change $\mathbf{e}'_{\omega }$ so that $h = cu^2$, with $c \in \mathbf{F}_3$, at the expense of possibly losing the diagonal form of $\widehat{\gamma }_4$. But with $h = cu^2$ and $\widehat{\gamma }_4(\mathbf{e}'_{\omega }) = -\sqrt {-1}\mathbf{e}'_{\omega } + h_{\gamma _4}(u)\mathbf{e}_{\omega }$, the conditions (9.3.1) imply $h_{\gamma _4} = -h_{\gamma _4}^3$, and so $h_{\gamma _4}(u) = (\sqrt {-1})a$ for some $a \in \mathbf{F}_3$. Then $\widehat{\gamma }_4^4 = 1$ forces $a = 0$, so $\widehat{\gamma }_4$ still has diagonal form.

It is easy to check that $N(\mathbf{e}'_{\omega }) = 0$, so $N\circ \phi _1 = 0$. Thus, we must have

$$\begin{equation} \widehat{\gamma _3^{\pm 1}}\circ \phi '_1 = \phi '_1\circ \widehat{\gamma _3^{\pm 1}} {\tag{9.3.2}} \end{equation}$$

on $\operatorname {\mathcal{M}}'_1$. Since the wild descent data has to be of the form

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_{\omega }) = (1\pm u^{18})\mathbf{e}_{\omega },\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_{\omega }) = (1\pm u^{18})\mathbf{e}'_{\omega } + h_{\pm 1}\mathbf{e}_{\omega } \end{equation*}$$

for some $h_{\pm 1} \in \mathbf{F}_3[u]/u^{36}$, evaluation of (9.3.2) on $u^2\mathbf{e}'_{\omega } + cu^2\mathbf{e}_{\omega } \in \operatorname {\mathcal{M}}_1$ gives $h_{\pm 1} = (1\pm u^{18})h_{\pm 1}^3$, so $h_{\pm 1} = c_{\pm 1}(1\pm u^{18})$ for some $c_{\pm 1} \in \mathbf{F}_3$. The relation $\widehat{\gamma _3^{\pm 1}}\circ \widehat{\gamma }_4\circ \widehat{\gamma _3^{\pm 1}}\circ \widehat{\gamma _4^3}(\mathbf{e}'_{\omega }) = \mathbf{e}'_{\omega }$ forces $c_{\pm 1} = 0$.

We now have described all possibilities in terms of the single parameter $c \in \mathbf{F}_3$, and it is straightforward to use Corollary 5.6.2 to check that all of these examples are in fact well defined. Generic splittings over unramified extensions of $\mathbf{Q}_3$ correspond to the maps

$$\begin{equation*} \overline{\mathbf{F}}_3 \otimes _{\mathbf{F}_3} \operatorname {\mathcal{M}}(0,1) \longrightarrow \overline{\mathbf{F}}_3 \otimes _{\mathbf{F}_3} \operatorname {\mathcal{M}} \end{equation*}$$

given by

$$\begin{equation*} \mathbf{e}\longmapsto au^3\mathbf{e}_{\omega } + u(u^2\mathbf{e}'_{\omega } + cu^2\mathbf{e}_{\omega }), \end{equation*}$$

where $a \in \overline{\mathbf{F}}_3$ satisfies $a^3 = a + c$. Such generic splittings can be defined over $\mathbf{Q}_3$ (i.e. without extending the residue field) if and only if $c = 0$.

■

Lemma 9.3.4.

(1)

The map of groups$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}(\operatorname {\mathcal{M}}'_{6,{\mathbf{1}}}, \operatorname {\mathcal{M}}'_{6,{\mathbf{1}}}) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(1,1) \end{equation*}$$

is an isomorphism.

Explicitly, the group $\operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}(\operatorname {\mathcal{M}}'_{6,{\mathbf{1}}}, \operatorname {\mathcal{M}}'_{6,{\mathbf{1}}})$ is parametrised by pairs $(c,c') \in \mathbf{F}_3^2$ corresponding to$$\begin{equation*} \operatorname {\mathcal{M}}= (\mathbf{F}_3[u]/u^{36})\mathbf{e}_1 \oplus (\mathbf{F}_3[u]/u^{36})\mathbf{e}'_1,\,\,\, \operatorname {\mathcal{M}}_1 = \langle u^6\mathbf{e}_1, u^6\mathbf{e}'_1 + (cu^2 + c'u^6)\mathbf{e}_1 \rangle , \end{equation*}$$

with$$\begin{equation*} \phi _1(u^6\mathbf{e}_1) = \mathbf{e}_1, \,\,\, \phi _1(u^6\mathbf{e}'_1 + (cu^2 + c'u^6)\mathbf{e}_1) = \mathbf{e}'_1, \end{equation*}$$

$$\begin{equation*} N(\mathbf{e}_1) = 0,\,\,\, N(\mathbf{e}'_1) = -c u^{24}\mathbf{e}_1 \end{equation*}$$

and descent data$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_1) = -\sqrt {-1}\mathbf{e}_1,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_1) = -\sqrt {-1}\mathbf{e}'_1, \end{equation*}$$

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_1) = \mathbf{e}_1,\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_1) = \mathbf{e}'_1 + c(\pm u^6 \pm u^{18} - u^{24} \pm u^{30})\mathbf{e}_1. \end{equation*}$$

The classes in $\operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(1,1)$ which split over an unramified extension of $\mathbf{Q}_3$ correspond to the pairs with $c = 0$.

(2)

The map of groups$$\begin{equation*} \operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}(\operatorname {\mathcal{M}}'_{6,{\mathbf{\omega }}}, \operatorname {\mathcal{M}}'_{6,{\mathbf{\omega }}}) \longrightarrow \operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(\omega ,\omega ) \end{equation*}$$

is an isomorphism.

