Proof. This follows from [Reference Howard and PappasHP17, § 6.2.1 and Proposition 6.2.2]. ◻

Proof. The first equivalence follows from Lemma 2.11.2. The second equivalence holds because $\mathbf{v}$ is $\unicode[STIX]{x1D6F7}$ -invariant.◻

Proof. The statements immediately follow from the definition. ◻

Proof. By flat descent, to show that the closed formal subschemes on the two sides of (3.2.1.1) agree, it suffices to show that they have the same $k$ -points and the same formal completion at every $k$ -point (cf. [Reference Bueltel and PappasBP17, 5.2.7]). The claim then follows from the observation that both the $k$ -points and the formal completions have purely group-theoretic description.

In fact, the $k$ -points of $\operatorname{RZ}^{\flat }=\operatorname{RZ}_{G^{\flat }}$ , $\operatorname{RZ}(\mathbb{X}_{0}^{\flat },\unicode[STIX]{x1D706}_{0}^{\flat })=\operatorname{RZ}_{H}$ and $\operatorname{RZ}=\operatorname{RZ}_{G}$ have the group-theoretic description as the affine Deligne–Lusztig sets (2.11.1.2) associated to the groups $G^{\flat }=\operatorname{GSpin}(V^{\flat })$ , $H=\operatorname{GSp}(C(V^{\flat }),\unicode[STIX]{x1D713}^{\flat })$ and $G=\operatorname{GSpin}(V)$ , respectively. Since $G^{\flat }=H\cap G$ inside $\operatorname{GL}(C(V))$ , we know that both sides of (3.2.1.1) have the same $k$ -points. Fix a $k$ -point $x\in \operatorname{RZ}^{\flat }(k)$ . Then, by [Reference Howard and PappasHP17, 3.2.12], $\widehat{\operatorname{RZ}}_{G^{\flat },x}$ can be identified with $U_{G^{\flat }}^{\unicode[STIX]{x1D707}_{x},\wedge }$ , where $\unicode[STIX]{x1D707}_{x}:\mathbb{G}_{m,W}\rightarrow G_{W}^{\flat }$ gives a filtration that lifts the Hodge filtration for $x$ , $U_{G^{\flat }}^{\unicode[STIX]{x1D707}_{x}}\subseteq G^{\flat }$ is the unipotent radical of the opposite parabolic group defined by $\unicode[STIX]{x1D707}_{x}$  [Reference Howard and PappasHP17, 3.1.6], and $U_{G^{\flat }}^{\unicode[STIX]{x1D707}_{x},\wedge }$ is its formal completion along its identity section over $W$ . Similarly, we can identify $\widehat{\operatorname{RZ}}_{H,x}$ and $\widehat{\operatorname{RZ}}_{G,x}$ with $U_{H}^{\unicode[STIX]{x1D707}_{x},\wedge }$ and $U_{G}^{\unicode[STIX]{x1D707}_{x},\wedge }$ . Again, because $G^{\flat }=H\cap G$ , we know that the formal completions at $x$ of both sides of (3.2.1.1) agree inside $U_{\operatorname{GL}(C(V))}^{\unicode[STIX]{x1D707}_{x},\wedge }$ .◻

Proof. Let $X^{\flat }$ be the universal $p$ -divisible group over $\operatorname{RZ}^{\flat }$ and $\unicode[STIX]{x1D70C}^{\flat }$ be the universal quasi-isogeny. Then it follows from the commutative diagram (3.2.0.1) that the image of $(X^{\flat },\unicode[STIX]{x1D70C})$ under $\unicode[STIX]{x1D6FF}$ is given by the $p$ -divisible group $(X^{\flat }\,\oplus \,X^{\flat },\unicode[STIX]{x1D70C}^{\flat }\,\oplus \,\unicode[STIX]{x1D70C}^{\flat })$ . Since $x_{n}$ has norm 1, right multiplication by $x_{n}$ swaps the two factors $C(V^{\flat })$ and $C(V^{\flat })x_{n}$ . It follows that the quasi-endomorphism

$$\begin{eqnarray}(\unicode[STIX]{x1D70C}^{\flat }\oplus \unicode[STIX]{x1D70C}^{\flat })\circ x_{n}\circ (\unicode[STIX]{x1D70C}^{\flat }\oplus \unicode[STIX]{x1D70C}^{\flat })^{-1}:(X^{\flat }\oplus X^{\flat })\rightarrow (X^{\flat }\oplus X^{\flat })\end{eqnarray}$$

(uniquely determined by the rigidity of quasi-isogenies) simply swaps the two factors, which is an actual endomorphism (i.e. the swapping) of $X^{\flat }\,\oplus \,X^{\flat }$ . By Definition 2.14.1 of ${\mathcal{Z}}(x_{n})$ , we have $\unicode[STIX]{x1D6FF}(\operatorname{RZ}^{\flat })\subseteq {\mathcal{Z}}(x_{n})$ .

