Irreducibility of Newton strata in ${\mathrm{GU}}(1,n-1)$ Shimura varieties

Abstract

Let $L$ be a quadratic imaginary field, inert at the rational prime $p$. Fix an integer $n\ge 3$, and let $\mathcal{M}$ be the moduli space (in characteristic $p$) of principally polarized abelian varieties of dimension $n$ equipped with an action by $\mathcal{O}_L$ of signature $(1,n-1)$. We show that each Newton stratum of $\mathcal{M}$, other than the supersingular stratum, is irreducible.

1. Introduction

For a complex abelian variety $X$, the isomorphism class of its $p$-torsion group scheme $X[p]$ and of its $p$-divisible group $X[p^\infty ]$ depend only on the dimension of $X$. In contrast, in characteristic $p$, there are different possibilities for the corresponding isomorphism (or even isogeny) class. Each such invariant provides a stratification of a family of abelian varieties in positive characteristic.

The isogeny class of $X[p^\infty ]$ is called the Newton polygon of $X$. The goal of the present paper is to prove that the space of abelian varieties with given Newton polygon and a certain, specified endomorphism structure is irreducible.

More precisely, let $L$ be a quadratic imaginary field, inert at the rational prime $p$. Fix an integer $n \ge 3$, and let ${\mathcal{M}}$ be the moduli space (over ${\mathbb{F}}_{p^2}$) of principally polarized abelian varieties of dimension $n$ equipped with an action by ${\mathcal{O}}_L$ of signature $(1,n-1)$. Our main result is:

Theorem 1.1.

Let $\xi \not = \sigma$ be an admissible Newton polygon for ${\mathcal{M}}$ which is not supersingular. Then the corresponding stratum ${\mathcal{M}}^\xi$ is irreducible.

The proof of Theorem 1.1 is modelled on, but considerably easier than, that of 4, Theorem A. This is possible because the Newton and Ekedahl-Oort stratifications of ${\mathcal{M}}$ are much simpler than those of ${\mathcal{A}}_g$.

In the special case where $L = {\mathbb{Q}}(\zeta _3)$ and $n$ is $3$ or $4$, ${\mathcal{M}}$ essentially coincides with a component of the moduli space of cyclic triple covers of the projective line. Theorem 1.1 provides a crucial base case for forthcoming work of Ozman, Pries and Weir on such covers 11, and that work was the initial impetus for the present study.

For a topological space $T$, let $\Pi _0(T)$ denote the set of irreducible components of $T$. If $T \subset {\mathcal{M}}$, then ${\overline{T}}$ is its closure in ${\mathcal{M}}$. The symbol $k$ will denote an arbitrary algebraically closed field of characteristic $p$.

2. Background on $\mathcal{M}$

2.1. Moduli spaces

Let ${\mathcal{M}}$ be the moduli stack (over ${\mathcal{O}}_L/p \cong {\mathbb{F}}_{p^2}$) of principally polarized abelian varieties of dimension $n$ with an action by ${\mathcal{O}}_L$ of signature $(1,n-1)$. Somewhat more precisely, ${\mathcal{M}}(S)$ consists of isomorphism classes of data $(X,\iota ,\lambda )$, where $X \rightarrow S$ is an abelian variety of relative dimension $n$, $\iota : {\mathcal{O}}_L \rightarrow \operatorname {End}_S(X)$ is an embedding taking $1_L$ to $\operatorname {id}_X$ such that $\operatorname {Lie}(X)$, as a module over ${\mathcal{O}}_L\otimes {\mathcal{O}}_S$, has signature $(1,n-1)$, and $\lambda : X \rightarrow X^\vee$ is a principal polarization such that if $(\dagger )$ is the induced Rosati involution on $\operatorname {End}(X)$, then for each $a\in {\mathcal{O}}_L$ one has $\iota ({\overline{a}}) = \iota (a)^{(\dagger )}$. It is standard that $\dim {\mathcal{M}} = 1\cdot (n-1) = n-1$.

In fact, ${\mathcal{M}}$ is the moduli stack attached to the Shimura (pro-)variety constructed from a certain group $\operatorname {G}$, as follows.

Let $V$ be an $n$-dimensional vector space over $L$, equipped with a Hermitian pairing of signature $(1,n-1)$. Let $\operatorname {G}/{\mathbb{Q}}$ be the group of unitary similitudes of $V$, and let $\operatorname {U}$ be the unitary group of $V$. Fix a hyperspecial subgroup ${\mathbb{K}}_p \subset \operatorname {G}({\mathbb{Q}}_p)$. For each sufficiently small open compact subgroup ${\mathbb{K}}^p \subset G({\mathbb{A}}^p_f)$, there is a moduli space ${\mathcal{M}}_{{\mathbb{K}}^p} = {\mathcal{M}}_{{\mathbb{K}}_p{\mathbb{K}}^p}$ of abelian varieties of dimension $n$ as above with ${\mathbb{K}}^p$ structure; see 7 for more details. If ${\mathbb{K}}^p$ is sufficiently small, then ${\mathcal{M}}_{{\mathbb{K}}^p}$ is a smooth, quasiprojective variety, and ${\mathcal{M}}$ may be constructed as the quotient of any ${\mathcal{M}}_{{\mathbb{K}}^p}$ by an appropriate finite group.