Explicitly, the group $\operatorname {Ext}^1_{{\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}}(\operatorname {\mathcal{M}}'_{6,{\mathbf{\omega }}}, \operatorname {\mathcal{M}}'_{6,{\mathbf{\omega }}})$ is parametrised by pairs $(c,c') \in \mathbf{F}_3^2$ corresponding to$$\begin{equation*} \operatorname {\mathcal{M}}= (\mathbf{F}_3[u]/u^{36})\mathbf{e}_\omega \oplus (\mathbf{F}_3[u]/u^{36})\mathbf{e}'_\omega ,\,\,\, \operatorname {\mathcal{M}}_1 = \langle u^6\mathbf{e}_\omega , u^6\mathbf{e}'_\omega + (cu^2 + c'u^6)\mathbf{e}_\omega \rangle , \end{equation*}$$

with$$\begin{equation*} \phi _1(u^6\mathbf{e}_\omega ) = \mathbf{e}_\omega , \,\,\, \phi _1(u^6\mathbf{e}'_\omega + (cu^2 + c'u^6)\mathbf{e}_\omega ) = \mathbf{e}'_\omega , \end{equation*}$$

$$\begin{equation*} N(\mathbf{e}_\omega ) = 0,\,\,\, N(\mathbf{e}'_\omega ) = -c u^{24}\mathbf{e}_\omega \end{equation*}$$

and descent data$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_\omega ) = \sqrt {-1}\mathbf{e}_\omega ,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_\omega ) = \sqrt {-1}\mathbf{e}'_\omega , \end{equation*}$$

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_\omega ) = \mathbf{e}_\omega ,\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_\omega ) = \mathbf{e}'_\omega + c(\pm u^6 \pm u^{18} - u^{24} \pm u^{30})\mathbf{e}_\omega . \end{equation*}$$

The classes in $\operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(\omega ,\omega )$ which split over an unramified extension of $\mathbf{Q}_3$ correspond to the pairs with $c = 0$.

Proof.

We treat the first part of the lemma; replacing $\sqrt {-1}$ with $-\sqrt {-1}$ throughout gives the proof of the second part.

As usual, we can find an ordered $\mathbf{F}_3[u]/u^{36}$-basis $\mathbf{e}_1$, $\mathbf{e}'_1$ of $\operatorname {\mathcal{M}}$ so that

$$\begin{equation*} \operatorname {\mathcal{M}}_1 = \langle u^6\mathbf{e}_1, u^6\mathbf{e}'_1 + h\mathbf{e}_1\rangle ,\,\,\, \phi _1(u^6\mathbf{e}_1) = \mathbf{e}_1,\,\,\, \phi _1(u^6\mathbf{e}'_1 + h\mathbf{e}_1) = \mathbf{e}'_1, \end{equation*}$$

and $\widehat{\gamma }_4(\mathbf{e}_1) = -\sqrt {-1}\mathbf{e}_1$, $\widehat{\gamma }_4(\mathbf{e}'_1) = -\sqrt {-1}\mathbf{e}'_1$. The conditions $\widehat{\gamma }_4(\operatorname {\mathcal{M}}'_1) \subseteq \operatorname {\mathcal{M}}'_1$ and $\widehat{\gamma }_4\circ \phi '_1 = \phi '_1 \circ \widehat{\gamma }_4$ on $\operatorname {\mathcal{M}}'_1$ amount to

$$\begin{equation*} h(u) \equiv -h(-\sqrt {-1}u) \bmod u^{18}. \end{equation*}$$

Since $\{u^6t - u^6 t^3|t \in \mathbf{F}_3[u]/u^{36}\}$ consists of multiples of $u^7$, we can change the choice of $\mathbf{e}'_1$ so that

$$\begin{equation*} h = cu^2 + c'u^6 \end{equation*}$$

for some $c, c' \in \mathbf{F}_3$, where we may a priori lose the diagonal form of $\widehat{\gamma }_4$. But the same kind of calculation as in Lemma 9.3.3 shows $\widehat{\gamma }_4(\mathbf{e}'_1) = -\sqrt {-1}\mathbf{e}'_1 + a\sqrt {-1}\mathbf{e}_1$ for some $a \in \mathbf{F}_3$, so the condition $\widehat{\gamma }_4^4 = 1$ forces $a = 0$ (i.e. $\widehat{\gamma }_4$ still has diagonal action).

It is straightforward to compute the asserted formula for $N$, and then the wild descent data can be computed exactly as in our previous computations of wild descent data; this yields the formulas

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_1) = \mathbf{e}_1,\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_1) = \mathbf{e}'_1 + (c_{\pm 1} + c(\pm u^6 \pm u^{18} - u^{24} \pm u^{30}))\mathbf{e}_1, \end{equation*}$$

where $c_{\pm 1} \in \mathbf{F}_3$. Modulo $u$, the linear action of $\gamma _3^{\pm 1}\gamma _4\gamma _3^{\pm 1}\gamma _4^3$ sends $\mathbf{e}'_1$ to $\mathbf{e}'_1 - c_{\pm 1} \mathbf{e}_1$, but $\gamma _3^{\pm 1}\gamma _4\gamma _3^{\pm 1}\gamma _4^3 = 1$, so $c_{\epsilon } = 0$ for $\epsilon = \pm 1$. Thus, we obtain the asserted list of possibilities. The well-definedness of these examples follows from Lemma 5.2.2 and Corollary 5.6.2.

It is easy to see that there is a non-zero map $\overline{\mathbf{F}}_3 \otimes _{\mathbf{F}_3} \operatorname {\mathcal{M}}(0,1) \rightarrow \overline{\mathbf{F}}_3 \otimes _{\mathbf{F}_3} \operatorname {\mathcal{M}}$ if and only if $c = 0$, in which case such non-zero maps are precisely those induced by

$$\begin{equation*} \mathbf{e}\longmapsto a u^9 \mathbf{e}_1 + u^3(u^6\mathbf{e}'_1 + c'u^6\mathbf{e}_1), \end{equation*}$$

where $a \in \overline{\mathbf{F}}_3$ satisfies $a^3 = a + c'$. The verification that $c = c' = 0$ corresponds to being in the kernel of our map of $\operatorname {Ext}^1$’s is now clear, since $X^3 = X + c'$ has a solution in $\mathbf{F}_3$ if and only if $c' = 0$.

