2.1 Similitude groups and variants

We begin by introducing algebraic groups over $F_0$:

\begin{gather*} G' :=\operatorname{Res}_{F/F_0}(\mathrm{GL}_{n-1} \times \mathrm{GL}_n), \\ H_1' := \operatorname{Res}_{F/F_0} \mathrm{GL}_{n-1}, \quad H_2' := \mathrm{GL}_{n-1}\times\mathrm{GL}_n, \quad H_{1,2}' := H_1' \times H_2'. \end{gather*}

Next let $W$ be a non-degenerate $F/F_0$-hermitian space of dimension $n \geq 2$. We fix a non-isotropic vector $u \in W$, which we call the special vector. We denote by $W^{\flat }$ the orthogonal complement of $u$ in $W$. We define another four algebraic groups over $F_0$:

\begin{align*} G &:= \mathrm{U}(W), \quad H := \mathrm{U}(W^{\flat}),\\ G_W&:= H \times G, \quad H_W := H \times H. \end{align*}

We systematically use the symbol $c$ to denote the similitude factor of a point on a unitary similitude group. We further define the following algebraic groups over $\mathbb{Q}$:

\begin{align*} Z^{\mathbb{Q}} &:= \{ z \in \operatorname{Res}_{F/\mathbb{Q}} \mathbb{G}_m \,|\, \operatorname{Nm}_{F/F_0}(z) \in \mathbb{G}_m\},\\ H^{\mathbb{Q}} &:= \{ h \in \operatorname{Res}_{F_0/\mathbb{Q}} \mathrm{GU}(W^{\flat}) \,|\, c(h)\in \mathbb{G}_m \},\\ G^{\mathbb{Q}} &:= \{ g \in \operatorname{Res}_{F_0/\mathbb{Q}} \mathrm{GU}(W) \,|\, c(g)\in \mathbb{G}_m \},\\ \widetilde H &:= Z^{\mathbb{Q}} \times_{\mathbb{G}_m} H^{\mathbb{Q}} = \{(z,h) \in Z^{\mathbb{Q}} \times H^{\mathbb{Q}} \,|\, \operatorname{Nm}_{F/F_0}(z) = c(h)\},\\ \widetilde G &:= Z^{\mathbb{Q}} \times_{\mathbb{G}_m} G^{\mathbb{Q}} = \{(z,g) \in Z^{\mathbb{Q}} \times G^{\mathbb{Q}} \,|\, \operatorname{Nm}_{F/F_0}(z) = c(g)\},\\ \widetilde{HG} &:= \widetilde H \times_{Z^{\mathbb{Q}}} \widetilde G = Z^{\mathbb{Q}} \times_{\mathbb{G}_m} H^{\mathbb{Q}} \times_{\mathbb{G}_m} G^{\mathbb{Q}} \\ &\phantom{:}= \{(z,h,g) \in Z^{\mathbb{Q}} \times H^{\mathbb{Q}} \times G^{\mathbb{Q}} \,|\, \operatorname{Nm}_{F/F_0}(z) = c(h) = c(g)\}. \end{align*}

Note that $Z^{\mathbb{Q}}$ is naturally a central subgroup of $H^{\mathbb{Q}}$ and $G^{\mathbb{Q}}$, and these inclusions give rise to product decompositions

(2.1)

We also record that the decomposition $W = W^{\flat } \oplus Fu$ gives rise to natural closed embeddings of algebraic groups:

(2.2)

Thus, in terms of the product decompositions in (2.1), the embeddings $\widetilde H {\hookrightarrow} \widetilde G$ and $\widetilde H {\hookrightarrow} \widetilde {HG}$ in (

) are obtained by applying the functor $Z^{\mathbb{Q}} \times -$ to the embeddings $\operatorname {Res}_{F_0/\mathbb{Q}} H {\hookrightarrow} \operatorname {Res}_{F_0/\mathbb{Q}} G$ and $\operatorname {Res}_{F_0/\mathbb{Q}} H {\hookrightarrow} \operatorname {Res}_{F_0/\mathbb{Q}} (H \times G)$, respectively.

2.2 Orbit matching

The following lemma is obvious.

Lemma 2.1 The natural projections in (2.1) induce isomorphisms

\[ \widetilde G / \widetilde H {\tilde{\to}} \operatorname{Res}_{F_0/\mathbb{Q}} G/ \operatorname{Res}_{F_0/\mathbb{Q}} H \]

and

\[ \widetilde H \backslash \widetilde{HG} / \widetilde H {\tilde{\to}} \operatorname{Res}_{F_0/\mathbb{Q}} H \backslash \operatorname{Res}_{F_0/\mathbb{Q}} G_W / \operatorname{Res}_{F_0/\mathbb{Q}} H. \]

There is a natural injection of orbit spaces of regular semisimple elements,

\[ H(F_{0, v})\backslash G_W(F_{0, v})_\mathrm{rs}/H(F_{0, v}) {\hookrightarrow} H'_1(F_{0,v})\backslash G'(F_{0, v})_\mathrm{rs}/H'_2(F_{0, v}), \]

for any place $v$ of $F_0$ (see [Reference Rapoport, Smithling and ZhangRSZ17, § 2] for the case where $v$ is a non-archimedean place not split in $F$; the case of archimedean places is completely analogous). If $v$ is split in $F$, we define matching as in [Reference ZhangZha14a, § 2]. In brief, we identify $H(F_{0,v})$ with $\mathrm {GL}_{n-1}(F_{0,v})$ and $G(F_{0,v})$ with $\mathrm {GL}_{n}(F_{0,v})$. This gives a natural way of matching regular semisimple elements, a homogeneous version of the matching relation of [Reference ZhangZha12a].

Using Lemma 2.1, we obtain an injection for every prime number $p$,

(2.3)\begin{equation} \widetilde H(\mathbb{Q}_p) \backslash \widetilde{HG}(\mathbb{Q}_p)_\mathrm{rs} / \widetilde H(\mathbb{Q}_p) {\hookrightarrow} \prod_{v\mid p} H'_1(F_{0, v})\backslash G'(F_{0, v})_\mathrm{rs}/ H'_2(F_{0, v}). \end{equation}

3.1 The Shimura data

In this subsection we define Shimura data for some of the groups introduced in § 2. We assume that the hermitian space $W$ has the following signatures at the archimedean places of $F_0$: for a distinguished element $\varphi _0 \in \Phi$ the signature of $W_{\varphi _0}$ is $(1,n-1)$, and for all other $\varphi \in \Phi$ the signature of $W_\varphi$ is $(0,n)$. We also assume that the special vector $u$ is totally negative, that is, that $(u, u)_\varphi <0$ for all $\varphi$.