Conversely, over ${\mathcal{Z}}(x_{n})$ the universal $p$ -divisible group $X$ admits an action of $C(x_{n})^{\text{op}}\otimes C(V)$ , where $C(x_{n})$ is the Clifford algebra of the rank-one quadratic space $\mathbb{Z}_{p}x_{n}$ . Notice that

$$\begin{eqnarray}C(x_{n})^{\text{op}}\otimes C(V)\cong (C(x_{n})^{\text{op}}\otimes C(x_{n}))\oplus (C(x_{n})^{\text{op}}\otimes C(V^{\flat })).\end{eqnarray}$$

It follows that over ${\mathcal{Z}}(x_{n})$ the universal $p$ -divisible group $X$ admits an action of $C(x_{n})^{\text{op}}\otimes C(x_{n})$ , which is isomorphic to the matrix algebra $M_{2}(\mathbb{Z}_{p})$ . The two natural idempotents of $M_{2}(\mathbb{Z}_{p})$ then decompose $X$ into a direct sum of the form $X^{\flat }\,\oplus \,X^{\flat }$ . Hence, ${\mathcal{Z}}(x_{n})\subseteq \unicode[STIX]{x1D719}(\operatorname{RZ}(\mathbb{X}_{0}^{\flat },\unicode[STIX]{x1D706}_{0}^{\flat }))\cap \operatorname{RZ}$ . The latter is equal to $\unicode[STIX]{x1D6FF}(\operatorname{RZ}^{\flat })$ by (3.2.1.1) and hence ${\mathcal{Z}}(x_{n})\subseteq \unicode[STIX]{x1D6FF}(\operatorname{RZ}^{\flat })$ .◻

Proof. Since $p^{\mathbb{Z}}$ is central in $J_{b}(\mathbb{Q}_{p})$ , we know that $\operatorname{RZ}^{g}$ is stable under $p^{\mathbb{Z}}$ . The morphism $\unicode[STIX]{x1D6FF}:\operatorname{RZ}^{\flat }\rightarrow \operatorname{RZ}$ is equivariant with respect to the natural inclusion $J_{b^{\flat }}(\mathbb{Q}_{p})\rightarrow J_{b}(\mathbb{Q}_{p})$ , and the morphism $J_{b^{\flat }}\rightarrow J_{b}$ restricts to the identity between the centers $\mathbb{G}_{m}$ of $J_{b^{\flat }}$ and of $J_{b}$ . It follows that $\unicode[STIX]{x1D6FF}$ is equivariant for the $p^{\mathbb{Z}}$ action and so $\unicode[STIX]{x1D6FF}(\operatorname{RZ}^{\flat })$ is stable under $p^{\mathbb{Z}}$ . We now prove the statements (1) to (4).

1. (1) For a point $L^{\flat }\in p^{\mathbb{Z}}\backslash \operatorname{RZ}^{\flat }(k)$ , we can write $L^{\flat }=h^{\flat }\unicode[STIX]{x1D707}^{\flat }(p^{-1}).V_{W}^{\flat }\subseteq V_{K}^{\flat }$ for some $h^{\flat }\in G^{\flat }(K)$ . Then its image under $\unicode[STIX]{x1D6FF}$ is given by $L=h\unicode[STIX]{x1D707}(p^{-1}).V_{W}\subseteq V_{K}$ , where $h$ is the image of $h^{\flat }$ in $G(K)$ . By $V=V^{\flat }\,\oplus \,\mathbb{Z}_{p}x_{n}$ and the compatibility between $h,\unicode[STIX]{x1D707}$ and $h^{\flat },\unicode[STIX]{x1D707}^{\flat }$ , we know that $L=L^{\flat }\,\oplus \,Wx_{n}$ .