■

9.4. Completion of the proof of Theorem 4.6.1

Everything in Theorem 4.6.1 is now clear except for the third assertion, which we now prove. Let $({\mathcal{G}}',\{ [g] \})$ be as in the third part of that theorem. We may suppose that ${\mathcal{G}}'={\mathcal{G}}\times _{{\mathcal{O}}_F} {\mathcal{O}}_{F'}$ for some ${\mathcal{G}}_{/{\mathcal{O}}_F}$. The filtration on ${\overline{\rho }}\otimes _{\mathbf{F}_3} k$ gives a filtration

$$\begin{equation*} (0) \longrightarrow {\mathcal{G}}_\omega \longrightarrow {\mathcal{G}}\longrightarrow {\mathcal{G}}_1 \longrightarrow (0), \end{equation*}$$

which is compatible with the descent data over $\mathbf{Q}_3$. According to Lemma 5.2.3 we have ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}_\omega ) \cong {\operatorname {\mathcal{M}}}(k;r_\omega ,f_\omega )$ and ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}_1) \cong {\operatorname {\mathcal{M}}}(k;r_1,f_1)$ for some $0 \leq r_1,r_\omega \leq 12$ and some $f_1,f_\omega \in k[u]/u^{36}$. We will let $\chi$ denote either $1$ or $\omega$. In particular ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}_\chi )_1 =u^{r_\chi } {\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}_\chi )$ for $\chi =1,\omega$. From this one can conclude that if ${\mathcal{H}}$ is a subquotient of ${\mathcal{G}}_\chi$, then ${\operatorname {\mathcal{M}}}_\pi ({\mathcal{H}})_1=u^{r_\chi }{\operatorname {\mathcal{M}}}_\pi ({\mathcal{H}})$. Quite generally, for any Breuil module $\operatorname {\mathcal{M}}$ over ${\mathcal{O}}_F$ with $\operatorname {\mathcal{M}}_1 = u^r\operatorname {\mathcal{M}}$ and any short exact sequence of Breuil modules

$$\begin{equation*} 0 \longrightarrow \operatorname {\mathcal{M}}' \longrightarrow \operatorname {\mathcal{M}}\longrightarrow \operatorname {\mathcal{M}}'' \longrightarrow 0, \end{equation*}$$

we must also have

$$\begin{equation*} \operatorname {\mathcal{M}}'_1 = u^r\operatorname {\mathcal{M}}',\,\,\, \operatorname {\mathcal{M}}''_1 = u^r\operatorname {\mathcal{M}}''. \end{equation*}$$

Indeed, $\operatorname {\mathcal{M}}\rightarrow \operatorname {\mathcal{M}}''$ is a surjection taking $\operatorname {\mathcal{M}}_1$ onto $\operatorname {\mathcal{M}}''_1$, so the assertion for $\operatorname {\mathcal{M}}''$ is clear. Since $\operatorname {\mathcal{M}}'$ is an $\mathbf{F}_3[u]/u^{36}$-module direct summand of $\operatorname {\mathcal{M}}$ and

$$\begin{equation*} \operatorname {\mathcal{M}}'_1 = \operatorname {\mathcal{M}}' \cap \operatorname {\mathcal{M}}_1 = \operatorname {\mathcal{M}}' \cap u^r \operatorname {\mathcal{M}}, \end{equation*}$$

the assertion for $\operatorname {\mathcal{M}}'$ is likewise clear. We conclude that $({\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}}_\chi )', \{ \widehat{g} \})$ admits a filtration with successive quotients $\operatorname {\mathcal{M}}'_{r_{\chi },\chi }$. Thus $r_\chi \in \{ 2,6,10 \}$.

Consider a fixed surjection of $\mathbf{F}_3[G_3]$-modules

$$\begin{equation*} {\overline{\rho }}\otimes k \longrightarrow \hspace{-1.44em} \longrightarrow {\overline{\rho }}. \end{equation*}$$

This gives rise to a finite flat ${\mathcal{O}}_F$-group scheme ${\mathcal{H}}$ with descent data on ${\mathcal{H}}'={\mathcal{H}}\times {\mathcal{O}}_{F'}$ over $\mathbf{Q}_3$ corresponding to ${\overline{\rho }}$ and an epimorphism

$$\begin{equation*} {\mathcal{G}}\longrightarrow \hspace{-1.44em} \longrightarrow {\mathcal{H}} \end{equation*}$$

compatible with descent data. Consider the commutative diagram

$$\begin{equation*} \begin{matrix} 0\rightarrow &\operatorname {\mathcal{M}}_\pi ({\mathcal{H}}_1)&\rightarrow & \operatorname {\mathcal{M}}_\pi ({\mathcal{H}})&\rightarrow &\operatorname {\mathcal{M}}_\pi ({\mathcal{H}}_\omega )&\rightarrow 0\\ &\downarrow &&\downarrow &&\downarrow &\\ 0\rightarrow &\operatorname {\mathcal{M}}_\pi (\mathcal{G}_1)&\rightarrow & \operatorname {\mathcal{M}}_\pi (\mathcal{G})&\rightarrow &\operatorname {\mathcal{M}}_\pi (\mathcal{G}_{\omega })&\rightarrow 0 \end{matrix} \end{equation*}$$

where the top row corresponds to the non-split filtration of ${\overline{\rho }}$. The middle vertical map is an isomorphism of the source onto an $\mathbf{F}_3[u]/u^{36}$-module direct summand of the target, so the left vertical map is as well, because an injection of $\mathbf{F}_3[u]/u^{36}$ into a free $\mathbf{F}_3[u]/u^{36}$-module must be an identification with such a direct summand (consider torsion). This forces $\operatorname {\mathcal{M}}_\pi ({\mathcal{H}}_1)' = \operatorname {\mathcal{M}}_{r_1,{\mathbf{1}}}'$ and so, by Proposition 9.2.1, we see that $r_1 \neq 2$. Repeating the analogous argument applied to a submodule ${\overline{\rho }}\subseteq {\overline{\rho }}\otimes k$ one sees that $r_\omega \neq 10$.

Thus $({\mathcal{G}}',\{ [g]\})$ is weakly filtered by $\{ {\mathcal{G}}_{s,1}, {\mathcal{G}}_{r,\omega } \}$ for $(r,s)=(2,6)$, $(6,10)$, $(2,10)$ or $(6,6)$, as desired.