We first define Shimura data $(G^{\mathbb{Q}}, \{h_{G^{\mathbb{Q}}}\})$ and $(H^{\mathbb{Q}}, \{h_{ H^{\mathbb{Q}}}\})$ (cf. [Reference Pappas and RapoportPR09, § 1.1]). Using the canonical inclusions $G^{\mathbb{Q}}_\mathbb{R}\subset \prod _{\varphi \in \Phi } \mathrm {GU}(W_\varphi )$ and $H^{\mathbb{Q}}_\mathbb{R}\subset \prod _{\varphi \in \Phi } \mathrm {GU}(W^{\flat }_\varphi )$, it suffices to define the components $h_{G^{\mathbb{Q}}, \varphi }$ of $h_{G^{\mathbb{Q}}}$ and $h_{H^{\mathbb{Q}}, \varphi }$ of $h_{H^{\mathbb{Q}}}$. We define matrices

\[ J_\varphi := \begin{cases} \operatorname{diag}(1, (-1)^{(n-1)}), & \varphi=\varphi_0,\\ \operatorname{diag}( -1, -1, \dotsc, -1), & \varphi\in\Phi\smallsetminus\{\varphi_0\}, \end{cases} \]

and we define the matrix $J_\varphi ^{\flat }$ by removing a $-1$ from $J_\varphi$. We may then choose bases $W_\varphi \simeq \mathbb{C}^{n}$ and $W^{\flat }_\varphi \simeq \mathbb{C}^{n-1}$ such that $u \otimes 1 \in W_\varphi$ identifies with a multiple of $(0^{(n-1)},1)$, such that the inclusion $W_\varphi ^{\flat } \subset W_\varphi$ is compatible with the inclusion $\mathbb{C}^{n-1} \subset \mathbb{C}^{n-1} \oplus \mathbb{C} = \mathbb{C}^{n}$, and such that the hermitian forms on $W_\varphi$ and $W_\varphi ^{\flat }$ have respective normal forms

\[ (x, y)_\varphi = {^t}{x} J_\varphi \overline y \quad\text{and}\quad (x^\flat, y^\flat)_\varphi = {^t}{x}{^\flat} J^\flat_\varphi \overline y^\flat. \]

We then define the component maps

\[ h_{G^{\mathbb{Q}},\varphi}\colon \mathbb{C}^{\times}\longrightarrow \mathrm{GU}(W_\varphi)(\mathbb{R}) \quad\text{and}\quad h_{H^{\mathbb{Q}},\varphi}\colon \mathbb{C}^{\times}\longrightarrow \mathrm{GU}(W_\varphi^{\flat})(\mathbb{R}) \]

to be induced by the respective $\mathbb{R}$-algebra homomorphisms

By definition of $\Phi$, the form $x,y \mapsto \operatorname {tr}_{\mathbb{C}/\mathbb{R}} \varphi (\sqrt \Delta ){{}^{-1}}(h_{G^{\mathbb{Q}},\varphi }(\sqrt {-1}) x, y)_\varphi$ is symmetric and positive definite on $W_\varphi$ for each $\varphi \in \Phi$, and similarly for $W^{\flat }_\varphi$.

We next define Shimura data $(\widetilde H, \{h_{\widetilde H}\})$, $(\widetilde G, \{h_{\widetilde G}\})$, and $(\widetilde {HG}, \{h_{\widetilde {HG}}\})$. For this, note that $\Phi$ induces an identification

\[ Z^{\mathbb{Q}}(\mathbb{R}) \cong \{(z_\varphi)\in (\mathbb{C}^{\times})^{\Phi} \,|\, |z_\varphi| = |z_{\varphi'}| \ \text{for all}\ \varphi,\varphi' \in \Phi \}. \]

In terms of this identification, we define $h_{Z^{\mathbb{Q}}}\colon \mathbb{C}^{\times } \to Z^{\mathbb{Q}}(\mathbb{R})$ to be the diagonal embedding, pre-composed with complex conjugation. We then obtain the desired Shimura data by defining the Shimura homomorphisms

\[ h_{\widetilde H}\colon \mathbb{C}^{\times} {\buildrel {(h_{Z^{\mathbb Q}},h_{H^{\mathbb Q}})} \over \longrightarrow} \widetilde H(\mathbb{R}), \quad h_{\widetilde G}\colon \mathbb{C}^{\times} {\buildrel {(h_{Z^{\mathbb Q}},h_{G^{\mathbb Q}})} \over \longrightarrow} \widetilde G(\mathbb{R}), \quad h_{\widetilde{HG}}\colon \mathbb{C}^{\times} {\buildrel {(h_{Z^{\mathbb Q}},h_{H^{\mathbb Q}},h_{G^{\mathbb Q}})} \over \longrightarrow} \widetilde{HG}(\mathbb{R}). \]

It is easy to see that $(\widetilde H, \{h_{\widetilde H}\})$, $(\widetilde G, \{h_{\widetilde G}\})$, and $(\widetilde {HG}, \{h_{\widetilde {HG}}\})$ have common reflex field $E \subset \mathbb{C}$ characterized by

(3.1)\begin{equation} \operatorname{Aut}(\mathbb{C}/E) = \{\sigma\in\operatorname{Aut}(\mathbb{C})\mid \sigma \circ\Phi=\Phi \text{ and } \sigma\circ\varphi_0=\varphi_0 \}. \end{equation}

We therefore obtain canonical models over $E$ of the Shimura varieties

\[ \mathrm{Sh}_{K_{\widetilde H}}(\widetilde H, \{h_{\widetilde H}\}), \quad \mathrm{Sh}_{K_{\widetilde G}}(\widetilde G, \{h_{\widetilde G}\}), \quad \mathrm{Sh}_{K_{\widetilde{HG}}}(\widetilde{HG}, \{h_{\widetilde{HG}}\}) , \]

where ${K_{\widetilde H}}$ (respectively ${K_{\widetilde G}}$, ${K_{\widetilde {HG}}}$) varies through the open compact subgroups of $\widetilde H(\mathbb{A}_f)$ (respectively $\widetilde G(\mathbb{A}_f)$, $\widetilde {HG}(\mathbb{A}_f)$).

The morphisms (2.2) are obviously compatible with the Shimura data $\{h_{\widetilde H}\}$ and $\{h_{\widetilde G}\}$, (respectively, $\{h_{\widetilde H}\}$ and $\{h_{\widetilde {HG}}\}$). We therefore obtain injective morphisms of Shimura varieties, that is, injective morphisms of pro-varieties, in the sense of [Reference DeligneDel71, Proposition 1.15]:

(3.2)\begin{equation} \mathrm{Sh}(\widetilde H, \{h_{\widetilde H}\}) \hookrightarrow \mathrm{Sh}(\widetilde G, \{h_{\widetilde G}\}) \quad\text{and}\quad \mathrm{Sh}(\widetilde H, \{h_{\widetilde H}\}) \hookrightarrow \mathrm{Sh}(\widetilde{HG}, \{h_{\widetilde{HG}}\}). \end{equation}

Remark 3.2 The above Shimura varieties are related to other Shimura varieties as follows.

(i) The pair $(Z^{\mathbb{Q}}, \{h_{Z^{\mathbb{Q}}}\})$ is a Shimura datum, and there are morphisms of Shimura data

\[ (\widetilde H, \{h_{\widetilde H}\})\longrightarrow (Z^{\mathbb{Q}}, \{h_{Z^{\mathbb{Q}}}\}), \quad (\widetilde G, \{h_{\widetilde G}\})\longrightarrow (Z^{\mathbb{Q}}, \{h_{Z^{\mathbb{Q}}}\}), \quad (\widetilde{HG}, \{h_{\widetilde{HG}}\})\longrightarrow (Z^{\mathbb{Q}}, \{h_{Z^{\mathbb{Q}}}\}) \]

induced by the natural projections to $Z^{\mathbb{Q}}$. These induce morphisms of Shimura varieties

(3.3)\begin{equation} \begin{gathered} \mathrm{Sh}(\widetilde H, \{h_{\widetilde H}\})\longrightarrow \mathrm{Sh}(Z^{\mathbb{Q}}, \{h_{Z^{\mathbb{Q}}}\}),\\ \mathrm{Sh}(\widetilde G, \{h_{\widetilde G}\})\longrightarrow \mathrm{Sh}(Z^{\mathbb{Q}}, \{h_{Z^{\mathbb{Q}}}\}),\\ \mathrm{Sh}(\widetilde{HG}, \{h_{\widetilde{HG}}\})\longrightarrow \mathrm{Sh}(Z^{\mathbb{Q}}, \{h_{Z^{\mathbb{Q}}}\}), \end{gathered} \end{equation}

which identify

\[ \mathrm{Sh}(\widetilde{HG}, \{h_{\widetilde{HG}}\}) \cong \mathrm{Sh}(\widetilde H, \{h_{\widetilde H}\}) \times_{\mathrm{Sh}(Z^{\mathbb{Q}}, \{h_{Z^{\mathbb{Q}}}\})} \mathrm{Sh}(\widetilde G, \{h_{\widetilde G}\}). \]

The reflex field of $(Z^{\mathbb{Q}}, \{h_{Z^{\mathbb{Q}}}\})$ is the subfield $E_\Phi$ of $E$.