2. (2) Suppose that $L$ is a special lattice with $x_{n}\in L$ . Since $x_{n}$ has norm 1, we know that $L=L^{\prime }\,\oplus \,Wx_{n}$ is the direct sum of $Wx_{n}$ and its orthogonal complement $L^{\prime }$ in $L$ . One can check that $L^{\prime }\subseteq V_{K}^{\flat }$ is also a special lattice. This finishes the proof in view of item (1).

3. (3) This is clear since $\operatorname{RZ}^{g}(k)$ is the fixed locus of $g$ .

4. (4) For a point $L\in p^{\mathbb{Z}}\backslash (\unicode[STIX]{x1D6FF}(\operatorname{RZ}^{\flat }(k))\cap \operatorname{RZ}^{g}(k))$ , by items (1) (3), we have $L=L^{\flat }\,\oplus \,Wx_{n}$ and $gL=L$ . It follows from $x_{n}\in L$ that $gx_{n},\ldots ,g^{n-1}x_{n}\in L$ and so $L\supseteq L(g)_{W}$ . Conversely, if a point $L\in \operatorname{RZ}(k)$ satisfies $gL=L$ and $L\supset L(g)_{W}$ , then $L\in p^{\mathbb{Z}}\backslash (\unicode[STIX]{x1D6FF}(\operatorname{RZ}^{\flat }(k))\cap \operatorname{RZ}^{g}(k))$ by items (2) and (3). ◻

Proof. By Lemma 3.4.2, it suffices to show that the $k$ -points of the right-hand side are in bijection with special lattices $L$ such that $gL=L$ and $L\supseteq L(g)_{W}$ . Notice that any special lattice $L$ is self-dual, so the condition $L\supseteq L(g)_{W}$ is equivalent to the condition $L\subseteq L(g)_{W}^{\vee }$ . Since $\unicode[STIX]{x1D6EC}(L)_{W}$ is the minimal $\unicode[STIX]{x1D6F7}$ -invariant lattice containing $L$ (§ 2.12) and $L(g)_{W}^{\vee }$ is $\unicode[STIX]{x1D6F7}$ -invariant, we know that the condition $L\subseteq L(g)_{W}^{\vee }$ is equivalent to the condition $\unicode[STIX]{x1D6EC}(L)\subseteq L(g)^{\vee }$ . The result now follows from taking $\bar{g}$ -invariants and $g$ -invariants of the two sides of the bijection (2.12.1.1).◻

Proof. (1) Suppose that $X^{\bar{g}}$ is non-empty. Then it is a general fact about Deligne–Lusztig varieties that $\bar{g}$ must be semisimple [Reference LusztigLus11, 5.9(a)]. Let $T(w)\subseteq \operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})$ be a torus of Coxeter type (associated to $w=w^{+}$ or $w^{-}$ ) and $B(w)\supseteq T(w)$ be a Borel. Assume that $\bar{g}$ is semisimple. Then we know from [Reference Deligne and LusztigDL76, Proposition 4.7] that $X^{\bar{g}}$ is a disjoint union of Deligne–Lusztig varieties $X_{T^{\prime }\subseteq B^{\prime }}$ for the group $G^{\prime }=Z^{\circ }(\bar{g})$ and the pairs

$$\begin{eqnarray}(T^{\prime },B^{\prime })=(hT(w)h^{-1},hB(w)h^{-1}\cap G^{\prime }),\end{eqnarray}$$

where $h$ runs over classes $G^{\prime }(\mathbb{F}_{p})\backslash \operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})(\mathbb{F}_{p})$ such that $\bar{g}\in hT(w)h^{-1}$ . Therefore, $X^{\bar{g}}$ is non-empty if and only if there exists $h\in \operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})(\mathbb{F}_{p})$ such that $\bar{g}\in hT(w)h^{-1}$ , if and only if $\bar{g}$ is contained in a maximal torus of Coxeter type (as so is $T(w)$ ).