9.5. Completion of the proof of Theorem 4.6.3

Write $A_N$ for $\mathbf{F}_3[[T]]/(T^N)$. For $(r,s)=(2,6)$, $(6,10)$ and $(2,10)$, we will define a Breuil module ${\operatorname {\mathcal{M}}}_{N,(r,s)}$ over ${\mathcal{O}}_F$ and descent data $\{ \widehat{g}\}$ for $\operatorname {Gal}(F'/\mathbf{Q}_3)$ on ${\operatorname {\mathcal{M}}}_{N,(r,s)}'={\operatorname {\mathcal{M}}}_{N,(r,s)} \otimes _{\mathbf{F}_3} \mathbf{F}_9$ such that ${\operatorname {\mathcal{M}}}_{N,(r,s)}$ and $({\operatorname {\mathcal{M}}}_{N,(r,s)}',\{ \widehat{g} \})$ have compatible actions of $A_N$ (and $\widehat{\gamma }_2=1 \otimes \operatorname {Frob}_3$). More specifically set $t=2$, $6$ or $8$ according as $(r,s)=(2,6)$, $(6,10)$ or $(2,10)$. Viewing ${\overline{\rho }}$ as an extension class, it corresponds to a particular pair $(c,c_1)\in \mathbf{F}_3^2$ in Proposition 9.2.1. Fix these values. Motivated by the idea of deforming the formulae in Proposition 9.2.1, we are led to define

$$\begin{equation*} \operatorname {\mathcal{M}}_{N,(r,s)} = (A_N[u]/u^{36})\mathbf{e}_1 \oplus (A_N[u]/u^{36})\mathbf{e}'_{\omega }, \end{equation*}$$

$$\begin{equation*} (\operatorname {\mathcal{M}}_{N,(r,s)})_1 = \langle u^{s}\mathbf{e}_1, u^r\mathbf{e}'_{\omega } + (c + T)u^t\mathbf{e}_1\rangle \end{equation*}$$ with

$$\begin{equation*} \phi _1(u^{s}\mathbf{e}_1) = \mathbf{e}_1,\,\,\, \phi _1(u^r\mathbf{e}'_{\omega } + (c + T)u^t\mathbf{e}_1) = \mathbf{e}'_{\omega }. \end{equation*}$$

It is straightforward to check that $N \circ \phi _1 = 0$ on ${\operatorname {\mathcal{M}}}_{N,(r,s)}$. We may define $A_N$-linear descent data on ${\operatorname {\mathcal{M}}}_{N, (r,s)}'$ by setting $\widehat{\gamma }_2 = 1 \otimes \operatorname {Frob}_3$ and using the following formulae.

(1)

When $(r,s)=(2,6)$, set$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_1) = -\sqrt {-1}\mathbf{e}_1,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_{\omega }) = -\sqrt {-1}\mathbf{e}'_{\omega }, \end{equation*}$$

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_1) = \mathbf{e}_1,\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_{\omega }) = (1 \pm u^{18})(\mathbf{e}'_{\omega } \pm c_1 u^{6}\mathbf{e}_1). \end{equation*}$$

(2)

When $(r,s)=(6,10)$, set$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_1) = \sqrt {-1}\mathbf{e}_1,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_{\omega }) = \sqrt {-1}\mathbf{e}'_{\omega }, \end{equation*}$$

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_1) = (1\mp u^{18})\mathbf{e}_1,\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_{\omega }) = (\mathbf{e}'_{\omega } \pm c_1 u^{6}\mathbf{e}_1). \end{equation*}$$

(3)

When $(r,s)=(2,10)$, set$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_1) = \sqrt {-1}\mathbf{e}_1,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_{\omega }) = -\sqrt {-1}\mathbf{e}'_{\omega }, \end{equation*}$$

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_1) = (1\mp u^{18})\mathbf{e}_1,\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_{\omega }) = (1 \pm u^{18})(\mathbf{e}'_{\omega } \pm c_1 u^{12}\mathbf{e}_1). \end{equation*}$$

It is readily checked that this defines an object of ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}$ with an action of $A_N$. Let ${\mathcal{G}}_{N,(r,s)}$ and $({\mathcal{G}}_{N,(r,s)}',\{ [g]\})$ be the corresponding finite flat ${\mathcal{O}}_F$-group scheme and finite flat ${\mathcal{O}}_{F'}$-group scheme with descent data.

If $1 \leq M < N$, then we have a short exact sequence in ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}$

$$\begin{equation*} (0) \longrightarrow {\operatorname {\mathcal{M}}}_{M,(r,s)}' \longrightarrow {\operatorname {\mathcal{M}}}_{N,(r,s)}' \longrightarrow {\operatorname {\mathcal{M}}}_{N-M,(r,s)}' \longrightarrow (0), \end{equation*}$$

where the first map is induced by multiplication by $T^{N-M}$. The case $M=1$ shows that

$$\begin{equation*} ({\mathcal{G}}_{N,(r,s)},\{ [g] \})_{\mathbf{Q}_3}/T({\mathcal{G}}_{N,(r,s)}, \{ \widehat{g} \})_{\mathbf{Q}_3} \end{equation*}$$

corresponds to ${\overline{\rho }}$. Thus we get a surjection of $A_N[G_3]$-modules $A_N^2 \rightarrow \hspace{-.86em} \rightarrow {\mathcal{G}}_{N,(r,s)}(\overline{\mathbf{Q}}_3)$, which must in fact be an isomorphism (count orders). Thus $({\mathcal{G}}_{N,(r,s)},\{ [g] \})_{\mathbf{Q}_3}$ defines a deformation $\rho _{N,(r,s)}$ of ${\overline{\rho }}$ to $A_N^2$. For $N \geq 2$ we have $\rho _{N,(r,s)} \bmod T^2 \cong \rho _{2,(r,s)}$.