(ii) There are morphisms of Shimura data

\[ (\widetilde H, \{h_{\widetilde H}\})\longrightarrow (H^{\mathbb{Q}}, \{h_{H^{\mathbb{Q}}}\}) \quad\text{and}\quad (\widetilde G, \{h_{\widetilde G}\})\longrightarrow (G^{\mathbb{Q}}, \{h_{G^{\mathbb{Q}}}\}), \]

both induced by the natural projections, which induce morphisms of Shimura varieties

\[ \mathrm{Sh}(\widetilde H, \{h_{\widetilde H}\})\longrightarrow \mathrm{Sh}(H^{\mathbb{Q}}, \{h_{H^{\mathbb{Q}}}\}) \quad\text{and}\quad \mathrm{Sh}(\widetilde G, \{h_{\widetilde G}\})\longrightarrow \mathrm{Sh}(G^{\mathbb{Q}}, \{h_{G^{\mathbb{Q}}}\}). \]

(iii) One may also introduce Shimura data

\[ (\operatorname{Res}_{F_0/\mathbb{Q}} H, \{h_{H}\}), \quad (\operatorname{Res}_{F_0/\mathbb{Q}} G, \{h_{ G}\}), \quad (\operatorname{Res}_{F_0/\mathbb{Q}}(H\times G), \{h_{H\times G}\}), \]

where $h_H$ (respectively $h_G$, $h_{H \times G}$) is defined by composing $h_{\widetilde H}$ (respectively $h_{\widetilde G}$, $h_{\widetilde {HG}}$) with the projection to the second factor in (2.1). Note that the restrictions of $h_H$, $h_G$, and $h_{H\times G}$ to the subgroup $\mathbb{R}^{\times }$ of $\mathbb{C}^{\times }$ are trivial. In particular, the corresponding Shimura varieties are not of PEL type. These are the Shimura varieties that appear in Gan, Gross, and Prasad [Reference Gan, Gross and PrasadGGP12, § 27]. The product decompositions in (2.1) induce product decompositions of Shimura data,

(3.4)\begin{align} (\widetilde H, \{h_{\widetilde H}\}) &= (Z^{\mathbb{Q}},\{h_{Z^{\mathbb{Q}}}\}) \times (\operatorname{Res}_{F/F_0} H,\{h_H\}),\notag\\ (\widetilde G, \{h_{\widetilde G}\}) &= (Z^{\mathbb{Q}},\{h_{Z^{\mathbb{Q}}}\}) \times (\operatorname{Res}_{F/F_0} G,\{h_G\}),\\ (\widetilde{HG}, \{h_{\widetilde{HG}}\}) &= (Z^{\mathbb{Q}},\{h_{Z^{\mathbb{Q}}}\}) \times (\operatorname{Res}_{F_0/\mathbb{Q}}(H\times G), \{h_{H\times G}\}).\notag \end{align}

In [Reference Gan, Gross and PrasadGGP12, § 27], the Shimura variety $\mathrm {Sh}(\operatorname {Res}_{F_0/\mathbb{Q}}G, \{h_{G}\})$ is considered over its reflex field, which is $F$, embedded into $\mathbb{C}$ via $\varphi _0$. By contrast, in the present paper, we consider the Shimura variety $\mathrm {Sh}(\widetilde G, \{h_{\widetilde G}\})$ over $E$, which is the join of the reflex fields of the two factors in the product decomposition (3.4). The natural projections then induce morphisms of Shimura varieties,

(3.5)\begin{equation} \begin{gathered} \mathrm{Sh}(\widetilde H, \{h_{\widetilde H}\})\longrightarrow \mathrm{Sh}(\operatorname{Res}_{F_0/\mathbb{Q}}H, \{h_H\}),\\ \mathrm{Sh}(\widetilde G, \{h_{\widetilde G}\})\longrightarrow \mathrm{Sh}(\operatorname{Res}_{F_0/\mathbb{Q}}G, \{h_{G}\}),\\ \mathrm{Sh}(\widetilde{HG}, \{h_{\widetilde{HG}}\})\longrightarrow \mathrm{Sh}(\operatorname{Res}_{F_0/\mathbb{Q}}(H\times G), \{h_{H\times G}\}). \end{gathered} \end{equation}

Remark 3.3 Let us finally make our Shimura varieties more concrete. We consider the case of $\mathrm {Sh}(\widetilde G, \{h_{\widetilde G}\})$; the other Shimura varieties are analogous. In terms of the product decomposition $\widetilde G_\mathbb{R} \cong Z^{\mathbb{Q}}_\mathbb{R} \times \prod _{\varphi \in \Phi } \mathrm {U}(W_\varphi )$ induced by (2.1), the conjugacy class $\{h_{\widetilde G}\}$ is the product of $\{h_{Z^{\mathbb{Q}}}\}$ with the $\mathrm {U}(W_\varphi )(\mathbb{R})$-conjugacy class $\{h_{G,\varphi }\}$ for each $\varphi \in \Phi$, where $h_{G,\varphi }$ denotes the $\varphi$-component of the cocharacter $h_G$ defined in Remark 3.2(iii). The conjugacy class $\{h_{Z^{\mathbb{Q}}}\}$ consists of a single element; so does $\{h_{G,\varphi }\}$ for $\varphi \neq \varphi _0$, since in this case $h_{G,\varphi }$ is the trivial cocharacter. For $\varphi = \varphi _0$, in terms of the basis for $W_{\varphi _0}$ chosen above, $h_{G,\varphi _0}$ is the cocharacter

\[ h_{G,\varphi_0} \colon z \mapsto \operatorname{diag}(z/\overline z, 1,\ldots, 1). \]

The conjugacy class $\{h_{G,\varphi _0}\}$ then identifies with the open subset ${{\mathcal {D}}}_{\varphi _0} \subset \mathbb{P}(W_{\varphi _0})(\mathbb{C})$ of positive-definite lines for the hermitian form (send $h \in \{h_{G,\varphi _0}\}$ to the $-1$-eigenspace of $h({\sqrt {-1}})$; we remark that ${{\mathcal {D}}}_{\varphi _0}$ is also isomorphic to the open unit ball in $\mathbb{C}^{n-1}$). Thus, for $K_{\widetilde G} \subset \widetilde G(\mathbb{A}_f)$ an open compact subgroup, we obtain the presentation

\[ \mathrm{Sh}_{K_{\widetilde G}}(\widetilde G, \{h_{\widetilde G}\}) (\mathbb{C}) = \widetilde G(\mathbb{Q}) \backslash [{{\mathcal{D}}}_{\varphi_0}\times \widetilde G(\mathbb{A}_f)/K_{\widetilde G}] , \]

where the action of $\widetilde G(\mathbb{Q})$ is diagonal by the translation action on $\widetilde G(\mathbb{A}_f)$ and by the action on ${{\mathcal {D}}}_{\varphi _0}$ given via

\[ \widetilde G(\mathbb{Q})\longrightarrow \widetilde G_{\mathrm{ad}}(\mathbb{R})\longrightarrow \textrm{PU}(W_{\varphi_0})(\mathbb{R}). \]

3.2 The moduli problem over $E$

In this subsection we define moduli problems on the category of schemes over $\operatorname {Spec} E$ for the three Shimura varieties above. Since these are almost identical in each case, let us do this for $\mathrm {Sh}(\widetilde G, \{h_{\widetilde G}\})$, and only indicate briefly the modifications needed for the other two (mostly for $\mathrm {Sh}(\widetilde {HG}, \{h_{\widetilde {HG}}\})$). We will only consider open compact subgroups ${K_{\widetilde G}}\subset \widetilde G(\mathbb{A}_f)$ which, with respect to the product decomposition (2.1), are of the form

(3.6)\begin{equation} K_{\widetilde G}= K_{Z^{\mathbb{Q}}} \times K_G, \end{equation}

where $K_G\subset G(\mathbb{A}_{F_0,f})$ is an open compact subgroup and $K_{Z^{\mathbb{Q}}}\subset {Z^{\mathbb{Q}}}(\mathbb{A}_f)$ is the unique maximal compact subgroup

(3.7)\begin{equation} K_{Z^{\mathbb{Q}}} := Z^{\mathbb{Q}}(\widehat{\mathbb{Z}}) = \{z\in (O_F\otimes \widehat{\mathbb{Z}})^{\times} \,|\, \operatorname{Nm}_{F/F_0}(z)\in \widehat{\mathbb{Z}}^{\times}\}. \end{equation}

(Note that $Z^{\mathbb{Q}}$ is defined over $\operatorname {Spec} \mathbb{Z}$ in an obvious way.)

