2.1.1 Let $G \rightarrow G'$ be a homomorphism, and $S$ a finite set with an action of $G$. We say that the action of $G$ on $S$ lifts to $G'$ if there is an action of $G'$ on $S$, which induces the given action of $G$ on $S$. We call such a $G'$-action a lifting of the $G$-action on $S$. We say that the action of $G$ on $S$ virtually lifts to $G'$ if there is a finite index subgroup $G'' \subset G',$ containing the image of $G,$ such that the action of $G$ on $S$ lifts to $G''$.

For a positive integer $n$ we denote by $S^n$ the $n$-fold product equipped with the diagonal action of $G,$ and by $\pi _i: S^n \rightarrow S$, $i=1,\ldots , n$, the projections.

Proof. Let $N \subset G'$ be the kernel of $G' \rightarrow {\rm Aut}(S),$ and $G'' \subset G'$ the subgroup generated by $N$ and the image of $G$. Since $S$ is finite, $N$ and hence $G''$ has finite index in $G^{\prime}$, and so $G^{\prime\prime}$ satisfies the conditions in (i).

For (ii), suppose that the action of $G$ on $S$ virtually lifts to $G'$. By (i), after replacing $G'$ by a finite index subgroup containing the image of $G,$ we may assume that the action of $G$ on $S$ lifts to $G',$ and that $G$ and $G'$ have the same image in ${\rm Aut}(S)$. Then $T$ is $G'$-stable, so the action of $G$ on $T$ lifts to $G'$. Conversely, if the action of $G$ on $T$ virtually lifts to $G',$ then, by (i), we may assume that the action of $G$ on $T$ lifts to $G',$ and that $G$ and $G'$ have the same image in ${\rm Aut}(T)$. In particular, any element of $G'$ acts on ${\rm Aut}(T)$ via an element of ${\rm Aut}(S)$. Since $\bigcup _{i=1}^n \pi _i(T) = S$ this element of ${\rm Aut}(S)$ is uniquely determined. The uniqueness implies that $G' \rightarrow {\rm Aut}(T)$ factors through a homomorphism $G' \rightarrow {\rm Aut}(S),$ which lifts the action of $G$ on $S$.

2.1.3 Let $X$ be a separated, locally Noetherian scheme and $f:Y \rightarrow X$ a finite étale cover. We will say that $f$ is Galois if any connected component of $Y$ is Galois over its image.

Suppose that $X$ is connected, and let $\bar x$ be a geometric point of $X$. Let $F_{\bar x}$ denote the functor which assigns to any finite étale cover $Y \rightarrow X$ the underlying topological space $|Y_{\bar x}|$ of its fiber over $\bar x$. Then ${\rm Aut}(F_{\bar x})$ is called the étale fundamental group and is denoted by $\pi _1(X,\bar x),$ [Reference GrothendieckSGA1, Exp. V, § 7]. It follows from [Reference GrothendieckSGA1, Exp. V, § 7] that the association $Y \mapsto S(Y) : = |Y_{\bar x}|$ induces a bijection between isomorphism classes of finite étale covers, and of finite sets with $\pi _1(X,\bar x)$-action. If $X$ is connected and $S$ is a finite set with an action of $\pi _1(X,\bar x),$ we denote by $Y(S)$ the corresponding finite étale cover of $X$.

In the next three lemmas we consider a map of separated, locally Noetherian schemes $g: X \rightarrow X'$ and a finite étale cover $f': Y' \rightarrow X'$ equipped with an isomorphism $Y \overset \sim \longrightarrow Y'\times _{X'}X$. If $X$ and $X'$ are connected, and $\bar x$ again denotes the geometric point of $X'$ induced by $\bar x,$ then such a $Y'$ exists if and only if the action of $\pi _1(X,\bar x)$ on $S(Y)$ lifts to $\pi _1(X',\bar x)$.

Proof. We may assume that $X$ and $X'$ are connected. We apply Lemma 2.1.2(i), to $\pi _1(X,\bar x) \rightarrow \pi _1(X',\bar x)$ and $S = S(Y) = S(Y')$. Let $h: X'' \rightarrow X'$ be the finite étale map corresponding to the $\pi _1(X',\bar x)$-set $\pi _1(X',\bar x)/G''$. Since this set has a $\pi _1(X,\bar x)$ fixed point, $g$ factors through $X''$. By construction, the images of $\pi _1(X'',\bar x)$ and $\pi _1(X,\bar x)$ in ${\rm Aut}(S(Y')) = {\rm Aut}(S(Y))$ are equal. Denote this image by $H,$ and for $s \in S(Y)$ denote the stabilizer by $H_s$. Since $Y\rightarrow X$ is Galois, $H_s$ is a normal subgroup which does not depend on $s,$ which implies that $Y'' \rightarrow X''$ is Galois.

2.1.5 Let $A$ be a finite ring. By an $A$-local system ${\mathcal {F}}$ on $X$ we will mean an étale sheaf of $A$-modules which is locally isomorphic to the constant $A$-module $A^n$ for some $n$. Such an ${\mathcal {F}}$ is representable by a finite étale map $Y({\mathcal {F}}) \rightarrow X$. This is clear étale locally on $X,$ and follows from étale descent in general. If $X$ is connected, and equipped with a geometric point $\bar x,$ then the association ${\mathcal {F}} \mapsto |Y({\mathcal {F}})|_{\bar x}$ induces a bijection between isomorphism classes of $A$-local systems and conjugacy classes of representations $\pi _1(X,\bar x) \rightarrow {\rm GL}_n(A)$. Indeed, this follows easily from the bijection explained in 2.1.3.

For $X \rightarrow X_1$ a map of schemes, and $\mathcal {G}$ an $A$-local system on $X_1,$ we will denote by $\mathcal {G}|_X$ the pullback of $\mathcal {G}$ to $X$.

Proof. We may assume that $X$ and $X'$ are connected and choose $X''$ as in the proof of Lemma 2.1.4, using the construction in Lemma 2.1.2(i). Then the finite set $S(Y) = S(Y')$ naturally has the structure of a finite free $A$-module on which $\pi _1(X,\bar x)$ acts $A$-linearly. Since the images of $\pi _1(X'',\bar x)$ and $\pi _1(X,\bar x)$ in ${\rm Aut}(S(Y))$ are equal, $\pi _1(X'',\bar x)$ acts on $S(Y)$ $A$-linearly, which implies the statement of the lemma.

2.1.7 We continue to assume that $X$ is separated and locally Noetherian.

For any integer $N \geq 1$ we denote by $\mu _N$ the kernel of ${\mathbb {G}}_m \overset N \rightarrow {\mathbb {G}}_m$. This is a finite flat group scheme over ${\mathbb {Z}},$ and we denote by the same symbol its pullback to any scheme $X$. This pullback is étale if any only if $X$ is a ${\mathbb {Z}}[1/N]$-scheme.

Suppose $X$ is a ${\mathbb {Z}}[1/N]$-scheme. For any ${\mathbb {Z}}/N{\mathbb {Z}}$ algebra $A,$ we again denote by $\mu _N$ or $A(1)$ the étale sheaf $\mu _N\otimes _{{\mathbb {Z}}/N{\mathbb {Z}}}A$. For any nonnegative integer $i$ we write $A(i) = A(1)^{\otimes i}$. For $i$ negative we set $A(i)$ equal to the $A$-linear dual of $A(-i)$.