We also have an exact sequence

$$\begin{equation*} (0) \longrightarrow {\operatorname {\mathcal{M}}}_{s,1}' \otimes _{\mathbf{F}_3} A_N \longrightarrow ({\operatorname {\mathcal{M}}}_{N,(r,s)}', \{ \widehat{g}\}) \longrightarrow {\operatorname {\mathcal{M}}}_{r,\omega }' \otimes _{\mathbf{F}_3} A_N \longrightarrow (0) \end{equation*}$$

in ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3}$, from which we obtain an exact sequence of $A_N[G_3]$-modules

$$\begin{equation*} (0) \longrightarrow X_{\omega } \longrightarrow \rho _N \longrightarrow X_1 \longrightarrow (0). \end{equation*}$$

Note that $X_1 \cong \mathbf{F}_3^N$ and $X_\omega \cong \mathbf{F}_3(\omega )^N$ as $\mathbf{F}_3[G_3]$-modules. Moreover, this sequence must split as a sequence of $A_N$-modules. (Use, for instance, the kernel of $\rho _N(\sigma )-1$ for any $\sigma \in G_3 -G_{\mathbf{Q}_3(\sqrt {-3})}$.) Thus $X_1 \cong A_N$ and $X_\omega \cong A_N(\omega )$ as $A_N[G_3]$-modules, so $\det \rho _N = \omega$.

Finally, we must check that the exact sequence

$$\begin{equation*} (0) \longrightarrow {\overline{\rho }}\longrightarrow \rho _2 \longrightarrow {\overline{\rho }}\longrightarrow (0) \end{equation*}$$

is not split. We have maps of Breuil modules

$$\begin{equation*} \begin{array}{rcl} f_1:{\operatorname {\mathcal{M}}}_{s,1} & \longrightarrow & {\operatorname {\mathcal{M}}}_{2,(r,s)} \\\mathbf{e}& \longmapsto & \mathbf{e}_1 \end{array} \end{equation*}$$

and

$$\begin{equation*} \begin{array}{rcl} f_2:{\operatorname {\mathcal{M}}}_{2,(r,s)} & \longrightarrow & {\operatorname {\mathcal{M}}}_{r,\omega } \\\mathbf{e}_1 & \longmapsto & 0 \\T\mathbf{e}_1 & \longmapsto & 0 \\\mathbf{e}_\omega ' & \longmapsto & 0 \\T\mathbf{e}_\omega ' & \longmapsto & \mathbf{e}\end{array} \end{equation*}$$

compatible with descent data. These give rise to maps

$$\begin{equation*} f_1^*:\rho _2 \longrightarrow 1 \end{equation*}$$

and

$$\begin{equation*} f_2^*: \omega \longrightarrow \rho _2, \end{equation*}$$

such that the composites

$$\begin{equation*} {\overline{\rho }}\hookrightarrow \rho _2 \stackrel{f_1^*}{\longrightarrow } 1 \end{equation*}$$

and

$$\begin{equation*} \omega \stackrel{f_2^*}{\longrightarrow } \rho _2 \longrightarrow \hspace{-1.44em} \longrightarrow {\overline{\rho }} \end{equation*}$$

are non-zero.

To check that

$$\begin{equation*} (0) \longrightarrow {\overline{\rho }}\longrightarrow \rho _2 \longrightarrow {\overline{\rho }}\longrightarrow (0) \end{equation*}$$

is non-split, it suffices to check that

$$\begin{equation} (0) \longrightarrow \omega \longrightarrow \ker f_1^*/\operatorname {Im}f_2^* \longrightarrow 1 \longrightarrow (0) {\tag{9.5.1}} \end{equation}$$

is non-split. However, $\ker f_1^*/\operatorname {Im}f_2^*$ corresponds to an object $({\operatorname {\mathcal{N}}}', \{ \widehat{g} \})$ of ${\underline{\phi _1DD}}_{F'/\mathbf{Q}_3,{\mathcal{I}}}$ satisfying

$$\begin{equation*} \operatorname {\mathcal{N}}'= (\mathbf{F}_9[u]/u^{36})(T\mathbf{e}_1) \oplus (\mathbf{F}_9[u]/u^{36})\mathbf{e}'_{\omega },\,\,\, \operatorname {\mathcal{N}}'_1 = \langle u^{s}(T\mathbf{e}_1), u^r\mathbf{e}'_{\omega } + u^t(T\mathbf{e}_1)\rangle \end{equation*}$$

with

$$\begin{equation*} \phi _1(u^{s}(T\mathbf{e}_1)) = (T\mathbf{e}_1),\,\,\, \phi _1(u^r\mathbf{e}'_{\omega } + u^t(T\mathbf{e}_1)) = \mathbf{e}'_{\omega }. \end{equation*}$$

By Lemma 5.2.2, the sequence of Breuil modules with descent data

$$\begin{equation*} (0) \longrightarrow {\operatorname {\mathcal{M}}}_{s,1}' \longrightarrow {\operatorname {\mathcal{N}}}' \longrightarrow {\operatorname {\mathcal{M}}}_{r,\omega }' \longrightarrow (0) \end{equation*}$$

is not split. This sequence recovers (9.5.1) under generic fibre descent, so by Proposition 9.2.1

$$\begin{equation*} (0) \longrightarrow \omega \longrightarrow \ker f_1^*/\operatorname {Im}f_2^* \longrightarrow 1 \longrightarrow (0) \end{equation*}$$

is not split.

9.6. Completion of the proof of Theorem 4.6.2

Suppose first that $(r,s)=(2,6)$, $(6,10)$ or $(2,10)$. By Lemma 9.3.1

$$\begin{equation*} \theta _0: \operatorname {Ext}^1_{{\operatorname {\mathcal{S}}}_{i,(r,s)}}({\overline{\rho }}, {\overline{\rho }}) \longrightarrow H^1(G_3,\omega ) \end{equation*}$$

is the zero map. Lemma 9.3.3 then tells us that if $r\neq 6$, then

$$\begin{equation*} {\overline{\theta }}_1: \operatorname {Ext}^1_{{\operatorname {\mathcal{S}}}_{i,(r,s)}}({\overline{\rho }}, {\overline{\rho }}) \longrightarrow H^1(I_3,\mathbf{F}_3) \end{equation*}$$

is the zero map; while if $s\neq 6$, then

$$\begin{equation*} {\overline{\theta }}_\omega : \operatorname {Ext}^1_{{\operatorname {\mathcal{S}}}_{i,(r,s)}}({\overline{\rho }}, {\overline{\rho }}) \longrightarrow H^1(I_3,\mathbf{F}_3) \end{equation*}$$

is the zero map. Thus Theorem 4.7.5, and hence Theorem 4.6.2, follows in these cases.