Before doing this, let us first introduce an auxiliary moduli problem $M_0$ over $E$. In fact, for use in the construction of integral models later, we will define a moduli problem $\mathcal {M}_0$ over $O_E$ whose generic fiber will be $M_0$. For a locally noetherian $O_E$-scheme $S$, we define $\mathcal {M}_0(S)$ to be the groupoid of triples $(A_0, \iota _0, \lambda _0)$, where

• • $A_0$ is an abelian variety over $S$ with an $O_F$-action $\iota _0\colon O_F \to \operatorname {End}(A_0)$, which satisfies the Kottwitz condition of signature $((0, 1)_{\varphi \in \Phi })$, that is,

  (3.8)\begin{equation} \operatorname{char}(\iota(a)\mid\operatorname{Lie} A_0) = \prod_{\varphi\in\Phi}(T-\overline\varphi(a)) \quad\text{for all}\ a\in O_F; \end{equation}
  and

• • $\lambda _0$ is a principal polarization of $A_0$ whose Rosati involution induces on $O_F$, via $\iota _0$, the non-trivial Galois automorphism of $F/F_0$.

A morphism between two objects $(A_0,\iota _0,\lambda _0)$ and $(A'_0,\iota '_0,\lambda '_0)$ is an $O_F$-linear isomorphism $\mu _0\colon A_0 \to A'_0$ under which $\lambda _0'$ pulls back to $\lambda _0$. Then $\mathcal {M}_0$ is a Deligne–Mumford (DM) stack, finite and étale over $\operatorname {Spec} O_E$ (cf. [Reference HowardHow12, Proposition 3.1.2]).Footnote ⁶ (In fact, in [Reference HowardHow12, Proposition 3.1.2], $\mathcal {M}_0$ is defined over the ring of integers $O_{E_\Phi }$ in the reflex field of $\Phi$, which is contained in $O_E$; cf. Remark 3.2(i).) We let $M_0$ denote the generic fiber of $\mathcal {M}_0$.

Unfortunately, it may happen that $\mathcal {M}_0$ is empty. In order to circumvent this problem, we also introduce the following variant of $\mathcal {M}_0$ (cf. [Reference HowardHow12, Definition 3.1.1]). Fix a non-zero ideal $\mathfrak {a}$ of $O_{F_0}$. Then we introduce the DM stack $\mathcal {M}_0^{\mathfrak{a}}$ of triples $(A_0, \iota _0, \lambda _0)$ as before, except that we replace the condition that $\lambda _0$ is principal by the condition that $\ker \lambda _0=A_0[\mathfrak {a}]$. Then, again, $\mathcal {M}_0^{\mathfrak{a}}$ is finite and étale over $\operatorname {Spec} O_E$ (cf. [Reference HowardHow12, Proposition 3.1.2]).

If $\mathcal {M}_0^{\mathfrak{a}}$ is non-empty, then the complex fiber $\mathcal {M}^{\mathfrak{a}}_0 \otimes _{O_E} \mathbb{C}$ is isomorphic to a finite number of copies of $\mathrm {Sh}_{K_{Z^{\mathbb{Q}}}}(Z^{\mathbb{Q}}, \{h_{Z^{\mathbb{Q}}}\})$. More precisely, let $\mathcal {L}_\Phi ^{\mathfrak{a}}$ be the set of isomorphism classes of pairs $(\Lambda _0,\langle \text {,}\rangle _0)$ consisting of a locally free $O_F$-module $\Lambda _0$ of rank $1$ equipped with a non-degenerate alternating form $\langle \text {,}\rangle _0\colon \Lambda _0 \times \Lambda _0 \to \mathbb{Z}$ such that $\langle ax, y \rangle _0 = \langle x, \overline a y\rangle _0$ for all $x,y \in \Lambda _0$ and $a \in O_F$, such that $x \mapsto \langle \sqrt \Delta x, x \rangle _0$ is a negative-definite quadratic form on $\Lambda _0$, and such that the dual lattice $\Lambda _0^{\vee }$ of $\Lambda _0$ inside $\Lambda _0 \otimes _\mathbb{Z} \mathbb{Q}$ equals $\mathfrak {a}{{}^{-1}} \Lambda _0$. Then $\mathcal {L}_\Phi ^{\mathfrak{a}}$ is a finite set.Footnote ⁷ If $(\Lambda _0,\langle \text {,}\rangle _0)$ defines a class in $\mathcal {L}_\Phi ^{\mathfrak{a}}$, then using $\Phi$ to identify $F \otimes _\mathbb{Q} \mathbb{R} {\tilde{\to}} \mathbb{C}^{\Phi }$, and hence to define a complex structure on $\Lambda _0 \otimes _\mathbb{Z} \mathbb{R}$, the quotient $(\Lambda _0 \otimes _\mathbb{Z} \mathbb{R}) / \Lambda _0$ is a complex torus which defines a $\mathbb{C}$-point on $\mathcal {M}_0^{\mathfrak{a}}$. On the other hand, given a $\mathbb{C}$-point $(A_0,\iota _0,\lambda _0)$ on $\mathcal {M}_0^{\mathfrak{a}}$, the first homology group $H_1(A_0(\mathbb{C}),\mathbb{Z})$, endowed with its Riemann form induced by $\lambda _0$, defines a class in $\mathcal {L}_\Phi ^{\mathfrak{a}}$. In this way $\mathcal {L}_\Phi ^{\mathfrak{a}}$ is in bijection with the isomorphism classes of objects in $\mathcal {M}_0^{\mathfrak{a}}(\mathbb{C})$.

These inverse processes give rise to a decomposition of $\mathcal {M}_0^{\mathfrak{a}}$ into a union of Shimura varieties, in terms of the following equivalence relation on $\mathcal {L}_\Phi ^{\mathfrak{a}}$: define $\Lambda _0 \sim \Lambda _0'$ if $\Lambda _0 \otimes _\mathbb{Z} \widehat{\mathbb{Z}}$ and $\Lambda _0' \otimes _\mathbb{Z} \widehat{\mathbb{Z}}$ are $\widehat O_F$-linearly similar up to a factor in $\widehat{\mathbb{Z}}^{\times }$, and if $\Lambda _0 \otimes _\mathbb{Z} \mathbb{Q}$ and $\Lambda _0' \otimes _\mathbb{Z} \mathbb{Q}$ are $F$-linearly similar up to a factor in $\mathbb{Q}^{\times }$. We have the following lemma.