Proof. Replacing ${\mathcal {F}}$ by ${\mathcal {F}}(-j) : = {\mathcal {F}}\otimes _AA(-j)$, we may suppose that $j=0$. By Lemma 2.1.6, we may assume $Y''$ represents an $A$-local system and $g^*({\mathcal {F}}') \overset \sim \longrightarrow {\mathcal {F}}$ as $A$-local systems. Let ${\mathcal {F}}_1 \subset {\mathcal {F}}$ denote the sub $A$-local system corresponding to $A(j)^s,$ so that ${\mathcal {F}}_1$ corresponds to an $A$-submodule $S(Y')_1\subset S(Y')$.

Let $X''$ and $H$ be as in the proof of Lemma 2.1.4. Since the group $H$ acts trivially on $S(Y')_1,$ after replacing $X'$ by $X'',$ we may assume that ${\mathcal {F}}'$ is an extension of ${\mathcal {F}}'/A(i)^r$ by $A(j)^s,$ and $g^*({\mathcal {F}}') \overset \sim \longrightarrow {\mathcal {F}}$ is an isomorphism of extensions. A similar argument, applied to ${\mathcal {F}}/A(j)^s\otimes A(-i),$ shows that we may assume that ${\mathcal {F}}'$ is an extension of $A(i)^r$ by $A(j)^s,$ with $g^*({\mathcal {F}}') \overset \sim \longrightarrow {\mathcal {F}}$ as extensions.

2.2.1 Let $K$ be a field, $X$ a $K$-scheme of finite type, and $f: Y \rightarrow X$ a finite étale cover. The essential dimension [Reference Buhler and ReichsteinBR97, § 2] ${\rm ed}_K(Y/X)$ of $Y$ over $X$ is the smallest integer $e$ such that there exists a finite type $K$-scheme $W$ of dimension $e,$ a dense open subscheme $U \subset X,$ and a map $U \rightarrow W,$ such that $Y|_U$ is the pullback of a finite étale covering over $W$. The essential $p$-dimension ${\rm ed}_K(Y/X;p)$ is defined as the minimum of ${\rm ed}_K(Y\times _XE/E)$ where $E \rightarrow X$ runs over dominant, generically finite maps, which have degree prime to $p$ at all generic points of $X$.

Note that when $X$ is irreducible, the definition does not change if we consider only coverings with $E$ irreducible. Indeed, for any $E \rightarrow X,$ as above, one of the irreducible components of $E$ will have degree prime to $p$ over the generic point of $X$.

Suppose $X$ is connected. A Galois closure of $Y \rightarrow X$ is a union of Galois closures of the connected components of $Y$. If $Y_i \rightarrow X$, $i=1, \ldots r$, are finite étale, Galois and connected, then a composite of the $Y_i$ is a connected component of $Y_1\times _X\cdots \times _X Y_r$. Up to isomorphism, this does not depend on the choice of connected component. If we drop the assumption that $X$ is connected, we define the Galois closure and composite by making these constructions over each connected component of $X$.

Proof. We may assume that $X$ is connected. Let $\bar x$ be a geometric point for $X,$ and $S$ a $\pi _1(X,\bar x)$-set with $Y = Y(S)$. Suppose first that $Y$ is connected. Let $N \subset \pi _1(X,\bar x)$ be the (normal) subgroup which fixes $S$ pointwise. If $s \in S,$ and $\pi _1(X,\bar x)_s$ is the stabilizer of $s,$ then $N = \bigcap _{i=1}^n g_i\pi _1(X,\bar x)_sg_i^{-1}$ for a finite collection of elements $g_1,\ldots , g_n \in \pi _1(X,\bar x)$. Let $\tilde S$ denote the $\pi _1(X,\bar x)$-orbit of $(g_1\cdot s, \ldots , g_n\cdot s) \in S^n$. Then the stabilizer in $\pi _1(X,\bar x)$ of any point of $\tilde S$ is $N,$ so $\tilde Y = Y(\tilde S)$ is a Galois closure of $Y$.

Now we drop the assumption that $Y$ is connected, and let $Y_1, \ldots , Y_r$ denote the connected components of $Y,$ with $Y_i$ corresponding to a subset $S_i \subset S$ on which $\pi _1(X,\bar x)$ acts transitively. The above construction gives subsets $\tilde S_i \subset S_i^n,$ (we may assume without loss of generality that $n$ does not depend on $i$) with $\tilde Y_i = Y(\tilde S_i)$ a Galois closure of $Y_i$. If $\tilde S = \coprod _i \tilde S_i \subset S^n,$ then $\tilde Y = Y(\tilde S) = \coprod _i \tilde Y_i$ is a Galois closure for $Y$.

From the construction one sees that $S,$ $T = \tilde S$ and $G = \pi _1(X,\bar x)$ satisfy the conditions in Lemma 2.1.2(ii) (note that these conditions do not depend on the choice of $G'$). For any homomorphism $H \rightarrow G,$ these conditions continue to hold if we instead view $S$ and $T$ as $H$-sets.

Now let $E \rightarrow X$ be a map of $K$-schemes, $U \subset E$ a connected open subset, and $U \rightarrow E'$ a map of $K$-schemes. Suppose $U$ admits a geometric point $\bar y$ mapping to $\bar x$. Applying the above remark with $H = \pi _1(U,\bar y),$ Lemma 2.1.2(ii) implies that the action of $H$ on $S$ virtually lifts to $\pi _1(E',\bar y)$ if and only the action of $H$ on $T$ virtually lifts to $\pi _1(E',\bar y)$. As in the proof of Lemma 2.1.4, this implies that $Y|_U$ arises by pullback from a cover of $E''$ for some finite étale $E'' \rightarrow E',$ if and only if the same condition holds for $\tilde Y|_U$. Since $\bar x$ was an arbitrary geometric point of $X,$ this implies ${\rm ed}_K(Y|_E/E) = {\rm ed}_K(\tilde Y|_E/E)$.

Proof. Let $\bar x$ be a geometric point for $X,$ and let $S_i$ be a $\pi _1(X,\bar x)$-set with $Y_i = Y(S_i)$. Let $S = \coprod _i S_i$. Then $Y$ corresponds to a transitive $\pi _1(X,\bar x)$ set $T \subset S_1\times \dots \times S_r$ which surjects onto each $S_i$. Viewing $T \subset S^r,$ we see that $T$ satisfies the conditions of Lemma 2.1.2. The lemma now follows as in the proof of Lemma 2.2.2.

Proof. Consider the action of $\pi _1(X,\bar x)$ on $A^n$ corresponding to ${\mathcal {F}}$. The connected components of $Y({\mathcal {F}})$ correspond to the stabilizers $\pi _1(X)_s$ for $s \in A^n$. A Galois closure of such a component corresponds to

\[ \bigcap_{g \in \pi_1(X)} g\pi_1(X)_sg^{{-}1} = \bigcap_{g \in \pi_1(X)} \pi_1(X)_{gs}. \]

Thus the composite of such Galois closures corresponds to $\bigcap _s \pi _1(X)_s = {\rm ker}\, \rho _{{\mathcal {F}}}$.

The lemma now follows from Lemmas 2.2.2 and 2.2.3.