Now consider the case $(r,s)=(6,6)$. Choose $x \in H^1_{{\operatorname {\mathcal{S}}}_{i,(6,6)}}(G_3, \operatorname {ad}^0 {\overline{\rho }})$. Let ${\mathcal{G}}$ denote the corresponding rank $81$ finite flat ${\mathcal{O}}_F$-group scheme with descent data $\{ [g]\}$ on ${\mathcal{G}}'={\mathcal{G}}\times _{{\mathcal{O}}_F} {\mathcal{O}}_{F'}$. Set ${\operatorname {\mathcal{M}}}={\operatorname {\mathcal{M}}}_\pi ({\mathcal{G}})$. Let ${\mathcal{H}}\subset {\mathcal{G}}$ denote the closed subgroup scheme (with descent data) corresponding to the kernel of the map $({\mathcal{G}}',\{ [g]\})_{\mathbf{Q}_3} \rightarrow \hspace{-.86em} \rightarrow {\overline{\rho }}\rightarrow \hspace{-.86em} \rightarrow \mathbf{F}_3$ and let ${\operatorname {\mathcal{N}}}={\operatorname {\mathcal{M}}}_\pi ({\mathcal{H}})$. Then ${\operatorname {\mathcal{N}}}$ has $\mathbf{F}_3[u]/u^{36}$-basis $\mathbf{e}_\omega ,\mathbf{e}_1',\mathbf{e}_\omega '$ with respect to which

$$\begin{equation*} \operatorname {\mathcal{N}}_1 = \langle u^6\mathbf{e}_{\omega }, u^6\mathbf{e}'_1 + (b + b' u^4)\mathbf{e}_{\omega }, u^6\mathbf{e}'_{\omega } + (c + c'u^4)\mathbf{e}'_1 + f\mathbf{e}_{\omega }\rangle , \end{equation*}$$

where $b,b',c,c' \in \mathbf{F}_3$, $f \in \mathbf{F}_3[u]/u^{36}$ and $\phi _1$ sends the indicated generators of $\operatorname {\mathcal{N}}_1$ to $\mathbf{e}_{\omega }, \mathbf{e}'_1, \mathbf{e}'_{\omega }$ respectively. Also, the descent data has the form

$$\begin{equation*} \widehat{\gamma }_4(\mathbf{e}_{\omega }) = \sqrt {-1}\mathbf{e}_{\omega },\,\,\, \widehat{\gamma }_4(\mathbf{e}'_1) = -\sqrt {-1}\mathbf{e}'_1,\,\,\, \widehat{\gamma }_4(\mathbf{e}'_{\omega }) = \sqrt {-1}\mathbf{e}'_{\omega } + h_{\gamma _4}(u) \mathbf{e}_{\omega } \end{equation*}$$

for some $h_{\gamma _4} \in \mathbf{F}_9[u]/u^{36}$, and

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_{\omega }) = \mathbf{e}_{\omega },\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_1) = \mathbf{e}'_1 + (\pm b - b'(\pm u^{12} + u^{30}))\mathbf{e}_{\omega }, \end{equation*}$$

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_{\omega }) = \mathbf{e}'_{\omega } + (\pm c - c'(\pm u^{12} + u^{30}))\mathbf{e}'_1 + h_{\pm 1}\mathbf{e}_{\omega }, \end{equation*}$$ where $h_{\pm 1} \in \mathbf{F}_9[u]/u^{36}$. Also, as ${\overline{\rho }}$ is très ramifié, we see that $c \neq 0$ by Proposition 9.2.1 and Lemma 9.3.4.

The requirement that $u^{12}\operatorname {\mathcal{N}}\subseteq \operatorname {\mathcal{N}}_1$ forces $\operatorname {\mathcal{N}}_1$ to contain

$$\begin{align*} u^{12}\mathbf{e}'_{\omega } &= u^6(u^6\mathbf{e}'_{\omega } + (c + c'u^4)\mathbf{e}'_1 + f\mathbf{e}_{\omega }) - (c + c'u^4)(u^6\mathbf{e}'_1 + (b + b' u^4)\mathbf{e}_{\omega })\\ &\quad + ((b+b'u^4)(c+c'u^4) - f u^6)\mathbf{e}_{\omega }, \end{align*}$$

so $\operatorname {\mathcal{N}}_1$ must contain $(b+b'u^4)(c+c'u^4)\mathbf{e}_{\omega }$. As $c \ne 0$ we get $(b+b'u^4)\mathbf{e}_{\omega } \in \operatorname {\mathcal{N}}_1$, and since $\mathbf{e}_{\omega }, u^4\mathbf{e}_{\omega } \not \in \operatorname {\mathcal{N}}_1$, we must have $b = b' = 0$. We conclude that the natural map

$$\begin{equation*} \theta _0: \operatorname {Ext}^1_{{\operatorname {\mathcal{S}}}_{i,(6,6)}}({\overline{\rho }}, {\overline{\rho }}) \longrightarrow H^1(G_3,\omega ) \end{equation*}$$

is the zero map.

Let us further analyse ${\operatorname {\mathcal{N}}}$. Replacing $\mathbf{e}'_{\omega }$ by $\mathbf{e}'_{\omega } + t^3\mathbf{e}_{\omega }$ for $t \in \mathbf{F}_3[u]/u^{36}$ causes $f$ to be replaced by $f - u^6 t^3 + u^6 t$ and otherwise leaves our standardized form unchanged (except that $h_{\gamma _4}$ and $h_{\pm 1}$ may change). Using a suitable choice of such $t$, we may assume $f$ has degree at most 6. On the other hand,

$$\begin{align*} \widehat{\gamma }_4(u^6\mathbf{e}'_{\omega } + (c + c'u^4)\mathbf{e}'_1 + f \mathbf{e}_{\omega }) &= -\sqrt {-1}(u^6\mathbf{e}'_{\omega } + (c+c'u^4)\mathbf{e}'_1 + f\mathbf{e}_{\omega }) \\ &\quad + (\sqrt {-1}(f(u) + f(-\sqrt {-1}u)) - u^6 h_{\gamma _4}(u))\mathbf{e}_{\omega }, \end{align*}$$

so $\widehat{\gamma }_4(\operatorname {\mathcal{N}}'_1) \subseteq \operatorname {\mathcal{N}}'_1$ if and only if

$$\begin{equation*} f(u) + f(-\sqrt {-1}u) \equiv 0 \bmod u^6, \end{equation*}$$

which forces

$$\begin{equation*} f = a_2 u^2 + a_6 u^6 \end{equation*}$$

for some $a_2, a_6 \in \mathbf{F}_3$. From the wild descent data formulae derived in the proof of Lemma 9.3.4 we also see that $h_{\pm 1} \equiv 0 \bmod u^6$.