Lemma 3.4 The stack $\mathcal {M}_0^{\mathfrak{a}}$ admits a decomposition into open and closed substacks,

(3.9)\begin{equation} \mathcal{M}_0^{\mathfrak{a}} = \coprod_{\xi \in \mathcal{L}_\Phi^{\mathfrak{a}}/\sim} \mathcal{M}_0^{\mathfrak{a},\xi}, \end{equation}

induced by sending an object $(A_0,\iota _0,\lambda _0)$ in $\mathcal {M}_0^{\mathfrak{a}}(\mathbb{C})$ to the $\sim$-class of $H_1(A_0(\mathbb{C}),\mathbb{Z})$ endowed with its Riemann form.

Proof. Since $\mathcal {M}_0^{\mathfrak{a}}$ is finite and étale over $\operatorname {Spec} O_E$, it suffices to demonstrate the asserted decomposition in the generic fiber $M_0^{\mathfrak{a}}$,

(3.10)\begin{equation} M_0^{\mathfrak{a}} = \coprod_{\xi \in \mathcal{L}_\Phi^{\mathfrak{a}}/\sim} M_0^{\mathfrak{a},\xi}. \end{equation}

To establish (3.10), again by étaleness, it suffices to show that the $\sim$-class of $H_1(A_0(\mathbb{C}),\mathbb{Z})$ is constant on the $\operatorname {Aut}(\mathbb{C}/E)$-orbit of each object $(A_0,\iota _0,\lambda _0) \in M_0^{\mathfrak{a}}(\mathbb{C})$. So let $\sigma \in \operatorname {Aut}(\mathbb{C}/E)$. Then $\sigma$ identifies the Tate modules $\widehat T (A_0) {\tilde{\to}} \widehat T (A_0^{\sigma })$ compatibly with the Weil pairings on the two sides up to the similitude factor given by the image of $\sigma$ under the cyclotomic character $\operatorname {Aut}(\mathbb{C}) \to \widehat{\mathbb{Z}}^{\times }$. Hence, by the compatibility between singular homology and the Tate module, $H_1(A_0(\mathbb{C}),\widehat{\mathbb{Z}})$ and $H_1(A_0^{\sigma }(\mathbb{C}),\widehat{\mathbb{Z}})$ are $\widehat O_F$-linearly similar up to a factor in $\widehat{\mathbb{Z}}^{\times }$. On the other hand, the CM abelian variety $A_0$ is isomorphic to one defined over the algebraic closure of $E$ in $\mathbb{C}$, and hence by [Reference Chai, Conrad and OortCCO14, Theorem A.2.3.5] there exists a quasi-isogeny of complex abelian varieties $A_0 \to A_0^{\sigma }$ pulling $\lambda _0^{\sigma }$ back to a $\mathbb{Q}_{>0}$-multiple of $\lambda _0$. Hence $H_1(A_0(\mathbb{C}),\mathbb{Q})$ and $H_1(A_0^{\sigma }(\mathbb{C}),\mathbb{Q})$ are $F$-linearly similar up to a factor in $\mathbb{Q}^{\times }$, as desired.

If $\mathcal {M}_0^{\mathfrak{a}} \neq \emptyset$, then for fixed $\xi \in \mathcal {L}_\Phi ^{\mathfrak{a}}/{\sim }$, the complex fiber $\mathcal {M}_0^{\mathfrak {a},\xi } \otimes _{O_E} \mathbb{C}$ of the summand in (3.9) is canonically isomorphic to the Shimura variety $\mathrm {Sh}_{K_{Z^{\mathbb{Q}}}}(Z^{\mathbb{Q}}, \{h_{Z^{\mathbb{Q}}}\})$; this can be checked similarly to [Reference Kudla and RapoportKR14, Propositions 4.3, 4.4], or see [Reference DeligneDel71, (4.12), (4.18)–(4.20)].

In the following, we fix an ideal $\mathfrak {a}$ such that $\mathcal {M}_0^{\mathfrak{a}}$ is non-empty. We continue to denote the generic fiber of this stack by $M_0^{\mathfrak{a}}$. We also fix $\xi \in \mathcal {L}_\Phi ^{\mathfrak{a}}/{\sim }$. We now define a groupoid $M_{K_{\widetilde G}}(\widetilde G)$ fibered over the category of locally noetherian schemes over $E$. Here, for ease of notation, we have suppressed the ideal $\mathfrak {a}$ and the element $\xi$. For such a scheme $S$, the objects of $M_{K_{G}}(\widetilde G)(S)$ are collections $(A_0,\iota _0,\lambda _0,A,\iota ,\lambda ,\overline \eta )$, where

• • $(A_0, \iota _0, \lambda _0)$ is an object of $M^{\mathfrak {a}, \xi }_0(S)$;

• • $A$ is an abelian scheme over $S$ with an $F$-action $\iota \colon F\to \operatorname {End}^{\circ }(A)$ satisfying the Kottwitz condition of signature $((1, n-1)_{\varphi _0}, (0, n)_{\varphi \in \Phi \smallsetminus \{\varphi _0\}})$, that is,

  (3.11)\begin{equation} \operatorname{char}(\iota(a)\mid\operatorname{Lie} A) = (T-\varphi_0(a))(T-\varphi_0(\bar a))^{n-1}\prod_{\varphi\in\Phi\smallsetminus \{\varphi_0\}}(T-\varphi(\bar a))^{n} \quad\text{for all}\ a\in O_F; \end{equation}

• • $\lambda$ is a polarization $A \to A^{\vee }$ whose Rosati involution induces on $F$, via $\iota$, the non-trivial Galois automorphism of $F/F_0$; and

• • $\overline \eta$ is a ${K_{\widetilde G}}$-level structure, by which we mean a ${K_{G}}$-orbit (equivalently, a ${K_{\widetilde G}}$-orbit, where $K_{\widetilde G}$ acts through its projection $K_{\widetilde G}\to K_G$) of $\mathbb{A}_{F, f}$-linear isometries

  (3.12)\begin{equation} \eta\colon \widehat V(A_0, A)\simeq -W\otimes_F\mathbb{A}_{F, f}. \end{equation}

Here we regard the right-hand side of (3.12) as a constant smooth $\mathbb{A}_{F,f}$-sheaf on $S$, and

\[ \widehat V(A_0,A) := \operatorname{Hom}_{F}(\widehat V(A_0), \widehat V(A)) , \]

endowed with its natural $\mathbb{A}_{F, f}$-valued hermitian form $h_A$, defined by

(3.13)\begin{equation} h_A(x, y) := \lambda_0^{-1}\circ y^{\vee}\circ\lambda\circ x\in\operatorname{End}_{\mathbb{A}_{F, f}}(\widehat V(A_0))=\mathbb{A}_{F, f} \end{equation}

(cf. [Reference Kudla and RapoportKR14, § 2.3]). Over each connected component of $S$, upon choosing a geometric point $\overline s \to S$, we may view $\widehat V (A_0, A)$ as the space $\operatorname {Hom}_{\mathbb{A}_{F,f}}(\widehat V(A_{0,\overline s}), \widehat V(A_{\overline s}))$ endowed with its $\pi _1(S,\overline s)$-action, and in this way we require the orbit $\overline \eta$ to be $\pi _1(S,\overline s)$-invariant (cf. [Reference Kudla and RapoportKR14, Remark 4.2]). We note that the notion of level structure is independent of the choice of $\overline s$ on each connected component of $S$.

A morphism between two objects $(A_0,\iota _0,\lambda _0,A,\iota ,\lambda ,\overline \eta )$ and $(A'_0,\iota '_0,\lambda '_0,A',\iota ',\lambda ',\overline \eta ')$ is given by an isomorphism $\mu _0\colon (A_0, \iota _0, \lambda _0) {\tilde{\to}} (A'_0, \iota '_0, \lambda '_0)$ in $M_0^{\mathfrak {a},\xi }(S)$ and an $F$-linear quasi-isogeny $\mu \colon A \to A'$ pulling $\lambda '$ back to $\lambda$ and $\overline \eta '$ back to $\overline \eta$.