2.2. Newton polygons in $\mathcal{M}$

Newton and Ekedahl-Oort stratifications on $\operatorname {GU}(1,n-1)$ Shimura varieties are well understood 2. There are exactly $1 + {\lfloor n/2 \rfloor }$ (“admissible”) Newton polygons which occur, and the poset of admissible Newton polygons is actually totally ordered. Let $\sigma$ be the supersingular Newton polygon, so that $\sigma \preceq \xi$ for any admissible Newton polygon $\xi$ for ${\mathcal{M}}$. For a Newton polygon $\xi$, let ${\mathcal{M}}^\xi$ denote the locally closed locus corresponding to abelian varieties with Newton polygon $\xi$. Then ${\mathcal{M}}^\sigma$ is pure of dimension ${\lfloor \frac{n-1}{2} \rfloor }$. By purity 59, if $Z_\sigma \in \Pi _0({\mathcal{M}}^\sigma )$ and $\sigma \preceq \xi$, then there exists some $Z_\xi \in \Pi _0({\mathcal{M}}^\xi )$ such that $Z_\sigma \subseteq {\overline{Z_\xi }}$, the closure of $Z_\xi$ in ${\mathcal{M}}$.

The Newton stratification of ${\mathcal{M}}$ is described in 2, as follows. Each admissible Newton polygon is determined by its smallest slope. For each integer $1 \le j \le {\lfloor n/2 \rfloor }$, there is a Newton polygon $\xi _{2j}$ with smallest slope

$$\begin{equation*} \lambda (2j) = \frac{1}{2} - \frac{1}{2({\lfloor n/2 \rfloor }+1-j)}; \end{equation*}$$

then ${\mathcal{M}}^{\xi _{2j}}$ has codimension ${\lfloor n/2 \rfloor }-j$ in ${\mathcal{M}}$. (Admittedly, in many ways this normalization is more awkward than that of 2, in which ${\mathcal{M}}^{\xi _{2j}}$ is labeled ${\mathcal{M}}_{2({\lfloor n/2 \rfloor }-j)}$, but it will be more convenient for the deformation theory below.)

Away from the supersingular locus ${\mathcal{M}}^\sigma$, the Newton, Ekedahl-Oort, and final stratifications coincide; a $p$-divisible group is determined by its $\bmod p$ truncation 2, Theorem 5.3. This is recalled in greater detail in Section 2.3.

The Newton polygon and Ekedahl-Oort type of a polarized ${\mathcal{O}}_L$-abelian variety with prime-to-$p$ level structure do not depend on the level structure, and we set ${\mathcal{M}}_{{\mathbb{K}}^p}^\xi = {\mathcal{M}}_{{\mathbb{K}}^p}\times _{{\mathcal{M}}} {\mathcal{M}}^\xi$.

2.3. $p$-divisible groups

In contrast to the Siegel case, it is possible to write a finite, explicit collection of those principally quasipolarized $p$-divisible groups with ${\mathcal{O}}_L$-action which occur as $(X,\iota ,\lambda )[p^\infty ]$ for $(X,\iota ,\lambda ) \in {\mathcal{M}}(k)$. Following Wedhorn, we describe such $p$-divisible groups in terms of their covariant Dieudonné modules, as follows.

For $m \in {\mathbb{N}}$, let $M(m)$ be the following Dieudonné module:

•

As a $W(k)$-module, $M(m)$ admits basis $\{u_1, \cdots , u_m, v_1, \cdots , v_m\}$.

•

A display 1013 for $M(m)$ is$$\begin{align*} F u_1 &= (-1)^mv_m & v_1 &= Vu_m\\ Fv_2 &= u_1 & u_2 &= Vv_1 \\ F v_3 &= u_2 &u_3 &= Vv_2 \\ \vdots &&\vdots \\ F v_m &= u_{m-1} &u_m &= V v_{m-1}. \end{align*}$$

•

The two eigenspaces for the action of ${\mathcal{O}}_L$ on $M(m)$ are $\oplus W(k) u_i$ and $\oplus W(k) v_j$.

•

The quasipolarization is given by the symplectic pairing ${{\langle \cdot ,\cdot \rangle }}$, where$$\begin{align*} {{\langle u_i, v_j \rangle }}& = (-1)^i\delta _{ij},\\ {{\langle u_i,u_j \rangle }} = {{\langle v_i,v_j \rangle }} &= 0. \end{align*}$$

We also define the Dieudonné module $N$:

•

As a $W(k)$-module, $N$ admits basis $\{u_0,v_0\}$.

•

A display for $N$ is$$\begin{align*} Fv_0 &= -u_0 & u_0 &= Vv_0. \end{align*}$$

•

The quasipolarization is given by the symplectic pairing ${{\langle u_0,v_0 \rangle }}=1$.