Proof. Let $U_K \subset X_K$ be a dense open and $U_K \rightarrow W_K$ a morphism with ${\rm dim}\,W_K = {\rm ed}_K(Y_K/X_K)$ such that $Y_K|_{U_K}$ arises from a finite cover $f':Y'_K \rightarrow W_K$. Then $U_K,$ the finite cover $f'$ and the isomorphism $f^{\prime *} Y'_K \overset \sim \longrightarrow Y_K|_{U_K}$ are all defined over some finitely generated $K'$-algebra $R \subset K$. Specializing by a map $R \rightarrow K'$ produces the required data for the covering $Y \rightarrow X$.

2.2.6 It will be convenient to make the following definition. Suppose that $Y \rightarrow X$ is a finite étale covering of finite type $R$-schemes, where $R$ is a domain of characteristic $0$. Set $\overline {{\rm ed}}(Y/X) = {\rm ed}_{\bar K}(Y/X)$ where $\bar K$ is any algebraically closed field containing $R,$ and similarly for $\overline {{\rm ed}}(Y/X; p)$. By Lemma 2.2.5, this does not depend on the choice of $\bar K$.

Proof. To show $\overline {{\rm ed}}(Y/X; p) \geq \overline {{\rm ed}}(Y/H\to X; p),$ after shrinking $X,$ we may assume there is a map $f:X \rightarrow X'$ such that $Y = f^*Y'$ for a finite étale covering $Y' \rightarrow X',$ which may be assumed to be connected and Galois by Lemma 2.1.4. The Galois group of $Y'/X'$ is necessarily equal to $G,$ and we have $f^*(Y'/H) \overset \sim \longrightarrow Y/H$.

For the converse inequality when $p \nmid n,$ we may assume there is a map $f:X \rightarrow X'$ such that $Y/H = f^*Y'$ for a finite étale covering $Y' \rightarrow X',$ which we may again assume is connected and Galois with group $G/H$. The image, $c,$ of $Y' \rightarrow X'$ under

\[ H^1(X', G/H) \rightarrow H^2(X', H) \overset \sim \longrightarrow H^2(X', \mu_n) \]

is the obstruction to lifting $Y' \rightarrow X'$ to a $G$-covering. Here, for the final isomorphism, we are using that $K$ is algebraically closed of characteristic prime to $n$. Viewing $c$ as a Brauer class, we see that it has order dividing $n$. This implies that (after perhaps shrinking $X$ further) there is an étale covering $X_1' \rightarrow X'$ of order dividing a power $n$ such that $c|_{X_1}$ is trivial [Reference Farb and DennisFD93, Lemma 4.17]. In particular, $X_1' \rightarrow X'$ has order prime to $p$. Replacing $Y \rightarrow X \rightarrow X'$ by their pullbacks to $X_1',$ we may assume that $c = 0,$ and that $Y' = Y''/H$ for some Galois covering $Y'' \rightarrow X'$ with group $G$.

The difference between the $G$-coverings $Y \rightarrow X$ and $f^*Y'' \rightarrow X$ is measured by a class in $H^1(X,H)$. After replacing $X$ by the $H$-covering corresponding to this class, we may assume that this class is trivial, and so $Y \overset \sim \longrightarrow f^*Y''$. This shows that $\overline {{\rm ed}}(Y/X; p) \leq \overline {{\rm ed}}(Y/H\to X; p)$.

3.1.1 Now let $\tilde X = {\operatorname {Spec\,}} A$ be an affine ${\mathbb {Z}}_p$-scheme, and set $X = \tilde X\otimes {\mathbb {Q}}$. Let $n \geq 1,$ and consider the exact sequence of sheaves

\[ 1 \rightarrow \mu_{p^n} \rightarrow {\mathbb{G}}_m \overset{p^n}\rightarrow {\mathbb{G}}_m \rightarrow 1 \]

in the flat topology of $\tilde X$. Taking flat cohomology of this sequence and its restriction to $X$, we obtain the following commutative diagram with exact rows.

The group $H^1(\tilde X, {\mathbb {G}}_m)$ classifies line bundles on $\tilde X$. Hence, if $A$ is local it vanishes, and this can be used to classify extensions of ${\mathbb {Z}}/p^n{\mathbb {Z}}$ by $\mu _{p^n}$ as finite flat group schemes. We have

\[ {\rm Ext}^1_{\tilde X}({\mathbb{Z}}/p^n{\mathbb{Z}}, \mu_{p^n}) \overset \sim \longrightarrow H^1(\tilde X, \mu_{p^n}) \overset \sim \longrightarrow A^\times{/}(A^\times)^{p^n}. \]

Here and below, the group on the left denotes extensions as sheaves of ${\mathbb {Z}}/p^n{\mathbb {Z}}$-modules.

Similarly, we can classify extensions of ${\mathbb {Q}}_p/{\mathbb {Z}}_p$ by $\mu _{p^\infty } = \lim _n \mu _{p^n}$ as $p$-divisible groups. If $\mathcal {E}$ is such an extension, then $\mathcal {E}[p^n]$ is an extension of ${\mathbb {Z}}/p^n{\mathbb {Z}}$ by $\mu _{p^n}$ and we have

\[ \hat \theta_A: {\rm Ext}^1_{\tilde X}({\mathbb{Q}}_p/{\mathbb{Z}}_p,\mu_{p^\infty}) \overset \sim \longrightarrow \underleftarrow{{\rm lim}}_n A^\times{/}(A^\times)^{p^n}. \]

If $A$ is complete and local with residue field $k,$ then the right-hand side may be identified with $A^{\times ,1} \subset A^\times ,$ the subgroup of units which map to $1$ in $k^\times$. Thus we have

\[ \hat \theta_A: {\rm Ext}^1_{\tilde X}({\mathbb{Q}}_p/{\mathbb{Z}}_p,\mu_{p^\infty}) \overset \sim \longrightarrow A^{{\times},1}. \]

3.1.2 For the rest of this subsection we assume that $A = V[\![ x_1, \ldots , x_n ]\!]$. Then $H^1(X, {\mathbb {G}}_m) = 0$ by [Reference GrothendieckSGA2, XI, Thm 3.13], and we have the following commutative diagram.

3.1.3 We call an element of ${\rm Ext}^1_X({\mathbb {Z}}/p^n{\mathbb {Z}}, \mu _{p^n}) = H^1(X,\mu _{p^n})$ syntomic if it arises from an element of $A^\times ,$ or equivalently from a class in ${\rm Ext}^1_{\tilde X}({\mathbb {Z}}/p^n{\mathbb {Z}}, \mu _{p^n}),$ and we denote the subgroup of syntomic elements by ${\rm Ext}^{1,{\rm syn}}_X({\mathbb {Z}}/p^n{\mathbb {Z}}, \mu _{p^n}) \subset {\rm Ext}^1_X({\mathbb {Z}}/p^n{\mathbb {Z}}, \mu _{p^n})$

Proof. If $a \in A^\times$ is a $p^{n}$th power in $A[1/p]$ then it is a $p^{n}$th power in $A,$ as $A$ is normal. Hence the map $A^\times /(A^\times )^{p^n} \rightarrow A[1/p]^\times /(A[1/p]^\times )^{p^n}$ is injective, and the lemma follows from the description of ${\rm Ext}^1$s above.