(2) By part (1), we know that $X^{\bar{g}}$ is further finite if and only if all $X_{T^{\prime }\subseteq B^{\prime }}$ are zero dimensional, if and only if all $B^{\prime }=hBh^{-1}\cap G^{\prime }$ are tori. This happens exactly when $G^{\prime }=Z^{\circ }(\bar{g})$ itself is a torus, i.e. when $\bar{g}$ is regular. In this case, $G^{\prime }$ is a maximal torus of Coxeter type in $\operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})$ and the cardinality of $X^{\bar{g}}$ is equal to the cardinality of $N(T(w))(\mathbb{F}_{p})/T(w)(\mathbb{F}_{p})$ . The latter group is isomorphic to $(N(T(w))/T(w))^{\unicode[STIX]{x1D6F7}}$ by Lang’s theorem and hence is isomorphic to the $\unicode[STIX]{x1D6F7}$ -twisted centralizer of $w$ in the Weyl group $W(T)=N(T)/T$ :

$$\begin{eqnarray}Z_{\unicode[STIX]{x1D6F7}}(w):=\{x\in W(T):xw=w\unicode[STIX]{x1D6F7}(x)\}.\end{eqnarray}$$

The cardinality of $Z_{\unicode[STIX]{x1D6F7}}(w)$ is known as the Coxeter number of the group $\operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})$ , which is equal to $d=t_{\unicode[STIX]{x1D6EC}}/2$ since $\operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})$ is a non-split even orthogonal group [Reference LusztigLus76/77, 1.15].◻

Proof. By Proposition 3.4.4, we know that $p^{\mathbb{Z}}\backslash (\unicode[STIX]{x1D6FF}(\operatorname{RZ}^{\flat })\cap \operatorname{RZ}^{g})(k)$ is non-empty if and only if $p^{\mathbb{Z}}\backslash \operatorname{BT}_{\unicode[STIX]{x1D6EC}}^{g}$ is non-empty for some $\unicode[STIX]{x1D6EC}\in \operatorname{VL}(g)$ . For any $\unicode[STIX]{x1D6EC}\in \operatorname{VL}(g)$ , by definition we have a chain of inclusions of lattices

$$\begin{eqnarray}L(g)\subseteq \unicode[STIX]{x1D6EC}^{\vee }\subseteq \unicode[STIX]{x1D6EC}\subseteq L(g)^{\vee },\end{eqnarray}$$

which induces a filtration of $\mathbb{F}_{p}$ -vector spaces

$$\begin{eqnarray}0\subseteq \unicode[STIX]{x1D6EC}^{\vee }/L(g)\subseteq \unicode[STIX]{x1D6EC}/L(g)\subseteq \unicode[STIX]{x1D6FA}_{0}(g).\end{eqnarray}$$

It follows that the map $\unicode[STIX]{x1D6EC}\mapsto \unicode[STIX]{x1D6EC}^{\vee }/L(g)$ gives a bijection

(3.6.4.1) $$\begin{eqnarray}\operatorname{VL}(g)\cong \{\text{totally isotropic }\bar{g}\text{-invariant subspaces }U\subseteq \unicode[STIX]{x1D6FA}_{0}(g)\}.\end{eqnarray}$$

By the bijection (3.6.4.1), $\operatorname{VL}(g)$ is non-empty if and only if there is a totally isotropic $\bar{g}$ -invariant subspace $U$ of $\unicode[STIX]{x1D6FA}_{0}(g)$ . Such a subspace $U$ induces a filtration

(3.6.4.2) $$\begin{eqnarray}0\subseteq U\subseteq U^{\bot }\subseteq \unicode[STIX]{x1D6FA}_{0}(g).\end{eqnarray}$$

Since $U$ and $U^{\bot }$ are $\bar{g}$ -invariant, we obtain a decomposition of the characteristic polynomial

(3.6.4.3) $$\begin{eqnarray}P(T)=P_{1}(T)Q(T)P_{2}(T),\end{eqnarray}$$

where $P_{1}(T),Q(T)$ and $P_{2}(T)$ are respectively the characteristic polynomials of $\bar{g}$ acting on the associated graded $U$ , $U^{\bot }/U$ and $\unicode[STIX]{x1D6FA}_{0}(g)/U^{\bot }$ . Notice that the non-degenerate quadratic form on $\unicode[STIX]{x1D6FA}_{0}(g)$ identifies $\unicode[STIX]{x1D6FA}_{0}(g)/U^{\bot }$ with the linear dual of $U$ , from which we know that $P_{2}(T)=P_{1}^{\ast }(T)$ . Similarly, we know that $Q(T)=Q^{\ast }(T)$ , i.e.  $Q(T)$ is self-reciprocal.