•

The two eigenspaces for the action of ${\mathcal{O}}_L$ are $W(k)u_0$ and $W(k)v_0$.

With this notation in place, one has the following result of Bültel and Wedhorn:

Theorem 2.1 (2).

(a)

Suppose $1 \le j \le {\lfloor n/2 \rfloor }$. There exists an integer $r(j)$ such that if $(X,\iota ,\lambda ) \in {\mathcal{M}}^{\xi _{2j}}(k)$, then$$\begin{equation*} {\mathbb{D}}_*((X,\iota ,\lambda )[p^\infty ]) \cong M(2({\lfloor n/2 \rfloor }+1-j) \oplus N^{r(j)}. \end{equation*}$$

(b)

There exists an open dense subspace ${\mathcal{M}}^{\sigma \circ } \subset {\mathcal{M}}^\sigma$ such that if $(X,\iota ,\lambda ) \in {\mathcal{M}}^{\sigma \circ }(k)$, then$$\begin{equation*} {\mathbb{D}}_*((X,\iota ,\lambda )[p^\infty ]) \cong \begin{cases} M(n), & n\text{ odd}, \\ M(n-1) \oplus N, & n\text{ even}. \end{cases} \end{equation*}$$

2.4. Hecke operators

An inclusion ${\mathbb{K}}_1^p \hookrightarrow {\mathbb{K}}_2^p$ of open compact subgroups of $\operatorname {G}({\mathbb{A}}^p_f)$ induces a cover of Shimura varieties ${\mathcal{M}}_{{\mathbb{K}}_1^p} \rightarrow {\mathcal{M}}_{{\mathbb{K}}_2^p}$. More generally, an element $g \in \operatorname {G}({\mathbb{A}}^p_f)$ induces, for each open compact ${\mathbb{K}}^p$, a natural morphism ${\mathcal{M}}_{{\mathbb{K}}^p} \rightarrow {\mathcal{M}}_{g^{-1}{\mathbb{K}}^pg}$.

Let $z \in {\mathcal{M}}_{{\mathbb{K}}_0^p}(k)$. Its prime-to-$p$ (unitary) Hecke orbit, ${\mathcal{H}}^p(z)$, is defined as follows. Consider the pro-variety ${\widehat{{\mathcal{M}}}}_{{\mathbb{K}}_0^p} = \lim _{\stackrel{\leftarrow }{{\mathbb{K}}^p\subset {\mathbb{K}}_0^p}} {\mathcal{M}}_{{\mathbb{K}}^p}$. Choose a lift $\hat{z}$ of $z$ to ${\widehat{{\mathcal{M}}}}_{{\mathbb{K}}_0^p}$. Then ${\mathcal{H}}^p(z)$ is the projection to ${\mathcal{M}}_{{\mathbb{K}}_0^p}$ of $\operatorname {U}({\mathbb{A}}^p_f) \hat{z}$. (One can also construct the “similitude” Hecke orbit of $\hat{z}$ by replacing the orbit $\operatorname {U}({\mathbb{A}}^p_f)\hat{z}$ with $\operatorname {G}({\mathbb{A}}^p_f)\hat{z}$. However, the unitary Hecke orbit is both the output of 12, Theorem 4.6 and the input to 6, Theorem 1.4, and thus is better suited to the task at hand.)

3. Closures of Newton strata

Let $\xi$ be an admissible Newton polygon for ${\mathcal{M}}$ such that $\xi \not = \sigma$.

Lemma 3.1.

The locus ${\mathcal{M}}^\xi$ is smooth.

Proof.

The isomorphism class of $(X[p^\infty ],\iota [p^\infty ], \lambda [p^\infty ])$ for $(X,\iota ,\lambda ) \in {\mathcal{M}}^\xi (k)$ is independent of the choice of point (Theorem 2.1). By the Serre-Tate theorem, the formal neighborhoods of all points of ${\mathcal{M}}^\xi$ are thus isomorphic. Since ${\mathcal{M}}^\xi$ is by definition equipped with the reduced subscheme structure, it must be smooth.

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Lemma 3.2.

If $Z_\xi \in \Pi _0({\mathcal{M}}^\xi )$, then there exists $Z_\sigma \in \Pi _0({\mathcal{M}}^\sigma )$ such that $Z_\sigma \subset {\overline{Z}}_\xi$.

Proof.

We prove the following apparently stronger result. Suppose $\nu$ and $\xi$ are admissible Newton polygons with $\nu \prec \xi$, and $Z_\xi \in \Pi _0({\mathcal{M}}^\xi )$. We show that there exists $Z_\nu \in \Pi _0({\mathcal{N}}^\sigma )$ such that $Z_\nu \subset {\overline{Z}}_\xi$. It suffices to prove this statement under the assumption that $\nu$ is the immediate predecessor of $\xi$, so that $\dim {\mathcal{M}}^\nu = \dim {\mathcal{M}}^\xi -1$. The statement is trivially true if $\xi = \xi _{2{\lfloor n/2 \rfloor }}$ is the locus with positive $p$-rank; henceforth, we assume that $\xi$ is strictly smaller than $\xi _{2{\lfloor n/2 \rfloor }}$.