Proof. Let $b \in B[1/p]^\times$ be an element giving rise to $c$. Since $B$ is a unique factorization domain we may write $b = b_0\pi ^i$ with $b_0 \in B^\times$ and $i \in {\mathbb {Z}}$. Since $f^*(c)$ is syntomic, we may write $\pi ^i = a_0a^{p^n}$ with $a_0 \in A^\times$ and $a \in A[1/p]^\times$. Comparing the images of both sides in the group of divisors on $A,$ one sees that $p^n|i$. So $c$ arises from $b_0$.

3.1.6 Let ${\mathfrak m}_A$ be the maximal ideal of $A,$ and $\bar {\mathfrak m}_A$ its image in $A/\pi A$. The natural map

\[ \bar{\mathfrak m}_A/\bar {\mathfrak m}_A^2 \overset {a \mapsto 1+ a} \rightarrow k^\times\backslash (A/(\pi,{\mathfrak m}_A^2))^\times \]

is a bijection; both sides are $k$-vector spaces spanned by $x_1,\ldots , x_n$. We denote by $\theta _A$ the composite

\[ \theta_A: {\rm Ext}_X^{1,{\rm syn}} ({\mathbb{Z}}/p{\mathbb{Z}}, \mu_p) \overset \sim \longrightarrow A^\times{/}(A^\times)^p \rightarrow k^\times\backslash (A/(\pi,{\mathfrak m}_A^2))^\times \overset \sim \longrightarrow \bar{\mathfrak m}_A/\bar {\mathfrak m}_A^2. \]

Here we have used Lemma 3.1.4 to identify ${\rm Ext}_X^{1,{\rm syn}} ({\mathbb {Z}}/p{\mathbb {Z}}, \mu _p)$ and ${\rm Ext}_{\tilde X}^1 ({\mathbb {Z}}/p{\mathbb {Z}}, \mu _p)$.

Proof. By functoriality of the association $A \mapsto \theta _A,$ we have $\theta _A(f^*(L)) = f^*(\theta _B(L))$. Hence the $k$-span of $\theta _A(f^*(L))$ is contained in image of $\bar {\mathfrak m}_B/\bar {\mathfrak m}_B^2$. It follows that $\bar {\mathfrak m}_B/\bar {\mathfrak m}_B^2$ surjects onto $\bar {\mathfrak m}_A/\bar {\mathfrak m}_A^2$. Since $A$ and $B$ are complete local $V$-algebras, this implies that $B,$ which is a subring of $A,$ surjects onto $A$. Hence $f$ is an isomorphism.

3.2.1 Recall that an abelian scheme ${\mathcal {A}}$ over a ${\mathbb {Z}}_p$-scheme is called ordinary if the group scheme ${\mathcal {A}}[p^n]$ is ordinary for all $n \geq 1$. This is equivalent to requiring the condition for $n=1$.

Let $k$ be an algebraically closed field of characteristic $p > 0,$ and let $V$ be a complete discrete valuation ring with residue field $k,$ so that $W(k) \subset V$. Let ${\mathcal {A}}_0$ be an abelian scheme over $k$ of dimension $g$. We assume that ${\mathcal {A}}_0$ is ordinary. Since $k$ is algebraically closed, this implies that ${\mathcal {A}}_0[p^\infty ]$ is isomorphic to $({\mathbb {Q}}_p/{\mathbb {Z}}_p)^g \oplus \mu _{p^\infty }^g$.

Consider the functor $D_{{\mathcal {A}}_0}$ on the category of Artinian $V$-algebras $C$ with residue field $k,$ which attaches to $C$ the set of isomorphism classes of deformations of ${\mathcal {A}}_0$ to an abelian scheme over $C$. Recall [Reference KatzKat81, § 2] that $D_{{\mathcal {A}}_0}$ is equivalent to the functor which attaches to $C$ the set of isomorphism classes of deformations of ${\mathcal {A}}_0[p^\infty ],$ and that $D_{{\mathcal {A}}_0}$ is pro-representable by a formally smooth $V$-algebra $R$ of dimension $g^2,$ called the universal deformation $V$-algebra of ${\mathcal {A}}_0$.

Denote by ${\mathcal {A}}_R$ the universal (formal) abelian scheme over $R$. Note that although ${\mathcal {A}}_R$ is only a formal scheme over $R,$ the torsion group schemes ${\mathcal {A}}_R[p^n]$ are finite over $R,$ and so can be regarded as genuine $R$-schemes. Since ${\mathcal {A}}_0$ is ordinary the $p$-divisible group, ${\mathcal {A}}_R[p^\infty ] = \lim _n {\mathcal {A}}_R[p^n]$ is an extension of $({\mathbb {Q}}_p/{\mathbb {Z}}_p)^g$ by $\mu _{p^\infty }^g$. Hence ${\mathcal {A}}_R[p]$ is an extension of $({\mathbb {Z}}/p{\mathbb {Z}})^g$ by $\mu _p^g$. This extension class is given by a $g\times g$ matrix of classes $(c_{i,j})$ with $c_{i,j} \in {\rm Ext}_R^1({\mathbb {Z}}/p{\mathbb {Z}}, \mu _p)$.

Proof. Consider the isomorphism

\[ \hat \theta_R: {\rm Ext}_R^1({\mathbb{Q}}_p/{\mathbb{Z}}_p, \mu_{p^\infty}) \overset \sim \longrightarrow R^{{\times},1} \]

introduced in § 3.1. The universal extension of $p$-divisible groups over $R$ gives rise to a $g\times g$ matrix of elements $(\hat c_{i,j}) \in {\rm Ext}_R^1({\mathbb {Q}}_p/{\mathbb {Z}}_p, \mu _{p^\infty })$ which reduce to $(c_{i,j})$.

Let $L \subset \bar {\mathfrak m}_R/\bar {\mathfrak m}_R^2$ be the $k$-span of the images of the elements $\hat \theta _R(\hat c_{i,j})-1,$ or equivalently, the elements $\theta _R(c_{i,j})-1,$ and set $R' = k \oplus L \subset R/(\pi ,{\mathfrak m}_R^2)$. Using the isomorphism $\hat \theta _{R'},$ one sees that ${\mathcal {A}}_R[p^\infty ]|_{R/(\pi ,{\mathfrak m}_R^2)}$ is defined over $R'$. If $L \subsetneq \bar {\mathfrak m}_R/\bar {\mathfrak m}_R^2,$ then there exists a surjective map $R/(\pi ,{\mathfrak m}_R^2) \rightarrow k[x]/x^2$ which sends $L$ to zero. Specializing ${\mathcal {A}}_R[p^\infty ]|_{R/(\pi ,{\mathfrak m}_R^2)}$ by this map induces the trivial deformation of ${\mathcal {A}}_0[p^\infty ]$ (that is, the split extension of $({\mathbb {Q}}_p/{\mathbb {Z}}_p)^g$ by $\mu _{p^\infty }$) over ${\operatorname {Spec\,}} k[x]/x^2$. This contradicts the fact that $R$ pro-represents $D_{{\mathcal {A}}_0}$. Hence $L = \bar {\mathfrak m}_R/\bar {\mathfrak m}_R^2,$ which proves the lemma.

3.2.3 Let $A$ be a quotient of $R$ which is formally smooth over $V$. That is, $A$ is isomorphic as a complete $V$-algebra to $V[\![ x_1, \ldots x_n ]\!]$. As in § 3.1, we set $X = {\operatorname {Spec\,}} A[1/p]$ and $\tilde X = {\operatorname {Spec\,}} A$.