Let $\unicode[STIX]{x1D6EC}=L(g)+U^{\bot }$ be the $g$ -vertex lattice corresponding to $U$ under the bijection (3.6.4.1) and let $\unicode[STIX]{x1D6FA}_{0}=\unicode[STIX]{x1D6EC}/\unicode[STIX]{x1D6EC}^{\vee }$ and $\bar{g}_{0}\in \operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})(\mathbb{F}_{p})$ be the induced action of $\bar{g}$ on $\unicode[STIX]{x1D6FA}_{0}$ . Since the minimal polynomial of $\bar{g}$ is equal to its characteristic polynomial $P(T)$ (Remark 3.6.1), we know that the minimal polynomial of $\bar{g}_{0}$ is equal to its characteristic polynomial $Q(T)$ under the decomposition (3.6.4.3). If $\bar{g}_{0}$ is semisimple, then its eigenvalues are distinct. If $\bar{g}$ is further contained in a torus of Coxeter type, then we know that its eigenvalues belong to a single Galois orbit (Remark 3.5.3), so $Q(T)$ is irreducible. Conversely, if $Q(T)$ is irreducible, then clearly $\bar{g}_{0}$ is semisimple and contained in a torus of Coxeter type. Hence, we know that $\bar{g}_{0}$ is semisimple and contained in a torus of Coxeter type if and only if $Q(T)$ is irreducible.

Therefore, by Proposition 3.5.2(1), $\operatorname{BT}_{\unicode[STIX]{x1D6EC}}^{\bar{g}_{0}}$ is non-empty if and only if $Q(T)$ is irreducible. In this case, $\bar{g}_{0}$ is indeed regular semisimple and the cardinality of $p^{\mathbb{Z}}\backslash \operatorname{BT}_{\unicode[STIX]{x1D6EC}}^{\bar{g}_{0}}$ is equal to $2\cdot \#X^{\bar{g}_{0}}$ (due to two connected components), which is equal to $\dim _{\mathbb{F}_{p}}\unicode[STIX]{x1D6FA}_{0}=\deg Q(T)$ by Proposition 3.5.2(2).

Since $P_{2}(T)=P_{1}^{\ast }(T)$ , we know that the multiplicity of $R(T)$ in $P_{1}(T)P_{2}(T)$ is even for any self-reciprocal factor $R(T)$ . Hence, $Q(T)$ is the unique self-reciprocal monic irreducible factor of $P(T)$ such that $m(Q(T))$ is odd. Finally, the factorizations (3.6.4.3) with $P_{2}(T)=P_{1}^{\ast }(T)$ correspond bijectively to the filtrations (3.6.4.2). Notice such a factorization is given by a choice of the polynomial

$$\begin{eqnarray}P_{1}(T)=\mathop{\prod }_{R(T)\neq R^{\ast }(T)}R(T)^{e(R(T))}\cdot \mathop{\prod }_{R(T)=R^{\ast }(T)}R(T)^{\lfloor m(R(T))/2\rfloor },\end{eqnarray}$$

where $R(T)$ runs over all monic irreducible factors of $P(T)$ and $0\leqslant e(R(T))\leqslant m(R(T))$ . So, the total number of such factorizations is exactly given by

$$\begin{eqnarray}\mathop{\prod }_{R(T)\neq R^{\ast }(T)}(1+m(R(T))).\end{eqnarray}$$

The proof is finished. ◻

Proof. Assume that $v$ is orthogonal to $\unicode[STIX]{x1D709}$ . Then $v\unicode[STIX]{x1D709}=-\unicode[STIX]{x1D709}v$ , so $v$ preserves $\ker (\unicode[STIX]{x1D709})$ .

Conversely, assume that $v$ preserves $\ker (\unicode[STIX]{x1D709})$ . Write $q$ for the quadratic form and write $\langle \,,\rangle$ for the corresponding bilinear pairing. Since ${\mathcal{O}}\unicode[STIX]{x1D709}$ is a direct summand of $\mathbf{L}$ , there exists an ${\mathcal{O}}$ -module homomorphism $\mathbf{L}\rightarrow {\mathcal{O}}$ sending $\unicode[STIX]{x1D709}$ to $1$ . Since $\mathbf{L}$ is self-dual, we know that there exists $\unicode[STIX]{x1D701}\in \mathbf{L}$ representing such a homomorphism. Namely, we have

$$\begin{eqnarray}\langle \unicode[STIX]{x1D701},\unicode[STIX]{x1D709}\rangle =1.\end{eqnarray}$$