It is is slightly more convenient to work with fine moduli schemes. Let ${\mathbb{K}}^p \subset \operatorname {G}({\mathbb{A}}^p_f)$ be an open compact subgroup which is small enough that ${\mathcal{M}}_{{\mathbb{K}}^p}$ is a smooth, quasiprojective variety. Let $W_\xi \in \Pi _0(Z_\xi \times _{{\mathcal{M}}} {\mathcal{M}}_{{\mathbb{K}}^p})$ be an irreducible component of ${\mathcal{M}}_{{\mathbb{K}}^p}^\xi$ lying over $Z_\xi$. It suffices to show that the closure of $W_\xi$ in ${\mathcal{M}}_{{\mathbb{K}}^p}$ contains an irreducible component of ${\mathcal{M}}_{{\mathbb{K}}^p}^\nu$.

Let ${{\widetilde{{\mathcal{M}}}}}_{{\mathbb{K}}^p}$ be a toroidal compactification of ${\mathcal{M}}_{{\mathbb{K}}^p}$ (e.g., 8, 6.4.1.1). It is a smooth, projective variety. Let ${\widetilde{W}}_\xi$ be the closure of $W_\xi$ in ${{\widetilde{{\mathcal{M}}}}}_{{\mathbb{K}}^p}$, and let $\partial W_\xi = {\widetilde{W}}_\xi \smallsetminus W_\xi$. Newton strata (other than the supersingular stratum) coincide with Ekedahl-Oort strata, and the latter are known to be affine (e.g., 9). Because $W_\xi$ is positive dimensional, $\partial W_\xi$ is nonempty. The first slope of $\xi$ is positive, while the boundary of ${{\widetilde{{\mathcal{M}}}}}_{{\mathbb{K}}^p}$ parametrizes semiabelian varieties with nontrivial toric part. Consequently, $\partial W_\xi \cap ({{\widetilde{{\mathcal{M}}}}}_{{\mathbb{K}}^p}\smallsetminus {\mathcal{M}}_{{\mathbb{K}}^p})$ is empty, and $\partial W_\xi \subset {\mathcal{M}}_{{\mathbb{K}}^p}$. Again by purity (9), $\dim \partial W_\xi = \dim W_\xi -1$. By semicontinuity of Newton polygons, there is an a priori containment $\partial W_\xi \subseteq \bigcup _{\tau \prec \xi } {\mathcal{M}}_{{\mathbb{K}}^p}^{\tau }$. The result now follows from dimension counts: ${\mathcal{M}}_{{\mathbb{K}}^p}^\nu$ is pure of dimension $\dim W_\xi -1$, while if $\tau \prec \nu$, then $\dim {\mathcal{M}}_{{\mathbb{K}}^p}^\tau <\dim W_\nu = \dim \partial W_\xi$.

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Conversely,

Lemma 3.3.

If $Z_\sigma \in \Pi _0({\mathcal{M}}^\sigma )$, then there is a unique $Z_\xi \in \Pi _0({\mathcal{M}}^\xi )$ such that $Z_\sigma \subset {\overline{Z}}_\xi$.

Proof.

The existence of such a $Z_\xi$ follows from purity and dimension-counting. If there were two such components, then they would intersect along $Z_\sigma \cap {\mathcal{M}}^{\sigma \circ }$, which would contradict the smoothness shown in Lemma 4.1.

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4. Local calculations

Lemma 4.1.

Let $\xi$ be an admissible Newton polygon which is not supersingular, and suppose $z \in {\mathcal{M}}^{\sigma \circ }(k)$. Then ${{\overline{{\mathcal{M}}^\xi }}}$ is smooth at $z$.

Proof.

This follows directly from the explicit calculation (Lemmas 4.3 and 4.8) of the Newton stratification on the formal neighborhood ${\mathcal{M}}^{/z}$ of $z$.

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The necessary calculations are somewhat sensitive to the parity of $n$. We first work out the details when $n$ is odd and then indicate the changes necessary to accommodate even $n$.

4.1. The case of $n$ odd

Throughout this section, assume that $n$ is odd.

4.1.1. Explicit deformations

Suppose $z = (X,\iota ,\lambda ) \in {\mathcal{M}}^{\sigma \circ }(k)$. Our goal is to understand the Newton stratification on the formal neighborhood ${\mathcal{M}}^{/z} = \operatorname {Spf}{\widetilde{R}}$ of $z$ in ${\mathcal{M}}$. This will be accomplished using (covariant) Dieudonné theory. Suppose $z = (X,\iota ,\lambda ) \in {\mathcal{M}}^{\sigma \circ }(k)$. Then the Dieudonné module ${\mathbb{D}}_*(X[p^\infty ])$ is isomorphic to $M \coloneq M(n)$ (Theorem 2.1).