Proof. Using the notation of 3.2.1, we have that $\theta _R(\{c_{i,j}\}_{i,j})$ spans $\bar {\mathfrak m}_R/\bar {\mathfrak m}_R^2$ by Lemma 3.2.2. In particular, if we again denote by $c_{i,j}$ the restrictions of these classes to $A,$ then $\theta _A(\{c_{i,j}\}_{i,j})$ spans $\bar {\mathfrak m}_A/\bar {\mathfrak m}_A^2$.

Now by Lemma 3.1.5 the $g^2$ extension classes defining $\mathcal {L}$ are syntomic. So $\mathcal {L}$ arises from an extension of $({\mathbb {Z}}/p{\mathbb {Z}})^g$ by $\mu _p^g$ as finite flat group schemes over $\tilde Y$. If we denote by $(d_{i,j})$ the corresponding $g\times g$ matrix of elements of ${\rm Ext}_{\tilde Y}^1({\mathbb {Z}}/p{\mathbb {Z}}, \mu _p),$ then Lemma 3.1.4, together with the fact that $f^*\mathcal {L} \overset \sim \longrightarrow {\mathcal {A}}_R[p]|_X$, implies that $f^*(d_{i,j}) = c_{i,j}$. It follows that the elements $\theta _A(f^*(\{d_{i,j}\}))_{i,j}$ span $\bar {\mathfrak m}_A/\bar {\mathfrak m}_A^2,$ which implies that $f$ is an isomorphism by Lemma 3.1.7.

3.2.5 Fix an integer $g \geq 1$, a prime $p\geq 2$, and a positive integer $N \geq 2$ coprime to $p$. Consider the ring ${\mathbb {Z}}[\zeta _N][1/N],$ where $\zeta _N$ is a primitive $N$th root of $1$. Using the isomorphism ${\mathbb {Z}}/N{\mathbb {Z}} \underset {1 \mapsto \zeta _N}{\overset \sim \longrightarrow }\mu _N,$ for any ${\mathbb {Z}}[\zeta _N][1/N]$-scheme $T,$ and any principally polarized abelian scheme $A$ over $T,$ the $N$-torsion scheme $A[N]$ is equipped with the (alternating) Weil pairing

\[ A[N] \times A[N] \rightarrow {\mathbb{Z}}/N{\mathbb{Z}}. \]

We denote by ${\mathcal {A}}_{g,N}$ the ${\mathbb {Z}}[\zeta _N][1/N]$-scheme which is the coarse moduli space of principally polarized abelian schemes $A$ of dimension $g$ equipped with a symplectic basis of $A[N]$. When $N \geq 3,$ this is a fine moduli space which is smooth over ${\mathbb {Z}}[\zeta _N][1/N]$. For a ${\mathbb {Z}}[\zeta _N][1/N]$-algebra $B,$ we denote by ${\mathcal {A}}_{g,N/B}$ the base change of ${\mathcal {A}}_{g,N}$ to $B$. If no confusion is likely to result we sometimes denote this base change simply by ${\mathcal {A}}_{g,N}$.

Suppose that $N \geq 3,$ and let ${\mathcal {A}} \rightarrow {\mathcal {A}}_{g,N}$ be the universal abelian scheme. The $p$-torsion subgroup ${\mathcal {A}}[p] \subset {\mathcal {A}}$ is a finite flat group scheme over ${\mathcal {A}}_{g,N}$ which is étale over ${\mathbb {Z}}[\zeta _N][1/Np]$. Let $x \in {\mathcal {A}}_{g,N}$ be a point with residue field $\kappa (x)$ of characteristic $p,$ and ${\mathcal {A}}_x$ the corresponding abelian variety over $\kappa (x)$. The set of points $x$ such that ${\mathcal {A}}_x$ is ordinary is an open subscheme ${\mathcal {A}}_{g,N}^{\rm ord} \subset {\mathcal {A}}_{g,N}\otimes {\mathbb {F}}_p$. For any $N,$ we denote by ${\mathcal {A}}_{g,N}^{\rm ord} \subset {\mathcal {A}}_{g,N}\otimes {\mathbb {F}}_p$ the image of ${\mathcal {A}}_{g,NN'}$ for any $N' \geq 3$ coprime to $N$ and $p$.

We now denote by $k$ a perfect field of characteristic $p,$ and $K/W[1/p]$ a finite extension with ring of integers $\mathcal {O}_K$ and uniformizer $\pi$. We assume that $K$ is equipped with a choice of primitive $N$th root of $1,$ $\zeta _N \in K$. We remind the reader regarding the convention for the definition of $\overline {{\rm ed}}$ and $\overline {{\rm ed}}(\,\cdot \, ; p)$ introduced in 2.2.6.

Proof. It suffices to prove the theorem when $k$ is algebraically closed, which we assume from now on. Moreover, since $K$ is an arbitrary finite extension of $W[1/p],$ it is enough to show that ${\rm ed}_K({\mathcal {A}}[p]|_{{\mathcal {Z}}_K}/{\mathcal {Z}}_K;p) = {\rm dim}\,{\mathcal {Z}}_K.$ We may replace ${\mathcal {Z}}$ by a component whose special fiber meets the ordinary locus, and assume that ${\mathcal {Z}}_K$ and ${\mathcal {Z}}_k$ are geometrically connected.

Suppose that $\overline {{\rm ed}}({\mathcal {A}}[p]|_{{\mathcal {Z}}_K}/{\mathcal {Z}}_K; p) < {\rm dim}\,{\mathcal {Z}}_K$. Then there exists a dominant, generically finite map $U_K \rightarrow {\mathcal {Z}}_K$ of degree prime to $p$ at the generic points of $U_K,$ and a map $h: U_K \rightarrow Y_K$ to a finite type $K$-scheme $Y_K$ with ${\rm dim}\, Y_K < {\rm dim}\, {\mathcal {Z}}_K,$ such that ${\mathcal {A}}[p]|_{U_K}$ arises as the pullback of a finite étale covering of $Y_K$. We may assume that $Y_K$ is the scheme-theoretic image of $U_K$ under $h$. Next, after replacing both $U_K$ and $Y_K$ by dense affine opens, we may assume that both these schemes are affine corresponding to $K$-algebras $B_K$ and $C_K$ respectively, and that $U_K \rightarrow Y_K$ is flat.

Let $\tilde {\mathcal {Z}}$ be the normalization of ${\mathcal {Z}}$ in $U_K$. Let $\mathfrak p$ be the generic point of ${\mathcal {Z}}_k,$ and $\mathfrak q_1, \ldots , \mathfrak q_m$ the primes of $\tilde {\mathcal {Z}}$ over $\mathfrak p$. Since the degree of $\tilde {\mathcal {Z}} \rightarrow {\mathcal {Z}}$ over $\mathfrak p$ is prime to $p,$ for some $i$ the ramification degree $e(\mathfrak q_i/\mathfrak p)$ and the degree of the residue field extension $\kappa (\mathfrak q_i)/\kappa (\mathfrak p)$ are prime to $p$. In particular, the residue field extension is separable. By Abhyankar's Lemma, it follows that, after replacing $K$ by a finite extension, we may assume that $e(\mathfrak q_i/\mathfrak p) = 1$ for some $i,$ and that $\tilde {\mathcal {Z}} \rightarrow {\mathcal {Z}}$ is étale at $\mathfrak q_i$. Shrinking $U_K$ further if necessary, we may assume that there is an affine open ${\operatorname {Spec\,}} B = U \subset \tilde {\mathcal {Z}}$ such that $U_k \rightarrow {\mathcal {Z}}_k$ has dense image, $U\otimes K = U_K,$ and $U \rightarrow {\mathcal {Z}}$ is étale. In particular, $U$ is smooth over $\mathcal {O}_K$.