Now $\operatorname {\mathcal{M}}$ has an ordered basis $\mathbf{e}_1, \mathbf{e}_{\omega }, \mathbf{e}'_1, \mathbf{e}'_{\omega }$ with respect to which

$$\begin{multline} \operatorname {\mathcal{M}}_1 = \langle u^6\mathbf{e}_1, u^6\mathbf{e}_{\omega } + (c+c'u^4)\mathbf{e}_1, u^6\mathbf{e}'_1 + h \mathbf{e}_1,\\ u^6\mathbf{e}'_{\omega } + ({c}+{c}'u^4)\mathbf{e}'_1 + (a_2 u^2 + a_6u^6)\mathbf{e}_{\omega } + g\mathbf{e}_1\rangle , {\tag{9.6.1}} \end{multline}$$

where $g,h \in \mathbf{F}_3[u]/u^{36}$ and $\phi _1$ sends the indicated generators of $\operatorname {\mathcal{M}}_1$ to $\mathbf{e}_1, \mathbf{e}_{\omega }, \mathbf{e}'_1, \mathbf{e}'_{\omega }$. If we try to expand out $u^{12}\mathbf{e}'_{\omega }$ as a linear combination of the indicated generators of $\operatorname {\mathcal{M}}_1$, we find that

$$\begin{equation*} u^{12}\mathbf{e}'_{\omega } \equiv (({c}+{c}'u^4)h + ca_2 u^2)\mathbf{e}_1 \bmod \operatorname {\mathcal{M}}_1. \end{equation*}$$

It follows that $u^{12}\mathbf{e}_\omega ' \in {\operatorname {\mathcal{M}}}_1$ if and only if

$$\begin{equation*} (c+c'u^4)h+ca_2 u^2 \equiv 0 \bmod u^6. \end{equation*}$$

Since ${\overline{\rho }}$ is très ramifié, the last part of Proposition 9.2.1 tells us that $c \neq 0$. Thus, $u^{12}\mathbf{e}'_{\omega } \in \operatorname {\mathcal{M}}_1$ if and only if $h \equiv -a_2 u^2 \bmod u^6$. We can now use Lemma 9.3.4 to see that the wild descent data action is determined by

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_1) = \mathbf{e}_1,\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}_{\omega }) = \mathbf{e}_{\omega } + (\pm c - c'(\pm u^{12} + u^{30}))\mathbf{e}_1,\,\,\, \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_1) = \mathbf{e}'_1 + f_{\pm 1}\mathbf{e}_1 \end{equation*}$$

(with $f_{\pm 1} \equiv 0 \bmod u^6$), and

$$\begin{equation*} \widehat{\gamma _3^{\pm 1}}(\mathbf{e}'_{\omega }) = \mathbf{e}'_{\omega } + (\pm c - c'(\pm u^{12} + u^{30}))\mathbf{e}'_1 + h_{\pm 1}\mathbf{e}_{\omega } + g_{\pm 1}\mathbf{e}_1, \end{equation*}$$

where $g_{\pm 1} \in \mathbf{F}_9[u]/u^{36}$ and $h_{\pm 1} \equiv 0 \bmod u^6$.

We must have

$$\begin{equation} \widehat{\gamma _3^{\pm 1}}(u^6\mathbf{e}'_{\omega } + ({c}+{c}'u^4)\mathbf{e}'_1 + (a_2u^2 + a_6 u^6)\mathbf{e}_{\omega } + g(u)\mathbf{e}_1) \in {\operatorname {\mathcal{M}}}_1', {\tag{9.6.2}} \end{equation}$$

and this expression is easily computed to equal

$$\begin{align*} &u^6H_{\gamma _3^{\pm 1}}^6\cdot (\mathbf{e}'_{\omega } + (\pm {c} - {c}'(\pm u^{12} + u^{30}))\mathbf{e}'_1 + h_{\pm 1}\mathbf{e}_{\omega } + g_{\pm 1}\mathbf{e}_1) \\ &\quad + ({c}+{c}' u^4H_{\gamma _3^{\pm 1}}^4)(\mathbf{e}'_1 + f_{\pm 1}\mathbf{e}_1) \\ &\qquad + (a_2 u^2H_{\gamma _3^{\pm 1}}^2 + a_6u^6H_{\gamma _3^{\pm 1}}^6)( \mathbf{e}_{\omega } + (\pm c - c'(\pm u^{12} + u^{30}))\mathbf{e}_1) + g(uH_{\gamma _3^{\pm 1}})\mathbf{e}_1. \end{align*}$$

Remembering that $\langle u^6\mathbf{e}_1, u^{12}\operatorname {\mathcal{M}}' \rangle \subseteq \operatorname {\mathcal{M}}'_1$, (9.6.2) becomes

$$\begin{equation*} u^6(\mathbf{e}'_{\omega } \pm {c}\mathbf{e}'_1)\! +\! ({c}+{c}'u^4 H_{\gamma _3^{\pm 1}}^4)\mathbf{e}'_1 \!+\! (a_2 u^2 H_{\gamma _3^{\pm 1}}^2 + a_6 u^6)\mathbf{e}_{\omega } \pm a_2 c u^2 \mathbf{e}_1 \!+\! g(u)\mathbf{e}_1 \in {\operatorname {\mathcal{M}}}_1'. \end{equation*}$$

Using the explicit generators of ${\operatorname {\mathcal{M}}}_1$ given in (9.6.1) and recalling that $h\! \equiv \! -a_2 u^2 \bmod u^6$, this simplifies to

$$\begin{equation*} \pm a_2 c u^2\mathbf{e}_1 \in \operatorname {\mathcal{M}}'_1. \end{equation*}$$

Thus $a_2cu^2$ is divisible by $u^6$, so $a_2=0$.