It immediately follows that we have an ${\mathcal{O}}$ -module direct sum $\mathbf{L}=\unicode[STIX]{x1D709}^{\bot }\,\oplus \,{\mathcal{O}}\unicode[STIX]{x1D701}$ . Replacing $\unicode[STIX]{x1D701}$ by $\unicode[STIX]{x1D701}-(q(\unicode[STIX]{x1D701})/2)\unicode[STIX]{x1D709}$ , we may arrange that $\unicode[STIX]{x1D701}$ is isotropic. We have

$$\begin{eqnarray}q(\unicode[STIX]{x1D701}+\unicode[STIX]{x1D709})=2\langle \unicode[STIX]{x1D701},\unicode[STIX]{x1D709}\rangle =2\end{eqnarray}$$

and in $C(\mathbf{L})$ we have

$$\begin{eqnarray}q(\unicode[STIX]{x1D701}+\unicode[STIX]{x1D709})=\unicode[STIX]{x1D701}\unicode[STIX]{x1D709}+\unicode[STIX]{x1D709}\unicode[STIX]{x1D701}.\end{eqnarray}$$

Hence, in $C(\mathbf{L})$ , we have

(4.1.2.1) $$\begin{eqnarray}\unicode[STIX]{x1D709}\unicode[STIX]{x1D701}+\unicode[STIX]{x1D701}\unicode[STIX]{x1D709}=2.\end{eqnarray}$$

Write

$$\begin{eqnarray}v=v_{1}+\unicode[STIX]{x1D706}\unicode[STIX]{x1D701},\end{eqnarray}$$

with $v_{1}\in \unicode[STIX]{x1D709}^{\bot }$ and $\unicode[STIX]{x1D706}\in {\mathcal{O}}$ . By the first part of the proof we know that $v_{1}$ preserves $\ker (\unicode[STIX]{x1D709})$ . Therefore, $\unicode[STIX]{x1D706}\unicode[STIX]{x1D701}$ preserves $\ker (\unicode[STIX]{x1D709})$ . Note that $\unicode[STIX]{x1D709}\in \ker (\unicode[STIX]{x1D709})$ as $\unicode[STIX]{x1D709}$ is isotropic. It follows that, in $C(\mathbf{L})$ ,

This is possible only when $\unicode[STIX]{x1D706}=0$ and hence we have $v=v_{1}\in \unicode[STIX]{x1D709}^{\bot }$ .◻

Proof. The existence and construction of the bijection $f_{{\mathcal{O}}}$ and the property (1) are consequences of [Reference Madapusi PeraMad16, Proposition 5.16] and the global construction of $\operatorname{RZ}$ in [Reference Howard and PappasHP17] using the integral model of the $\operatorname{GSpin}$ Shimura variety. We explain this more precisely below.

Consider

$$\begin{eqnarray}\mathscr{S}=\mathscr{S}_{U_{p}U^{p}},\end{eqnarray}$$

the canonical integral model over $\mathbb{Z}_{(p)}$ of the Shimura variety associated to the $\operatorname{GSpin}$ Shimura datum associated to a quadratic space $V_{\mathbb{Q}}$ over $\mathbb{Q}$ , at a suitable level $U^{p}$ away from $p$ and a hyperspecial level $U_{p}$ at $p$ . See [Reference Howard and PappasHP17, § 7] or [Reference Madapusi PeraMad16] for more details on this notion. By [Reference Howard and PappasHP17, 7.2.3], we may assume that the following package of data:

• ∙ the Shimura datum associated to $V_{\mathbb{Q}}$ ;

• ∙ the Kuga–Satake Hodge embedding (see [Reference Howard and PappasHP17, 4.14]) of the Shimura datum into a $\operatorname{GSp}$ Shimura datum;

• ∙ the chosen hyperspecial level at $p$ ;

• ∙ an element $x_{00}\in \mathop{\varprojlim }\nolimits_{U^{p}}\mathscr{S}_{U^{p}U_{p}}(k)$

induces, in the fashion of [Reference Howard and PappasHP17, 3.1.4], the local unramified Shimura–Hodge datum that we used to define $\operatorname{RZ}$ . Let $\widehat{\mathscr{S}}$ be the formal scheme over $\mathbb{Z}_{p}$ obtained from $p$ -adic completion of $\mathscr{S}$ and let $\widehat{\mathscr{S}}_{W}$ be the base change to $W$ of $\widehat{\mathscr{S}}$ . Then, as in [Reference Howard and PappasHP17, 3.2.14], we have a morphism of formal schemes over $W$ :