Remark 3.6 (i) Let $r \colon \operatorname {Hom}(F,\mathbb{C}) \to \{0,1,n-1,n\}$, $\varphi \mapsto r_\varphi$ be the function defined by

(3.14)\begin{equation} r_\varphi := \begin{cases} 1, & \varphi = \varphi_0,\\ 0, & \varphi \in \Phi \smallsetminus \{\varphi_0\},\\ n-r_{\overline\varphi}, & \varphi \notin \Phi. \end{cases} \end{equation}

Then the Kottwitz condition (3.11) is

\[ \operatorname{char}(\iota(a)\mid\operatorname{Lie} A) = \prod_{\varphi\in\operatorname{Hom}(F,\mathbb{C})} (T-\varphi(a))^{r_\varphi} \quad\text{for all}\ a \in O_F \]

(cf. [Reference Rapoport and ZinkRZ17, (8.4)]).

(ii) The intervention of the sign on the right-hand side of (3.12) arises from our conventions on the signatures of $W$ at the archimedean places given at the beginning of § 3.1 and on the signatures in the Kottwitz conditions (3.8) and (3.11); cf. the proof of Proposition 3.7 below. If one took the opposite signatures for $W$ and in the two Kottwitz conditions, then no sign would be needed.

The following proposition is a special case of Deligne's description of Shimura varieties of PEL type.

Proof. We will content ourselves with exhibiting a map, which turns out to be an isomorphism,

\[ M_{K_{\widetilde G}}(\widetilde G)\otimes_{E}\mathbb{C} \longrightarrow \mathrm{Sh}_{K_{\widetilde G}}(\widetilde G, \{h_{\widetilde G}\}). \]

By Remark 3.2(iii), the target is the product of Shimura varieties

\[ \mathrm{Sh}_{K_{Z^{\mathbb{Q}}}}(Z^{\mathbb{Q}},\{h_{Z^{\mathbb{Q}}}\}) \times \mathrm{Sh}_{K_G}(\operatorname{Res}_{F/F_0} G,\{h_G\}). \]

For the map into the first factor, we simply compose the forgetful map $M_{K_{\widetilde G}}(\widetilde G)\otimes _{E}\mathbb{C} \to M_0^{\mathfrak {a}, \xi }\otimes _{E}\mathbb{C}$ with the isomorphism $M_0^{\mathfrak {a}, \xi }\otimes _{E}\mathbb{C} \simeq \mathrm {Sh}_{K_{Z^{\mathbb{Q}}}}(Z^{\mathbb{Q}},\{h_{Z^{\mathbb{Q}}}\})$.

To explain the map into the second factor, let $(A_0,\iota _0,\lambda _0,A,\iota ,\lambda ,\overline \eta )$ be a $\mathbb{C}$-valued point of $M_{K_{\widetilde G}}(\widetilde G)$. Let $\mathcal {H} := H_1(A, \mathbb{Q})$ and $\mathcal {H}_0 := H_1(A_0, \mathbb{Q})$. The polarization $\lambda$ endows $\mathcal {H}$ with a $\mathbb{Q}$-valued alternating form $\langle \text {,}\rangle$ satisfying $\langle \iota (a)x,y\rangle = \langle x, \iota (\overline a)y\rangle$ for all $a \in F$, and such that the induced form $x,y \mapsto \langle \sqrt {-1} \cdot x, y \rangle$ on $\mathcal {H} \otimes _\mathbb{Q} \mathbb{R}$ is symmetric and positive definite, where multiplication by $\sqrt {-1}$ is defined in terms of the right-hand side of the canonical isomorphism $\mathcal {H}_\mathbb{R} \cong \operatorname {Lie} A$. Similarly, $\lambda _0$ endows $\mathcal {H}_0$ with a Riemann form $\langle \text {,}\rangle _0$.

Let $V(A_0, A):=\operatorname {Hom}_F(\mathcal {H}_0, \mathcal {H})$. Then $V(A_0, A)$ is an $F$-vector space of the same dimension as $W$, and we make it into an $F/F_0$-hermitian space by defining the pairing $(\alpha ,\beta )$ to be the composite

where the checks denote $\mathbb{Q}$-linear duals and the last arrow is the inverse of $y \mapsto \langle -, y \rangle _0$; this composite is an $F$-linear endomorphism of the one-dimensional $F$-vector space $\mathcal {H}_0$, and hence identifies with an element in $F$. Clearly $V(A_0, A) \otimes _F \mathbb{A}_{F,f} \cong \widehat V(A_0, A)$ as hermitian spaces. Hence, by the existence of a level structure, $V(A_0, A) \otimes _F \mathbb{A}_{F,f} \simeq -W \otimes _F \mathbb{A}_{F,f}$. Furthermore, it is easy to see that the Kottwitz conditions (3.8) and (3.11) imply that $V(A_0, A)_\varphi$ has signature $(n-1,1)$ if $\varphi =\varphi _0$ and $(n,0)$ if $\varphi \in \Phi \smallsetminus \{\varphi _0\}$. Hence, by the Hasse principle for hermitian spaces, $V(A_0, A)$ and $-W$ are isomorphic. Choose an isometry $j\colon V(A_0, A){\tilde{\to}} -W$. Using the complex structures on $\mathcal {H}_{0,\mathbb{R}}$ and $\mathcal {H}_\mathbb{R}$, let $z \in \mathbb{C}^{\times }$ act on $V(A_0, A)$ by sending the $F$-linear map $\alpha$ to $z\alpha z{{}^{-1}}$. This defines a homomorphism $\mathbb{C}^{\times } \longrightarrow \mathrm {U}(V(A_0, A))(\mathbb{R})$, and composing this with $j_*\colon \mathrm {U}(V(A_0, A))(\mathbb{R}) {\tilde{\to}} \mathrm {U}(-W)(\mathbb{R}) = \mathrm {U}(W)(\mathbb{R})$ gives an element in $\{h_G\}$. The level structure $\overline \eta$ corresponds to an element of $(\operatorname {Res}_{F_0/\mathbb{Q}} G)(\mathbb{A}_f)/K_{G}$, and eliminating the choice of $j$ corresponds to dividing out by the action of $(\operatorname {Res}_{F_0/\mathbb{Q}} G)(\mathbb{Q})$.

An analogous description holds for the model $M_{K_{\widetilde H}}(\widetilde H)$ of the Shimura variety $\mathrm {Sh}_{K_{\widetilde H}}(\widetilde H, \{h_{\widetilde H}\})$ (replace $n$ by $n-1$, and $W$ by $W^{\flat }$).

There is also an analog for the Shimura variety $\mathrm {Sh}_{K_{\widetilde {HG}}}(\widetilde {HG}, \{h_{\widetilde {HG}}\})$. In this case we take the level subgroup to be of the form

(3.15)\begin{equation} K_{\widetilde{HG}} = K_{Z^{\mathbb{Q}}} \times K_H \times K_G, \end{equation}

where as always $K_{Z^{\mathbb{Q}}}$ is the subgroup (3.7), and $K_H \subset H(\mathbb{A}_{F_0,f})$ and $K_G \subset G(\mathbb{A}_{F_0,f})$ are open compact subgroups. The value of the corresponding moduli functor $M_{K_{\widetilde {HG}}}(\widetilde {HG})$ on a locally noetherian scheme $S$ over $E$ is the set of isomorphism classes of tuples

\[ (A_0,\iota_0,\lambda_0,A^{\flat},\iota^{\flat},\lambda^{\flat}, A, \iota,\lambda, \overline{(\eta^{\flat}, \eta)}) , \]

where the last entry is a pair of $\mathbb{A}_{F, f}$-linear isometries

\[ \eta^{\flat} \colon \widehat V(A_0, A^{\flat})\simeq -W^{\flat}\otimes_F\mathbb{A}_{F, f} \quad\text{and}\quad \eta\colon \widehat V(A_0, A)\simeq -W\otimes_F\mathbb{A}_{F, f} , \]

modulo the action of ${K_{H}\times K_G}$. In other words, the moduli functor $M_{K_{\widetilde {HG}}}(\widetilde {HG})$ is simply the fibered product $M_{K_{\widetilde H}}(\widetilde H) \times _{M^{\mathfrak {a},\xi }_0} M_{K_{\widetilde G}}(\widetilde G)$.