Now choose a finitely generated $\mathcal {O}_K$-subalgebra $C \subset C_K \cap B$ such that $C\otimes K = C_K$. This is possible as $C_K$ is finitely generated over $K$. Then $h$ extends to a map $h: U \rightarrow Y = {\operatorname {Spec\,}} C$. Let $J \supset (p)$ be an ideal of $C,$ and $Y_J \rightarrow Y$ the blow-up of $J$. Denote by $U_J$ the proper transform of $U$ by this blow-up. That is, $U_J$ is the closure of $U_K$ in $U\times Y_J$. By the Raynaud–Gruson Flattening Theorem [Reference Raynaud and GrusonRG71, Theorem 5.2.2], we can choose $J$ so that $U_J \rightarrow Y_J$ is flat. Since $U$ is normal, the map $U_J \rightarrow U$ is an isomorphism over the generic points of $U\otimes k$. Hence, after replacing $Y$ by an affine open in $Y_J,$ and shrinking $U,$ we may assume that $U \rightarrow Y$ is flat.

Shrinking $U$ further, we may assume that the special fiber $U_k$ maps to the ordinary locus of ${\mathcal {A}}_{g,N}$. Now let $\widehat B$ and $\widehat C$ denote the $p$-adic completions of $B$ and $C$ respectively, and set $\widehat U = {\operatorname {Spec\,}} \widehat B$ and $\widehat Y = {\operatorname {Spec\,}} \widehat C$.Footnote ² Since ${\mathcal {A}}[p]|_{\widehat U}$ is ordinary, there is a finite étale covering $\widehat U' = {\operatorname {Spec\,}} \widehat B' \rightarrow \widehat U$ such that ${\mathcal {A}}[p]|_{\widehat U'}$ is an extension of $({\mathbb {Z}}/p{\mathbb {Z}})^g$ by $\mu _p^g$. Hence by Lemmas 2.1.6 and 2.1.8, $\widehat U'_K \rightarrow \widehat Y_K$ factors through a finite étale map $\widehat Y'_K \rightarrow \widehat Y_K$ such that ${\mathcal {A}}[p]|_{\widehat U'_K}$ is the pullback of an extension ${\mathcal {F}}'$ of $({\mathbb {Z}}/p{\mathbb {Z}})^g$ by $\mu _p^g$ on $\widehat Y'_K$. As $\widehat U'$ is normal, we may assume $\widehat Y'_K$ is normal.

Let $\widehat Y' = {\operatorname {Spec\,}} \widehat C'$ be the normalization of $\widehat Y$ in $\widehat Y'_K$. As $\widehat U'$ is normal, we have

\[ \widehat U' \rightarrow \widehat Y' \rightarrow \widehat Y. \]

As $\widehat Y'$ is normal, $\widehat U' \rightarrow \widehat Y'$ is flat over the generic points of $\widehat Y'_k$. Hence, there exists $f_0 \in \widehat C'/\pi \widehat C'$ which is nowhere nilpotent on $\widehat Y'_k,$ and such that $\widehat U'_k \rightarrow \widehat Y'_k$ is flat over the complement of the support of the ideal $(f_0)$. Now let $f \in \widehat C'$ be a lift of $f_0,$ and let $\widehat C'' = \widehat {C'[1/f]}$ and $\widehat B'' = \widehat {B'[1/f]},$ the $p$-adic completionsFootnote ³ of $C'[1/f]$ and $B'[1/f]$. Let $\widehat U'' = {\operatorname {Spec\,}} B''$ and $\widehat Y'' = {\operatorname {Spec\,}} \widehat C''$. Then $\widehat U''$ is flat over $\widehat Y''$ by [Reference GrothendieckEGA, IV, 11.3.10.1]. Moreover, since $\widehat U \rightarrow \widehat Y$ is flat, the generic points of $\widehat U'_k$ map to generic points of $\widehat Y'_k$. So the image of $\widehat U''_k$ is dense in $\widehat U'_k,$ and in particular $\widehat U''(k)$ is nonempty.

Now choose a point $x \in \widehat U''(k),$ and denote by $y \in \widehat Y''(k)$ its image. We write $\mathcal {O}_{\widehat U'',x}$ and $\mathcal {O}_{\widehat Y'',y}$ for the complete local rings at $x$ and $y$. Since the maps

\[ \widehat U'' \rightarrow \widehat U' \rightarrow \widehat U \rightarrow {\mathcal{Z}} \]

are formally étale, $\mathcal {O}_{\widehat U'',x}$ is naturally isomorphic to the complete local ring at the image of $x$ in ${\mathcal {Z}}$. Let $R$ be the universal deformation $\mathcal {O}_K$-algebra of the abelian scheme ${\mathcal {A}}_x$. By condition (i) of the theorem, $\mathcal {O}_{\widehat U'',x}$ is naturally a quotient of $R$. The map $\mathcal {O}_{\widehat Y'',y} \rightarrow \mathcal {O}_{\widehat U'',x}$ satisfies the conditions of Lemma 3.2.4 (cf. [Reference GrothendieckEGA, IV, 17.5.3]), and it follows that this map is an isomorphism. In particular, this implies that

\[ {\rm dim}\, Y_K = {\rm dim}\, \mathcal{O}_{\widehat Y'',y} -1 = {\rm dim}\, \mathcal{O}_{\widehat U'',x} -1 = {\rm dim}\, {\mathcal{Z}}_K \]

which contradicts our initial assumption.

Proof. If $N \geq 3,$ the corollary follows from Theorem 3.2.6 and Lemma 2.2.4.

Suppose $N=1$ or $2$. Let $N' \geq 3$ be an integer coprime to $pN$. We may assume that $K$ is equipped with a primitive $pN^{\prime }$th root of $1,$ $\zeta _{pN'}$. The map of $K$-schemes ${\mathcal {A}}_{g,pN} \rightarrow {\mathcal {A}}_{g,N},$ is a covering with group ${\rm Sp}_{2g}(\mathbb {F}_p)/\{\pm 1\}$. Consider the maps

\[ {\mathcal{A}}_{g,pNN'} \rightarrow {\mathcal{A}}_{g,pN}\times_{{\mathcal{A}}_{g,N}}{\mathcal{A}}_{g,NN'} = : {\mathcal{A}}'_{g,pNN'} \rightarrow {\mathcal{A}}_{g,NN'}. \]

Then ${\mathcal {A}}'_{g,pNN'} \rightarrow {\mathcal {A}}_{g,NN'}$ again corresponds to a ${\rm Sp}_{2g}(\mathbb {F}_p)/\{\pm 1\}$ covering, and ${\mathcal {A}}_{g,pNN'} \rightarrow {\mathcal {A}}_{g,NN'}$ corresponds to a ${\rm Sp}_{2g}(\mathbb {F}_p)$ covering. When $p=2$ these two coverings coincide.