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}:\operatorname{RZ}\rightarrow \widehat{\mathscr{S}}_{W}.\end{eqnarray}$$

We know that $\unicode[STIX]{x1D6E9}$ maps $y_{00}$ to the $k$ -point of $\widehat{\mathscr{S}}_{W}$ induced by $x_{00}$ . Moreover, let

$$\begin{eqnarray}x_{0}:=\unicode[STIX]{x1D6E9}(y_{0})\in \widehat{\mathscr{S}}_{W}(k)=\mathscr{S}(k)\end{eqnarray}$$

and let $\widehat{U}$ be the formal completion of $\mathscr{S}$ at $x_{0}$ (or, what amounts to the same thing, the formal completion of $\widehat{\mathscr{S}}_{W}$ at $x_{0}$ ). By the construction of $\operatorname{RZ}$ in [Reference Howard and PappasHP17, § 3], we know that $\unicode[STIX]{x1D6E9}$ induces an isomorphism $\widehat{\operatorname{RZ}}_{y_{0}}\overset{{\sim}}{\longrightarrow }\widehat{U}$ .

In [Reference Madapusi PeraMad16], two crystals $\mathbf{H}_{\operatorname{cris}},\mathbf{L}_{\operatorname{cris}}$ are constructed on $\mathscr{S}_{k}$ . (In fact, [Reference Madapusi PeraMad16] works over $\mathbb{F}_{p}$ , but we always base change from $\mathbb{F}_{p}$ to $k$ .) Here $\mathbf{H}_{\operatorname{cris}}$ is by definition the first relative crystalline cohomology of the Kuga–Satake abelian scheme over $\mathscr{S}_{k}$ in the sense of [Reference Madapusi PeraMad16].Footnote ⁴ The specialization of $\mathbf{H}_{\operatorname{cris}}$ over $\operatorname{Spec}k$ via $x_{00}$ is identified with the Dieudonné module $C_{W}$ , which is the covariant Diedonné module of the $p$ -divisible group $\mathbb{X}_{0}$ considered in this article (and [Reference Howard and PappasHP17]) and the contravariant Diedonné module of the Kuga–Satake abelian variety at $x_{00}$ considered in [Reference Madapusi PeraMad16].Footnote ⁵ Moreover, the embedding $V{\hookrightarrow}\operatorname{End}_{\mathbb{Z}_{p}}(C)$ has a crystalline realization, which is a sub-crystal $\mathbf{L}_{\operatorname{cris}}$ of $\operatorname{End}(\mathbf{H}_{\operatorname{cris}})$ . For details, see [Reference Madapusi PeraMad16, § 4]. Among others, $\mathbf{L}_{\operatorname{cris}}$ has the following structures:

• ∙ its specialization $\mathbf{L}_{\operatorname{cris},x_{0}}$ to any $x_{0}\in \mathscr{S}(k)$ , viewed as a $W$ -module, has the structure of a $W$ -quadratic space;

• ∙ $\mathbf{L}_{\operatorname{cris},x_{0}}\,\otimes _{W}\,k$ contains a canonical isotropic line $\operatorname{Fil}^{1}(\mathbf{L}_{\operatorname{cris},x_{0}}\,\otimes _{W}\,k)$ .

By the definition of $\unicode[STIX]{x1D6E9}$ and the definition of the parametrization of $\operatorname{RZ}(k)$ by the affine Deligne–Lusztig set (cf. [Reference Howard and PappasHP17, § 2.4]), we know that when $y_{0}\in \operatorname{RZ}(k)$ corresponds to the special lattice $L$ under (2.11.1.1), the following statements are true.

1. (a) There is an isomorphism of Dieudonné modules $(gC)_{W}\overset{{\sim}}{\longrightarrow }\mathbf{H}_{\operatorname{cris},x_{0}}$ .

2. (b) There is a $W$ -linear isometry $(\unicode[STIX]{x1D6F7}L)_{W}\overset{{\sim}}{\longrightarrow }\mathbf{L}_{\operatorname{cris},x_{0}}$ under which $\operatorname{Fil}^{1}(\unicode[STIX]{x1D6F7}L)_{k}$ is identified with

   $$\begin{eqnarray}\operatorname{Fil}^{1}(\mathbf{L}_{\operatorname{ cris},x_{0}}\otimes _{W}k).\end{eqnarray}$$

3. (c) We have a commutative diagram

   
   where:

   • ∙ the right vertical map is induced by the map in (a);

   • ∙ the left vertical map is the map in (b);

   • ∙ the bottom horizontal map arises from the fact that $\mathbf{L}_{\operatorname{cris}}$ is a sub-crystal of $\operatorname{End}(\mathbf{H}_{\operatorname{cris}})$ .