In terms of these moduli problems, the injective morphisms (3.2) can be described as follows. After possibly scaling the special vector $u \in W$, we may and do assume that the norm $(u,u)$ is in $O_{F_0}$. Further, assume that ${K_{H}}\subset H(\mathbb{A}_{F_0,f})\cap {K_{ G}}$. Then the first morphism of Shimura varieties in (3.2) arises by base change from $E$ to $\mathbb{C}$ from the functor morphism

(3.16)

Here $\lambda _0(u)= -(u,u) \lambda _0$, which is indeed a polarization by [Reference ZinkZin82, Lemma 1.4] since $-(u,u)$ is totally positive. Furthermore, the isomorphism $\eta \colon \widehat V(A_0,A^{\flat }\times A_0) \simeq -W \otimes _F \mathbb{A}_{F, f} \bmod K_{G}$ is given, with respect to the decomposition

\[ \operatorname{Hom}_F( \widehat V(A_0), \widehat V(A^{\flat} \times A_0)) \cong \operatorname{Hom}_F(\widehat V(A_0), \widehat V(A^{\flat}) \oplus \widehat V(A_0)) \cong \widehat V(A_0,A^{\flat}) \oplus { \mathbb{A}}_{F, f}, \]

by the trivialization

\[ \eta^{\flat}\colon \widehat V(A_0,A^{\flat}) \simeq -W^{\flat} \otimes_F \mathbb{A}_{F, f} \bmod K_{ H} \]

in the first summand, and by identifying the basis element $1$ in the second summand with $u \otimes 1 \in -W \otimes _F \mathbb{A}_{F,f}$. The morphism (

) is finite and unramified.

The second morphism in (3.2) then arises from the graph morphism of (3.16),

(3.17)\begin{equation} M_{K_{\widetilde H}} (\widetilde H) \longrightarrow M_{K_{\widetilde{HG}}}(\widetilde{HG}) = M_{K_{\widetilde H}}(\widetilde H) \times_{M^{\mathfrak{a},\xi}_0} M_{K_{\widetilde G}}(\widetilde G). \end{equation}

The morphism (3.17) is a closed embedding.

4.1 Hyperspecial level at $v_0$

In this case we assume that the place $v_0$ is unramified over $p$, and that $v_0$ either splits in $F$ or is inert in $F$ and the hermitian space $W_{v_0}$ is split. We also assume $p \neq 2$ if there is any $v \in \mathcal {V}_p$ which is non-split in $F$.Footnote ⁸ We will define smooth models over $O_{E,(\nu )}$.

For each $v\in \mathcal {V}_{p}$, choose a vertex lattice $\Lambda _{v}$ in the $F_v/F_{0,v}$-hermitian space $W_{v}$. By our assumptions on $v_0$ in the previous paragraph, we may and do take $\Lambda _{v_0}$ to be self-dual. Recalling the subgroup $K_{\widetilde G} = K_{Z^{\mathbb{Q}}} \times K_G$ from (3.6), we take $K_G$ to be of the form

\[ K_G = K_G^{p} \times K_{G,p}, \]

where $K_G^{p} \subset G(\mathbb{A}_{F_0,f}^{p})$ is arbitrary, and where the level subgroup at $p$ is the product

(4.1)\begin{equation} K_{G,p} :=\prod_{v\in \mathcal{V}_p}K_{G, v} \subset G(F_0\otimes \mathbb{Q}_p)=\prod_{v\in \mathcal{V}_p}G(F_{0, v}), \end{equation}

with $K_{G, v}$ the stabilizer of $\Lambda _v$ in $G(F_{0,v})$.

We formulate a moduli problem over $\operatorname {Spec} O_{E,(\nu )}$ as follows. To each locally noetherian $O_{E,(\nu )}$-scheme $S$, we associate the groupoid of tuples $(A_0,\iota _0,\lambda _0,A,\iota ,\lambda ,\overline \eta ^{p})$, where $(A_0,\iota _0,\lambda _0)$ is an object of $\mathcal {M}^{\mathfrak {a}, \xi }_0(S)$. Furthermore:

• • $(A,\iota )$ is an abelian scheme over $S$ with an $O_{F}\otimes \mathbb{Z}_{(p)}$-action $\iota$ satisfying the Kottwitz condition (3.11) of signature $((1, n-1)_{\varphi _0}, (0, n)_{\varphi \in \Phi \smallsetminus \{\varphi _0\}})$.

• • $\lambda$ is a polarization on $A$ whose Rosati involution induces on $O_{F}\otimes \mathbb{Z}_{(p)}$ the non-trivial Galois automorphism of $F/F_0$, subject to the following condition. Note that the action of the ring $O_{F_0}\otimes \mathbb{Z}_p \cong \prod _{v \in \mathcal {V}_p} O_{F_0,v}$ on the $p$-divisible group $A[p^{\infty }]$ induces a decomposition of $p$-divisible groups,

  (4.2)\begin{equation} A[p^{\infty}] = \prod_{v \in \mathcal{V}_p} A[v^{\infty}]. \end{equation}
  Since $\operatorname {Ros}_\lambda$ is trivial on $O_{F_0}$, $\lambda$ induces a polarization $\lambda _v \colon A[v^{\infty }] \to A^{\vee }[v^{\infty }] \cong A[v^{\infty }]^{\vee }$ for each $v$. The condition we impose is that $\ker \lambda _v$ is contained in $A[\iota (\pi _v)]$ of rank $\#(\Lambda _v^{*}/ \Lambda _v)$ for each $v \in \mathcal {V}_p$.

• • $\overline \eta ^{p}$ is a $K_G^{p}$-orbit of $\mathbb{A}_{F,f}^{p}$-linear isometries

  (4.3)\begin{equation} \eta^{p}\colon \widehat V^{p}(A_0,A) \simeq -W \otimes_F \mathbb{A}_{F,f}^{p}, \end{equation}
  where
  \[ \widehat V^{p}(A_0,A) := \operatorname{Hom}_{F}(\widehat V^{p}(A_0), \widehat V^{p}(A)), \]
  and the hermitian form on $\widehat V^{p}(A_0,A)$ is the evident prime-to-$p$ analog of (3.13).

We also impose for each $v\neq v_0$ over $p$ the sign condition and the Eisenstein condition. Let us explain these conditions.

The sign condition at $v$ is only non-empty when $v$ does not split in $F$, in which case it requires that at every point $s$ of $S$,

(4.4)\begin{equation} {{\mathrm{inv}}}^{r}_v(A_{0,s}, \iota_{0,s}, \lambda_{0,s}, A_s, \iota_s, \lambda_s) = {{\mathrm{inv}}}_v(-W_v). \end{equation}

Here the left-hand side is the sign factor defined in (A.5) and (A.8) in Appendix A (in the definition of (A.8), one may use the embedding $\widetilde \nu$ fixed at the beginning of this section). The right-hand side is the Hasse invariant of the hermitian space $-W_v$ defined above in (1.4). Note that by Proposition A.1, the left-hand side of (4.4) is a locally constant function in $s$.