Let ${\mathcal {Z}}_{N'} = {\mathcal {Z}} \times _{{\mathcal {A}}_{g,N}}{\mathcal {A}}_{g,NN'}$. Our assumption on the automorphisms of ${\mathcal {A}}_{\bar \eta }$ implies that at the generic points of ${\mathcal {Z}}_k,$ the map ${\mathcal {Z}}_{N'} \rightarrow {\mathcal {Z}}$ is étale. Thus, after replacing ${\mathcal {Z}}$ by a fiberwise dense open, we may assume that ${\mathcal {Z}}_{N'}$ is smooth over $\mathcal {O}_K$. The observations of the previous paragraph, Lemma 2.2.7 when $p >2,$ Lemma 2.2.4 and Theorem 3.2.6 imply that we have

(3.2.8)\begin{align} \overline{{\rm ed}}({\mathcal{A}}_{g,pN}|_{{\mathcal{Z}}_K}/{\mathcal{Z}}_K; p) &\geq \overline{{\rm ed}}({\mathcal{A}}'_{g,pNN'}|_{{\mathcal{Z}}_{N',K}}/{\mathcal{Z}}_{N',K}; p) \nonumber\\ &= \overline{{\rm ed}}({\mathcal{A}}_{g,pNN'}|_{{\mathcal{Z}}_{N',K}}/{\mathcal{Z}}_{N',K}; p)\nonumber\\ & = \overline{{\rm ed}}({\mathcal{A}}[p]|_{{\mathcal{Z}}_{N',K}}/{\mathcal{Z}}_{N',K}; p)\nonumber\\ & ={\rm dim}\, {\mathcal{Z}}_K.\end{align}

Proof. A fortiori it suffices to consider the case when $n=p$ is prime, which is a special case of Corollary 3.2.7.

3.3.1 Let $g \geq 2,$ and let ${\mathcal {M}}_g$ denote the coarse moduli space of smooth, proper, genus $g$ curves. For any integer $n,$ let ${\mathcal {M}}_g[n]$ denote the ${\mathbb {Z}}[\zeta _n][1/n]$-scheme which is the coarse moduli space of pairs $(C,{\mathcal {B}})$ consisting of a proper smooth curve $C$ of genus $g$ together with a choice $\mathcal {B}$ of symplectic basis for $J(C)[n],$ where $J(C)$ denotes the Jacobian of $C$. For $n \geq 3$ this is a fine moduli space which is smooth over ${\mathbb {Z}}[\zeta _n][1/n]$ [Reference Deligne and MumfordDM69].

Proof. Let $N \geq 3$ be an integer. There is a natural map of ${\mathbb {Z}}[\zeta _N][1/N]$-schemes $\varpi : {\mathcal {M}}_g[N] \rightarrow A_{g,N}$ taking a curve to its Jacobian. For $(C,\mathcal {B})$ in ${\mathcal {M}}_g[N]$ the pairs $(J(C), \mathcal {B})$ and $(J(C), -\mathcal {B})$ are isomorphic via $-1$ on $J(C)$. Thus $(C,\mathcal {B}) \mapsto (C,-\mathcal {B})$ is an involution $\Sigma$ of ${\mathcal {M}}_g[N],$ which is nontrivial, unless $g=2$. We denote by ${\mathcal {M}}_g[N]'$ the quotient of ${\mathcal {M}}_g[N]$ by this involution. When $g \geq 3,$ the fixed points of $\Sigma$ in any fiber of ${\mathcal {M}}_g[N]$ over ${\operatorname {Spec\,}} {\mathbb {Z}}[\zeta _N][1/N]$ are contained in a proper closed subset. Thus ${\mathcal {M}}_g[N]'$ is generically smooth over every point of ${\operatorname {Spec\,}} {\mathbb {Z}}[\zeta _N][1/N]$.

By [Reference Oort and SteenbrinkOS80, 1.11, 2.7, 2.8], the map of ${\mathbb {Z}}[\zeta _N][1/N]$-schemes ${\mathcal {M}}_g[N] \rightarrow {\mathcal {A}}_{g,N}$ induces a map ${\mathcal {M}}_g[N]' \rightarrow {\mathcal {A}}_{g,N}$ which is injective, an immersion if $g=2$ and an immersion outside the hyperelliptic locus if $g \geq 3$.

It suffices to prove the theorem with $n$ replaced by the prime factor $p$. Let $N \geq 3$ be coprime to $p,$ and let ${\mathcal {M}}_g[pN]'' = {\mathcal {M}}_{g}[p]\times _{{\mathcal {M}}_g}{{\mathcal {M}}_g[N]}'$. It is enough to show that $\overline {{\rm ed}}({\mathcal {M}}_g[pN]''/{\mathcal {M}}_g[N]'; p) = 3g-3$. If $p=2$ then ${\mathcal {M}}_g[pN]'' = {\mathcal {M}}_g[pN]'$ as coverings of ${\mathcal {M}}_g[N]',$ and if $p \geq 3$ we have natural degree $2$ maps of coverings of ${\mathcal {M}}_g[N]'$,

\[ {\mathcal{M}}_g[pN]'' \leftarrow {\mathcal{M}}_g[pN] \rightarrow {\mathcal{M}}_g[pN]'. \]

Thus, using Lemma 2.2.7 when $p \geq 3,$ it suffices to show that

\[ \overline{{\rm ed}}({\mathcal{M}}_g[pN]'/{\mathcal{M}}_g[N]';p) = 3g-3. \]

Since ${\mathcal {M}}_g[pN]' = {\mathcal {A}}_{g,pN}|_{{\mathcal {M}}_g[N]'},$ for example by comparing the degrees of these coverings, and ${\mathcal {M}}_g[N]'$ meets the ordinary locus in ${\mathcal {A}}_{g,N}\otimes {\mathbb {F}}_p$ [Reference Faber and van der GeerFvdG04, 2.3], the theorem now follows from Theorem 3.2.6.

3.3.3 We now prove the analogue of Theorem 3.3.2 for the moduli space of hyperelliptic curves. Let $S$ be a ${\mathbb {Z}}[1/2]$-scheme. Recall that a hyperelliptic curve over $S$ is a smooth proper curve $C/S$ of genus $g \geq 1,$ equipped with an involution $\sigma$ such that $P = C/\langle \sigma \rangle$ has genus $0$. Let $\mathcal {H}_g$ denote the coarse moduli space of genus $g$ hyperelliptic curves over ${\mathbb {Z}}$. It is classical (and not hard to see) that over ${\mathbb {Z}}[1/2]$ one has

\[ \mathcal{H}_g \cong {\mathcal{M}}_{0,2g+2}/S_{2g+2} \]

where ${\mathcal {M}}_{0,2g+2}$ is the moduli space of genus $0$ curves with $2g+2$ ordered marked points, and $S_{2g+2}$ is the symmetric group on $2g+2$ letters.

For any integer $n,$ let $\mathcal {H}_g[n]$ denote the ${\mathbb {Z}}[\zeta _n][1/n]$-scheme which is the coarse moduli space of pairs $(C,{\mathcal {B}})$ consisting of a hyperelliptic curve $C$ together with a symplectic basis $\mathcal {B}$ for $J(C)[n]$. As above, for $n \geq 3$ this is a fine moduli space which is smooth over ${\mathbb {Z}}[\zeta _n][1/n]$.