Deformations of $X[p^\infty ]$ are parametrized by $\operatorname {Hom}(VM/pM, M/VM)$; those which preserve the ${\mathcal{O}}_L$-structure are classified by $\operatorname {Hom}_{{\mathcal{O}}_L\otimes k}(VM/pM, M/VM)$ (e.g., 1). The display we have chosen gives coordinates on $VM/pM$ and $M/VM$:

$$\begin{align*} VM/pM &= k \{v_1, u_2, u_3, \cdots , u_n\}, \\ M/VM &= k \{u_1, v_2, v_3, \cdots , v_n\}. \end{align*}$$

Consequently,

$$\begin{align*} \operatorname {Hom}_{{\mathcal{O}}_L\otimes k}(VM/pM,M/VM) &= k\{ v_1^*v_2, v_1^*v_3, \cdots , v_1^*v_n, u_2^*u_1, u_3^*u_1, \cdots , u_n^*u_1\} \\ & \subset \operatorname {Hom}_k(VM/pM,M/VM)\\ & = (VM/pM)^* \otimes (M/VM), \end{align*}$$

and the universal equicharacteristic deformation ring of $(X[p^\infty ],\iota [p^\infty ])$ is

$$\begin{equation*} {\widetilde{R}}' = k\lBrack t(v_1v_2), t(v_1v_3), \cdots , t(v_1v_n), t(u_2u_1), \cdots , t(u_nu_1)\rBrack . \end{equation*}$$

For $t(xy) \in {\widetilde{R}}'$, let ${\underline{t}}(xy)$ be its Teichmüller lift to $W({\widetilde{R}}')$. Then ${\widetilde{X}}[p^\infty ]$ is displayed over ${\widetilde{R}}'$ by

$$\begin{align*} {\widetilde{F}} u_1 &= -v_n & v_1 &= {\widetilde{V}}(u_n + {\underline{t}}(u_nu_1)u_1)\\ {\widetilde{F}}v_2 &= u_1 & u_2 &= {\widetilde{V}}(v_1 +\sum _{2 \le j \le n} {\underline{t}}(v_1v_j)v_j)\\ {\widetilde{F}} v_3 &= u_2 + {\underline{t}}(u_2u_1)u_1& u_3 &= {\widetilde{V}}v_2 \\ \vdots &&\vdots \\ {\widetilde{F}} v_n &= u_{n-1}+ {\underline{t}}(u_{n-1}u_1)u_1 &u_n &= {\widetilde{V}} v_{n-1}. \end{align*}$$

The pairing ${{\langle \cdot ,\cdot \rangle }}$ extends to ${\widetilde{M}} = M \otimes _{W(k)} W({\widetilde{R}}')$ by linearity. We would like to identify the largest quotient ${\widetilde{R}}$ of ${\widetilde{R}}'$ to which ${{\langle \cdot ,\cdot \rangle }}$ extends as a pairing of Dieudonné modules, for then ${\mathcal{M}}^{/z} \cong \operatorname {Spf}{\widetilde{R}}$.

The quasipolarization extends to a ring $R$ if and only if, for each $x,y \in {\widetilde{M}}_R \coloneq {\widetilde{M}} \otimes _{W({\widetilde{R}}')}W(R)$, one has

$$\begin{equation*} {{\langle {\widetilde{F}}x,y \rangle }} = {{\langle x, {\widetilde{V}} \rangle }}^\sigma . \end{equation*}$$

Suppose $3 \le j \le n$, and let $(x,y) = (v_j, v_1 + \sum _{2 \le k \le n} {\underline{t}}(v_1v_k)v_k)$. Then

$$\begin{align*} {{\langle {\widetilde{F}} x, y \rangle }} &= {{\langle u_{j-1} + {\underline{t}}(u_{j-1}u_1)u_1, v_1 + \sum _{2 \le k \le n} {\underline{t}}(v_1v_k)v_k \rangle }} \\ &= {\underline{t}}(v_1v_{j-1}){{\langle u_{j-1}, v_{j-1} \rangle }} + {\underline{t}}(u_{j-1}u_1){{\langle u_1, v_1 \rangle }} \\ &= (-1)^{j-1}{\underline{t}}(v_1v_{j-1}) - {\underline{t}}(u_{j-1}u_1), \end{align*}$$

while

$$\begin{align*} {{\langle x, {\widetilde{V}}y \rangle }} &= {{\langle v_j,u_2 \rangle }} \\ &= 0. \end{align*}$$

Consequently, if ${\widetilde{M}}_R$ is quasipolarized by ${{\langle \cdot ,\cdot \rangle }}$, then for each $2 \le k \le n-1$, the image of $(-1)^k t(v_1v_k) - t(u_ku_1)$ in $R$ is zero.

Similarly, by considering $(x,y) = (v_1, v_2 + \sum {\underline{t}}(v_2v_j)v_j)$, we see that the image of $t(v_1v_n)$ in such an $R$ must be zero.