The image of the class $x$ in $\operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(\omega ,\omega )$ under $\theta _\omega$ corresponds to a finite flat ${\mathcal{O}}_F$-group scheme with Breuil module ${\operatorname {\mathcal{M}}}_x$ free of rank two over $\mathbf{F}_3[u]/u^{36}$ with basis $\mathbf{e}_\omega ,\mathbf{e}_\omega '$, and with

$$\begin{equation*} (\operatorname {\mathcal{M}}_x)_1 = \langle u^6\mathbf{e}_{\omega }, u^6\mathbf{e}'_\omega + a_6u^6 \mathbf{e}_\omega \rangle , \end{equation*}$$

where $\phi _1$ sends the indicated generators of $(\operatorname {\mathcal{M}}_x)_1$ to $\mathbf{e}_{\omega }$ and $\mathbf{e}_{\omega }'$ respectively. According to the proof of Lemma 9.3.4 this implies that the image of the class $x$ in $\operatorname {Ext}^1_{\mathbf{F}_3[G_3]}(\omega ,\omega )$ is split over an unramified extension of $\mathbf{Q}_3$. Thus,

$$\begin{equation*} {\overline{\theta }}_\omega :\operatorname {Ext}^1_{{\operatorname {\mathcal{S}}}_{i,(6,6)}}({\overline{\rho }}, {\overline{\rho }}) \longrightarrow H^1(I_3,\mathbf{F}_3) \end{equation*}$$

is the zero map. This completes the proof of Theorem 4.7.5, and hence of Theorem 4.6.2.

10. Corrigenda for CDT

We would like to take this opportunity to record a few corrections to CDT.

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Page 523, line $-10$: Insert “$K$-rational” after “For each”.

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Page 532, line $-6$: “The semisimplicity of $\sigma _n$ follows from that of $\sigma _1$” is false and should be deleted. This assertion was not used anywhere in the rest of the paper.

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Page 536, line 7: Replace $\operatorname {GL}_2(\mathbf{C})$ by $\operatorname {GL}_2(\mathbf{R})$.

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Page 538, line $-10$: Replace “of type $(S,\tau )$” by “such that $\rho |_{G_\ell }$ is of type $\tau$ and $\rho$ is of type $(S,\tau )$”.

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Page 539, lines 18–20: Replace each $\omega _1$ by $\eta _1$ and each $\omega _2$ by $\eta _2$.

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Page 541, line 14: Replace each of the three occurrences of $\mathbf{A}$ by $\mathbf{A}^\infty$.

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Page 544, line $-6$: “the discrete topology on $V_p$” should read “the $\ell$-adic topology on $M_p$”.

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Page 545, part 4 of Lemma 6.1.2: $V'$ should be assumed to be a normal subgroup of $V$.

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Page 546, line 1: We should have noted that the key component of this argument is very similar to the main idea of Kh.

•

§6.2: There are two significant errors in this section. The assertion “$\Gamma =\operatorname {SL}_2(\mathbf{Z}) \cap U_S$ satisfies the hypotheses of Theorem 6.1.1” is false and $\operatorname {Hom}(L_n,k)$ should be $L_n \otimes k$. The argument of this section can be repaired by making the following changes.

–

Page 546, lines 5 and 6: Replace “Setting $S=T({\overline{\rho }}) \cup \{ r\}$, we find that the group $\Gamma =\operatorname {SL}_2(\mathbf{Z}) \cap U_S$ satisfies the hypotheses of Theorem 6.1.1.” by “Set $S=T({\overline{\rho }}) \cup \{ r\}$; $U_S' = \prod _p U_{S,p}'$ where $U_{S,p}'=U_1(p^{c_p})$ if $p \in T({\overline{\rho }})$ and $U_{S,p}'=U_{S,p}$ otherwise; $V_S' = \prod _p V_{S,p}'$ where $V_{S,p}'=U_1(p^{c_p})$ if $p \in T({\overline{\rho }})$ and $V_{S,p}'=V_{S,p}$ otherwise; and $L_S'=\operatorname {Hom}_{{\mathcal{O}}[U_S'/V_S']} (M_\ell ,H^1(X_{V_S'},{\mathcal{O}}))[I_S']$. Then $\Gamma =\operatorname {SL}_2(\mathbf{Z}) \cap U_S'$ satisfies the hypotheses of Theorem 6.1.1.”

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Page 546, lines 7–13: Replace $Y_S$ by $Y_{U_S'}$, $\operatorname {Hom}(L_n,k)$ by $L_n \otimes k$, $M_S$ by $M_\ell$, ${\mathcal{F}}_S$ by ${\mathcal{F}}_{\operatorname {Hom}_{{\mathcal{O}}}(M_\ell ,{\mathcal{O}})}$ and $L_S$ by $L_S'$.

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Page 546, line 13: Replace “and ${\operatorname {\mathcal{N}}}_S$ is non-empty.” by “. Using the fact that Lemma 5.1.1 holds with $U_S'$ replacing $U_S$ and $\sigma _\ell$ replacing $\sigma _S$ and the discussion on page 541 we conclude that ${\operatorname {\mathcal{N}}}_S$ is non-empty.”

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Page 549, line $-15$: Replace $U_{\{ r,r' \} ,p }$ by $U_{\{ r \} ,p }$.

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Page 549, line $-11$: Replace $U_S'/U_{0,S}'$ by $V_0/V_1$.

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Page 552, line 4: The assertion is false in the case $\ell \geq 5$. It can be corrected by adding “and $j(E) \not \equiv 1728 \bmod \ell$ (which is true if, for instance, $E$ has potentially supersingular reduction and $\ell \equiv 1 \bmod 4$)” after “if $\ell \geq 5$”.

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Page 554, line 11: Replace “$j_E \in$” by “$E$ is isogenous to an elliptic curve with $j$-invariant in the set”.

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Page 554, line 11: Replace $5(29)^3/2^5$ by $-5(29)^3/2^5$.

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Page 554, line 17: Replace the parenthetical comment “(and $j=5(29)^3/2^5$)” by “(and isogenous to one with $j$-invariant $-5(29)^3/2^5$)”.

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Page 554, line $-5$: Replace $p$ by $q$ and $q$ by $p$.