Proof. There is a natural map of ${\mathbb {Z}}[1/2]$-schemes $\mathcal {H}_g \rightarrow {\mathcal {M}}_g$ which is generically an injective immersion on every fiber over ${\mathbb {Z}}[1/2],$ as a general hyperelliptic curve has only one nontrivial automorphism [Reference PoonenPoo00, Theorem 1]. By the Torelli theorem and [Reference Oort and SteenbrinkOS80, Corollary 3.2], the map ${\mathcal {M}}_g \rightarrow {\mathcal {A}}_g$ is also generically an injective immersion on every fiber over ${\mathbb {Z}}[1/2],$ and thus so is $\mathcal {H}_g \rightarrow {\mathcal {A}}_g$.

Now the Jacobian of a hyperelliptic curve over the generic point of $\mathcal {H}_g$ has automorphism group $\{\pm 1\}$ [Reference MatsusakaMat58, p. 790], and $\mathcal {H}_g$ meets the ordinary locus of ${\mathcal {A}}_g\otimes {\mathbb {F}}_p$ by [Reference Glass and PriesGP05, Theorem 1]. Hence the theorem follows from Corollary 3.2.7.

3.3.5 We remark that when $g=2,$ Theorem 3.3.4 extends to $p=2,$ as this is a special case of Theorem 3.3.2. However, an extension to $p=2$ is not possible when $g> 2$. To explain this, recall that for a finite group $G$ and a prime $p$, $\overline {{\rm ed}}(G;p)$ denotes the supremum of ${\rm ed}_K(Y/X;p)$ taken over all $G$-covers $Y/X$ of finite type $K$-schemes, for any algebraically closed field $K$ of characteristic $0$ (the definition being independent of $K$). The covering

\[ {\mathcal{M}}_{0,2g+2} \rightarrow {\mathcal{M}}_{0,2g+2}/S_{2g+2} \overset \sim \longrightarrow \mathcal{H}_g \]

is a component of $\mathcal {H}_g[2];$ for $g>2$, the cover is disconnected, and all components are isomorphic.Footnote ⁴ We conclude that

\begin{align*} \overline{{\rm ed}}(\mathcal{H}_g[2]/\mathcal{H}_g;2) &= \overline{{\rm ed}}({\mathcal{M}}_{0,2g+2}/\mathcal{H}_g;2) \\ & = \overline{{\rm ed}}(S_{2g+2};2)\\ & = g+1< 2g - 1 \end{align*}

where the second equality follows from the versality of ${\mathcal {M}}_{0,2g+2}$ for $S_{2g+2}$, and the third follows from [Reference Meyer and ReichsteinMR09, Cor. 4.2].

The lower bound $\overline {{\rm ed}}(\mathcal {H}_g[2]/\mathcal {H}_g;2) \geq g+1$ can actually be recovered using the techniques of this paper. The point is that although $\mathcal {H}_g \rightarrow {\mathcal {A}}_g$ is not generically an immersion in characteristic $2,$ one can show that the image of the map on tangent spaces at a generic point has dimension $g+1$. We are grateful to Aaron Landesman for showing us this calculation [Reference LandesmanLan19].

4.1.1 Let $F$ be a number field and $G = G^{{\rm ad}}$ an adjoint, connected reductive group over $F$. We fix algebraic closures $\bar F$ and $\bar F_v$ of $F$ and $F_v$ respectively, for every finite place $v$ of $F,$ as well as embeddings $\bar F \hookrightarrow \bar F_v$.

Recall [Reference Demazure and GrothendieckSGA3, XXIV, Theorem 1.3] that the automorphism group scheme of $G$ is an extension

(4.1.2)\begin{equation} 1 \rightarrow G \rightarrow {\rm Aut}(G) \rightarrow {\rm Out}(G) \rightarrow 1 \end{equation}

where ${\rm Out}(G)$ is a finite group scheme. If $G$ is split, then this extension is split and ${\rm Out}(G)$ is a constant group scheme which can be identified with the group of automorphisms of the Dynkin diagram of $G$.

We will also make use of the notion of the fundamental group $\pi _1(G)$ [Reference BorovoiBor96]. This is a finite abelian group equipped with a ${\rm Gal}(\bar F/F)$-action. As an étale sheaf on ${\operatorname {Spec\,}} F,$ one has $\pi _1(G)\otimes \mu _n \overset \sim \longrightarrow {\rm ker}(G^{{\rm sc}}\rightarrow G)$ where $G^{{\rm sc}}$ is the simply connected cover of $G,$ and $n$ is the order ${\rm ker}(G^{{\rm sc}} \rightarrow G)$. The following proposition is a variant of [Reference Harder and BorelHB78, Theorem B].

Proof. Recall the following facts about the cohomology of reductive groups over global and local fields [Reference KottwitzKot86]: Let $H$ be an adjoint connected reductive group over $F$. For any place $v$ of $F$, there is a map

\[ H^1(F_v, H) \rightarrow \pi_1(H)_{{\rm Gal}(\bar F_v/F_v)}, \]

which is an isomorphism if $v$ is finite. For any finite set of places $T$ of $F,$ consider the composite map

\[ \xi: \prod_{v \in T} H^1(F_v, H) \rightarrow \prod_{v\in T} \pi_1(H)_{{\rm Gal}(\bar F_v/F_v)} \rightarrow \pi_1(H)_{{\rm Gal}(\bar F/F)}. \]

Then by [Reference KottwitzKot86, § 2.2], $(x_v)_{v\in T} \in \prod _{v \in T} H^1(F_v, H)$ is in the image of $H^1(F,H)$ if $\xi ((x_v)) = 0$. Applying this to $T = S \cup \{v_0\}$ for some finite place $v_0 \notin S,$ we see that

(4.1.4)\begin{equation} H^1(F,H) \rightarrow \prod_{v \in S} H^1(F_v, H) \end{equation}

is surjective.

Now let $(x_v) \in \prod _{v \in S} H^1(F_v, {\rm Aut}(G)')$ and let $(\bar x_v) \in \prod _{v \in S} H^1(F_v, {\rm Out}(G)')$ be the image of $(x_v)$. By our assumptions on ${\rm Out}(G)',$ there exists $\bar x \in H^1(F, {\rm Out}(G)')$ mapping to $(\bar x_v)$. Since we are assuming $G$ is split, (4.1.2) is a split extension, so there is a $x \in H^1(F,{\rm Aut}(G)')$ mapping to $\bar x$. Let $H$ be the twist of $G$ by $x$. Recall that this means that if we choose a cocycle $x = (x_\sigma )_{\sigma \in {\rm Gal}(\bar F/F)}$ representing $x,$ then there is an isomorphism $\tau : G \overset \sim \longrightarrow H$ over $\bar F,$ such that, for $g \in G(\bar F)$ and $\sigma \in {\rm Gal}(\bar F/F),$ we have $\tau (\sigma (g)) = (\sigma (\tau (g)))^{x_{\sigma }}$. We have a commutative diagram

such that the vertical maps send $x$ and $(x|_{F_v})_v$ to the trivial classes in the bottom line. Thus it suffices to show that

\[ H^1(F,H) \rightarrow \prod_{v\in S} H^1(F_v, H) \]

is surjective, which we saw above.