The quotient ${\widetilde{R}}$ of ${\widetilde{R}}'$ by these relations is a smooth, local ring of dimension $n-1$, and thus we identify ${\widetilde{R}}$ with

$$\begin{equation*} {\widetilde{R}} = k\lBrack s_2, \cdots , s_n\rBrack , \end{equation*}$$

where $s_j$ is the image of $t(u_ju_1)$ in ${\widetilde{R}}$. We record these calculations as follows.

Lemma 4.2.

The formal neighborhood ${\mathcal{M}}^{/z}$ of $z$ is isomorphic to

$$\begin{equation*} {\widetilde{R}} = \operatorname {Spf}k\lBrack s_2, \cdots ,s_n\rBrack . \end{equation*}$$

Over ${\widetilde{R}}$, the Dieudonné module ${\widetilde{M}} = {\widetilde{M}}_{{\widetilde{R}}}$ of the universal deformation of $(X[p^\infty ], \iota [p^\infty ], \lambda [p^\infty ])$ is displayed by

$$\begin{align*} {\widetilde{F}} u_1 &= -v_n & v_1 &= {\widetilde{V}}(u_n - {\underline{s}}_nu_1)\\ {\widetilde{F}}v_2 &= u_1 & u_2 &= {\widetilde{V}}(v_1 +\sum _{2 \le j \le n} (-1)^j {\underline{s}}_jv_j)\\ {\widetilde{F}} v_3 &= u_2 + {\underline{s}}_2u_1& u_3 &= {\widetilde{V}}v_2 \\ \vdots &&\vdots \\ {\widetilde{F}} v_n &= u_{n-1}+ {\underline{s}}_{n-1}u_1 &u_n &= {\widetilde{V}} v_{n-1}. \end{align*}$$

4.1.2. Newton strata in local coordinates

In this choice of coordinates, it is easy to calculate the Newton stratification on ${\mathcal{M}}^{/z}$. For $1 \le j \le {\lfloor n/2 \rfloor }$, let

$$\begin{equation*} {\mathcal{M}}^{/z}_{<j} \coloneq {\mathcal{M}}^{/z} \cap ({\mathcal{M}}^\sigma \cup \bigcup _{1 \le i < j} {\mathcal{M}}^{\xi _{2i}}) \end{equation*}$$

be the locus in ${\mathcal{M}}^{/z}$ parametrizing those deformations whose first slope is strictly larger than $\lambda (2j)$.

Lemma 4.3.

Suppose $1 \le j \le {\lfloor n/2 \rfloor }$. Then

$$\begin{equation*} {\mathcal{M}}^{/z}_{<j} = \operatorname {Spf}\frac{k\lBrack s_2, \cdots , s_n\rBrack }{(s_{2j}, s_{2(j+1)}, \cdots , s_{2{\lfloor n/2 \rfloor }})} \subset {\mathcal{M}}^{/z} = \operatorname {Spf}k\lBrack s_2, \cdots , s_n\rBrack . \end{equation*}$$

Before proceeding with the proof, we construct a graph to encode part of the structure of (a deformation of) $M$. Initially, construct a graph $\Gamma$ as follows (see Figure 1). With a slight abuse of notation, let the vertex set be $\{u_1, \cdots , u_n, v_1, \cdots , v_n\}$. For $2 \le i \le n$, draw a (light) gray arrow from $v_i$ to $u_{i-1}$, to encode the fact that $Fv_i = u_{i-1}$. Similarly, draw a gray arrow from $u_1$ to $v_n$.

Also, for each $2 \le i \le n$, draw a black arrow from $u_i$ to $v_{i-1}$, to encode the fact that $Fu_i = pv_{i-1}$. Similarly, draw a black arrow from $v_1$ to $u_n$.

Note that $\Gamma$ is a (colored) cycle. In fact, starting from vertex $u_1$, one successively visits

$$\begin{equation*} \{u_1,v_n,u_{n-1}, v_{n-2},u_{n-3}, \cdots , v_1, u_n, v_{n-1}, u_{n-2}, \cdots , v_2, u_1\}. \end{equation*}$$

Now let $S$ be an integral domain equipped with a surjection $\phi :k\lBrack s_2, \cdots , s_n\rBrack \rightarrow S$, and let $K$ be the field of fractions of $S$. Construct a graph $\Gamma _S$ by (possibly) augmenting the edge set of $\Gamma$, as follows.

For each $2 \le i \le n-1$, if $\phi (s_i)\not = 0$, then add a gray edge from $v_j$ to $u_1$. (For the sake of completeness, if $\phi (s_n)\not =0$, then add a black edge from $v_1$ to $u_1$. For each $2 \le i \le n-1$, if $\phi (s_i)\not = 0$, then add a black edge from $u_2$ to $v_i$. These additional black edges will not affect the final calculation.)

