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