In the rest of the proof we make the identifications in (a) and (b) above and omit them from the notation. Abbreviate $\mathbf{H}:=\mathbf{H}_{\operatorname{cris},x_{0}}$ and $\mathbf{L}:=\mathbf{L}_{\operatorname{cris},x_{0}}$ .

Now, in [Reference Madapusi PeraMad16, Proposition 5.16], Madapusi Pera constructed a bijection

$$\begin{eqnarray}\widehat{U}({\mathcal{O}})\overset{{\sim}}{\longrightarrow }\{\text{isotropic lines in }\mathbf{L}_{{\mathcal{O}}}:=\mathbf{L}\otimes _{W}{\mathcal{O}}\text{ lifting }\operatorname{Fil}^{1}\mathbf{L}_{k}\}.\end{eqnarray}$$

Moreover, by the construction given in [Reference Madapusi PeraMad16, Proposition 5.16], the above bijection is functorial in ${\mathcal{O}}\in \mathscr{C}$ . We define $f_{{\mathcal{O}}}$ as the above bijection precomposed with the isomorphism $\unicode[STIX]{x1D6E9}:\widehat{\operatorname{RZ}}_{y_{0}}\overset{{\sim}}{\longrightarrow }\widehat{U}$ .

It remains to prove property (2). Note that $\mathbf{H}=gC_{W}$ is the covariant Dieudonné module of the $p$ -divisible group $X_{y_{0}}$ over $k$ determined by $y_{0}\in \operatorname{RZ}(k)$ . Given $y\in \widehat{\operatorname{RZ}}_{y_{0}}({\mathcal{O}})$ lifting $y_{0}$ , by Grothendieck–Messing theory (for covariant Dieudonné modules) we know that $y\in \widehat{{\mathcal{Z}}}_{y_{0}}$ if and only if the image of $\mathbf{v}$ in $\operatorname{End}_{{\mathcal{O}}}(\mathbf{H}_{{\mathcal{O}}})$ stabilizes $\operatorname{Fil}^{1}\mathbf{H}_{{\mathcal{O}}}\subset \mathbf{H}_{{\mathcal{O}}}$ , where $\operatorname{Fil}^{1}\mathbf{H}_{{\mathcal{O}}}$ is the Hodge filtration corresponding to the deformation from $k$ to ${\mathcal{O}}$ of the $X_{y_{0}}$ determined by $y$ . Now, as is stated in the proof of [Reference Madapusi PeraMad16, Proposition 5.16],Footnote ⁶ we know that $\operatorname{Fil}^{1}\mathbf{H}_{{\mathcal{O}}}$ is the kernel in $\mathbf{H}_{{\mathcal{O}}}$ of any ${\mathcal{O}}$ -generator $\unicode[STIX]{x1D709}$ of the isotropic line $f_{{\mathcal{O}}}(y)$ . Here $\unicode[STIX]{x1D709}\in \mathbf{L}_{{\mathcal{O}}}$ is viewed as an element of $\operatorname{End}_{{\mathcal{O}}}(\mathbf{H}_{{\mathcal{O}}})$ . By Lemma 4.1.2, $\mathbf{v}$ preserves $\operatorname{Fil}^{1}\mathbf{H}_{{\mathcal{O}}}=\ker \unicode[STIX]{x1D709}$ if and only if $\mathbf{v}$ is orthogonal to $\unicode[STIX]{x1D709}$ (inside $\mathbf{L}_{{\mathcal{O}}}$ ). Thus, $y\in \widehat{{\mathcal{Z}}}_{y_{0}}$ if and only if $f_{{\mathcal{O}}}(y)$ is orthogonal to the image of $\mathbf{v}$ in $\mathbf{L}_{{\mathcal{O}}}=(\unicode[STIX]{x1D6F7}L)_{{\mathcal{O}}}$ .◻

Proof. Consider the bijection $f_{{\mathcal{O}},\mathbf{v}}$ in Theorem 4.1.7(2) for ${\mathcal{O}}=k$ . Since the source of this bijection is non-empty, so is its target. The corollary then follows.◻