Let $C$ be a cycle or path in $\Gamma _S$. The length of $C$ is the number of edges in $C$, while the weight of $C$ is the number of black edges in $C$. Define the slope of $C$ to be

$$\begin{equation*} \lambda (C) = \frac{\mathrm{weight}(C)}{\mathrm{length}(C)}. \end{equation*}$$

Note that for the trivial deformation, corresponding to $\Gamma$ itself, we have $\lambda (\Gamma ) = \frac{n}{2n} = \frac{1}{2}$.

Lemma 4.4.

If $C \subset \Gamma _S$ is a cycle through $u_1$, then the smallest slope of the Newton polygon is at most $\lambda (C)$.

Proof.

It is harmless, and convenient, to replace $K$ by its perfection. Suppose there is a cycle $C$ of length $b$ and weight $a$; let ${\widetilde{N}}_K = W(K)\{u_2, \cdots , u_n, v_1, \cdots , v_n\}$. Then $F^b u_1 \in p^aW(K)\{u_1\} + {\widetilde{N}}_K$ but $F^bu_1 \not \in {\widetilde{N}}_K$, and ${\widetilde{M}}_K/{\widetilde{N}}_K$ is an $F$-$\sigma ^a$-crystal of slope at most $a/b$. Therefore, the smallest slope of ${\widetilde{M}}_K$ is at most $a/b$.

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Remark 4.5.

Let $B(K) = \operatorname {Frac}W(K)$; then the $B(K)[F]$-span of $u_1$ in ${\widetilde{M}}_K \otimes B(K)$ is all of ${\widetilde{M}}_K\otimes B(K)$. Therefore, one can in fact show that the smallest slope of ${\widetilde{M}}_K$ is

$$\begin{equation*} \min _{C \subset \Gamma \text{ a cycle through }u_1} \lambda (C). \end{equation*}$$

Lemma 4.6.

If $\phi (s_{2i})\not = 0$, then there is a cycle in $\Gamma _S$ of length $n+1-2j$ and weight $\frac{n-1}{2}-j$.

Proof.

In $\Gamma$, the unique path $P$ from $u_1$ to $v_{2j+1}$ has length $n+1-(2j+1)=n-2j$ and weight $\frac{n-(2j+1)}{2} = \frac{n-1}{2}-j$. If $\phi (s_{2j})\not = 0$, then in $\Gamma _S$ there is a cycle, obtained by concatenating $u_1$ to $P$, of length $\mathrm{length}(P)+1$ and weight $\mathrm{weight}(P)$.

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Lemma 4.7.

If the smallest slope of ${\widetilde{M}}_K$ is greater than $\lambda (2j)$, then

$$\begin{equation*} \phi (s_{2j}) = \phi (s_{2(j+1)}) = \cdots = \phi (2{\lfloor n/2 \rfloor }) =0. \end{equation*}$$

Proof.

The contrapositive follows immediately from Lemmas 4.6 and 4.4. If there is some $i \ge j$ with $\phi (s_{2i})\not = 0$, then the smallest slope of ${\widetilde{M}}_K$ is at most $\lambda (2i)$.

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Proof of Lemma 4.3.

By Lemma 4.7, the sought-for neighborhood ${\mathcal{N}}^{/z}_{<j}$ is the formal spectrum of a quotient of $R_{<j} \coloneq k\lBrack s_1, \cdots , s_n\rBrack /(s_{2j}, s_{2(j+1)}, \cdots , s_{2{\lfloor n/2 \rfloor }})$. We thus have ${\mathcal{N}}^{/z}_{<j} \hookrightarrow \operatorname {Spf}R_{<j} \hookrightarrow {\mathcal{M}}^{/z}$. Since both ${\mathcal{M}}^{/z}_{<j}$ and $\operatorname {Spf}R_{<j}$ have codimension ${\lfloor n/2 \rfloor }-j+1$, the result follows.

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4.2. The case of $n$ even

We now indicate the changes which must be made in order to perform the calculations of Section 4.1 in the case where $n$ is even.

Suppose $z= (X,\iota ,\lambda ) \in {\mathcal{M}}^{\sigma \circ }(k)$. The quasipolarized Dieudonné module $M$ of $X[p^\infty ]$, as a $p$-divisible group with ${\mathcal{O}}_L$-action, is $M(n-1)\oplus N$ (Theorem 2.1). A calculation exactly like the one in Section 4.1.1 shows that ${\mathcal{M}}^{/z} \cong \operatorname {Spf}{\widetilde{R}} = \operatorname {Spf}k\lBrack s_0, s_2, \cdots , s_{n-1}\rBrack$; the corresponding deformation ${\widetilde{M}}$ of $M$ is displayed by

$$\begin{align*} {\widetilde{F}} v_0 &= -u_0-{\underline{s}}_0u_1 & u_0 &= {\widetilde{V}} v_0 \\ {\widetilde{F}} u_1 &= -v_{n-1} & v_1 &= {\widetilde{V}}(u_{n-1} - {\underline{s}}_{n-1}u_1)\\ {\widetilde{F}}v_2 &= u_1 & u_2 &= {\widetilde{V}}(v_1 +\sum _{2 \le j \le n-1} (-1)^j {\underline{s}}_jv_j)\\ {\widetilde{F}} v_3 &= u_2 + {\underline{s}}_2u_1& u_3 &= {\widetilde{V}}v_2 \\ \vdots &&\vdots \\ {\widetilde{F}} v_{n-1} &= u_{n-2}+ {\underline{s}}_{n-2}u_1 &u_{n-1} &= {\widetilde{V}} v_{n-2}. \end{align*}$$

Construction and analysis of graphs $\Gamma$ and $\Gamma _S$, for quotients $S$ of ${\widetilde{R}}$, shows that Lemma 4.3 holds for even $n$ too:

Lemma 4.8.

Suppose $1 \le j \le n/2$. Then

$$\begin{equation*} {\mathcal{M}}^{/z}\cap (\bigcup _{1 \le i \le j} {\mathcal{M}}^{\xi _{2i}}) = \operatorname {Spf}\frac{k\lBrack s_0, s_2, s_3, \cdots , s_n\rBrack }{(s_{2j},s_{2(j+1)}, \cdots , s_{n})}. \end{equation*}$$

5. Hecke orbits for the supersingular locus

Lemma 5.1.

Let ${\mathbb{K}}^p\subset G({\mathbb{A}}^p_f)$ be a compact open subgroup. The $\operatorname {U}({\mathbb{A}}^p_f)$-Hecke operators act transitively on $\Pi _0({\mathcal{M}}_{{\mathbb{K}}^p}^\sigma )$.

Proof.

Let $(X,\iota ,\lambda ) = {\overline{\eta }}$ be a geometric generic point of ${\mathcal{M}}_{{\mathbb{K}}^p}^\sigma$. The central leaf ${\mathcal{C}}([{\overline{\eta }}])$, which in a general PEL Shimura variety context parametrizes those $(Y,\jmath ,\mu )$ with $(Y[p^\infty ], \jmath [p^\infty ],\mu [p^\infty ]) \cong (X[p^\infty ],\iota [p^\infty ],\lambda [p^\infty ])$, in this case coincides with (the union of a choice of geometric point over each generic point of) ${\mathcal{M}}^{\sigma \circ }$. There is an a priori inclusion ${\mathcal{H}}^p({\overline{\eta }}) \subseteq {\mathcal{C}}([{\overline{\eta }}])$. Since ${\overline{\eta }}$ is basic and $\operatorname {G}$, the reductive group defining ${\mathcal{M}}$, has a simply connected derived group, the prime-to-$p$ Hecke orbit of ${\overline{\eta }}$ coincides with the central leaf ${\mathcal{C}}([X[p^\infty ], \iota [p^\infty ], \lambda [p^\infty ]])$ 12, Theorem 4.6(1) and Remark 4.7(3).

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6. Irreducibility of Newton strata

Proof of Theorem 1.1.

Chai and Oort identify nine steps in their proof of 4, Theorem 3.1, which is the analogue for ${\mathcal{A}}_g$ of Theorem 1.1. We proceed here in a similar fashion. Fix an open compact subgroup ${\mathbb{K}}^p \subset \operatorname {G}({\mathbb{A}}^p_f)$; it suffices to prove that ${\mathcal{M}}_{{\mathbb{K}}^p}^\xi$is irreducible.

Steps 1-6

By Lemma 3.3, there is a well-defined map of sets$$\begin{equation*} \vcenter{\img[][102pt][12pt][{$\xymatrix{ \Pi_0(\stk M^\sigma) \ar[r]& \Pi_0(\stk M^\xi). }$}]{Images/imgfe3b0325e8d9c7c3e1961b7668ade5f5.svg}} \end{equation*}$$

It is surjective by Lemma 3.2. From it, we deduce the existence of a surjective$$\begin{equation*} \vcenter{\img[][110pt][11pt][{$\xymatrix{ \Pi_0(\stk M_{\level^p}^\sigma) \ar[r] & \Pi_0(\stk M_{\level^p}), }$}]{Images/img6fb8849e7e37805f81fa0a109ff10edf.svg}} \end{equation*}$$

visibly $\operatorname {U}({\mathbb{A}}^p_f)$-equivariant.

Steps 7-8

By Lemma 5.1, the action of $\operatorname {U}({\mathbb{A}}^p_f)$ on $\Pi _0({\mathcal{M}}_{{\mathbb{K}}^p}^\sigma )$ is transitive.

Step 9

Taken together, this shows that $\operatorname {U}({\mathbb{A}}^p_f)$ acts transitively on $\Pi _0({\mathcal{M}}^\zeta _{{\mathbb{K}}^p})$. By 6, Theorem 1.4, which is the PEL analogue of 3, ${\mathcal{M}}_{{\mathbb{K}}^p}^\xi$ is connected. Since ${\mathcal{M}}_{{\mathbb{K}}^p}^\xi$ is also smooth (Lemma 3.1), it is irreducible.■

Figure 1.

The graph $\Gamma$ for $n=7$.