The Eisenstein condition is only non-empty when the base scheme $S$ has non-empty special fiber. In this case, we may even base change via $\widetilde \nu \colon O_{E, (\nu )}\to \overline{\mathbb{Z}}_p$ (the ring of integers in $\overline{\mathbb{Q}}_p$) and pass to completions and assume that $S$ is a scheme over $\operatorname {Spf} \overline{\mathbb{Z}}_p$. Similarly to (4.2), there is a decomposition of the $p$-divisible group $A[p^{\infty }]$,

(4.5)\begin{equation} A[p^{\infty}] = \prod_{w\mid p} A[w^{\infty}] , \end{equation}

where the indices range over the places $w$ of $F$ lying over $p$. Since we assume that $p$ is locally nilpotent on $S$, there is a natural isomorphism

\begin{equation*} \operatorname{Lie} A\cong \operatorname{Lie} A[p^{\infty}]=\bigoplus_{w\mid p} \operatorname{Lie} A[w^{\infty}]. \end{equation*}

For each place $w$, by the Kottwitz condition (3.11), the $p$-divisible group $A[w^{\infty }]$ is of height $n \cdot [F_{w}:\mathbb{Q}_p]$ and dimension

(4.6)\begin{equation} \dim A[w^{\infty}] = \sum_{\varphi \in \operatorname{Hom}(F_w, \overline{\mathbb{Q}}_p)} r_\varphi. \end{equation}

Here $r_\varphi$ is as in (3.14), and we have used the embedding $\widetilde \nu \colon \overline{\mathbb{Q}} \to \overline{\mathbb{Q}}_p$ to identify

(4.7)\begin{equation} \operatorname{Hom}_\mathbb{Q}(F, \overline{\mathbb{Q}}) \simeq \operatorname{Hom}_\mathbb{Q}(F, \overline{\mathbb{Q}}_p) , \end{equation}

which in turn identifies

(4.8)\begin{equation} \{\varphi \in \operatorname{Hom}_{\mathbb{Q}}(F, \overline{\mathbb{Q}}) \,|\, w_\varphi=w \} \simeq \operatorname{Hom}_{\mathbb{Q}_p}(F_w, \overline{\mathbb{Q}}_p), \end{equation}

where $w_\varphi$ denotes the $p$-adic place in $F$ induced by $\widetilde \nu \circ \varphi$.

Now suppose that $w$ lies over a place $v$ different from $v_0$. Then the action of $F$ on $A[w^{\infty }]$ is of a banal signature type, in the sense that each integer $r_\varphi$ occurring in (4.6) is equal to $0$ or $n$ (cf. Appendix B). Let $\pi = \pi _w$ be a uniformizer in $F_w$, and let $F_w^{t} \subset F_w$ be the maximal unramified subextension of $\mathbb{Q}_p$. For each $\psi \in \operatorname {Hom}_{\mathbb{Q}_p}(F^{t}_w, \overline{\mathbb{Q}}_p)$, let

\[ A_\psi := \{\varphi\in \operatorname{Hom}_{\mathbb{Q}_p}(F_w, \overline{\mathbb{Q}}_p) \,|\, \varphi|_{F_w^{t}}=\psi \text{ and } r_\varphi=n \}. \]

Set

\[ Q_{A_\psi}(T):=\prod_{\varphi\in A_\psi}(T-\varphi(\pi))\in \overline{\mathbb{Z}}_p[T]. \]

Since we assume that $S$ is a scheme over $\operatorname {Spf} \overline{\mathbb{Z}}_p$, there is a natural isomorphism

\[ O_{F_w^{t}} \otimes_{\mathbb{Z}_p} \mathcal{O}_S {\tilde{\to}} \prod_{\psi \in \operatorname{Hom}_{\mathbb{Q}_p}(F_w^{t}, \overline{\mathbb{Q}}_p)} \mathcal{O}_S, \]

whose $\psi$-component is $\psi \otimes \mathrm {id}$. Hence the $O_{F_w^{t}}$-action on $\operatorname {Lie} A[w^{\infty }]$ induces a decomposition

(4.9)\begin{equation} \operatorname{Lie} A[w^{\infty}] \cong \bigoplus_{\psi \in \operatorname{Hom}_{\mathbb{Q}_p}(F_w^{t}, \overline{\mathbb{Q}}_p)} \operatorname{Lie}_\psi A[w^{\infty}]. \end{equation}

The Eisenstein condition at $v$ demands the identity of endomorphisms of $\operatorname {Lie}_\psi A[w^{\infty }]$,

(4.10)\begin{equation} Q_{A_\psi}(\iota(\pi) \mid \operatorname{Lie}_{\psi} A[w^{\infty}])=0 \quad \text{for each} \quad w \mid v \quad \text{and each} \quad \psi\in \operatorname{Hom}_{\mathbb{Q}_p}(F^{t}_w, \overline{\mathbb{Q}}_p). \end{equation}

This condition is the analog in our context of the condition with the same name in [Reference Rapoport and ZinkRZ17, (8.2)]. The Kottwitz condition implies that the Eisenstein condition at $v$ is automatically satisfied when the one or two places $w$ over $v$ are unramified over $p$ (cf. Lemma B.3).

A morphism between two objects $(A_0, \iota _0, \lambda _0, A, \iota , \lambda , \overline \eta ^{p})$ and $(A_0', \iota _0', \lambda _0', A', \iota ', \lambda ', \overline \eta '^{p})$ is given by an isomorphism $(A_0,\iota _0,\lambda _0) {\tilde{\to}} (A_0',\iota _0',\lambda _0')$ in $\mathcal {M}^{\mathfrak {a},\xi }_0(S)$ and a quasi-isogeny $A \to A'$ which induces an isomorphism

\[ A[p^{\infty}] {\tilde{\to}} A'[p^{\infty}] , \]

compatible with $\iota$ and $\iota '$, with $\lambda$ and $\lambda '$, and with $\overline \eta ^{p}$ and $\overline \eta ^{p\prime }$.

Proof. Representability and relative representability are standard (cf. [Reference KottwitzKot92, p. 391]). Smoothness follows as usual from the theory of local models (cf. [Reference Pappas, Rapoport and SmithlingPRS13]). More precisely, the local model for $\mathcal {M}_{K_{\widetilde G}}(\widetilde G)$ decomposes into a product of local models, one for each of the abelian schemes $A_0$ and $A$ in the moduli problem. The local model corresponding to $A_0$ is étale because $\mathcal {M}_0^{\mathfrak {a},\xi }$ is. Now, under the identifications (4.7) and (4.8), we have

(4.11)\begin{equation} \operatorname{Hom}_\mathbb{Q}(F,\overline{\mathbb{Q}}) \simeq \bigsqcup_{v \in \mathcal{V}_p} \operatorname{Hom}_{\mathbb{Q}_p}(F_v, \mathbb{Q}_p). \end{equation}

In this way the completion $E_\nu$ identifies with the join of the local reflex fields $E_{\Phi _v}$ and $E_{r|_v}$ in $\overline{\mathbb{Q}}_p$ as $v$ varies through $\mathcal {V}_p$, where $\Phi _v := \Phi \cap \operatorname {Hom}_{\mathbb{Q}_p}(F_v,\overline{\mathbb{Q}}_p)$ in terms of the identification (4.11), and where $r|_v\colon \operatorname {Hom}_{\mathbb{Q}_p}(F_v,\overline{\mathbb{Q}}_p) \longrightarrow \mathbb{Z}$ denotes the restriction of the function $r$ to the $v$-summand on the right-hand side of (4.11). The local model $M$ corresponding to $A$ then decomposes as

(4.12)\begin{equation} M = \prod_{v \in \mathcal{V}_p} M_v \times_{\operatorname{Spec} O_{E_{r|_v}}} \operatorname{Spec} O_{E_\nu}. \end{equation}

Here for $v \neq v_0$, by the Kottwitz condition (3.11), $M_v = M(F_v/F_{0,v},r|_v,\Lambda _v)$ is a banal local model, that is, of the form defined in Appendix B. Hence $M_v = \operatorname {Spec} O_{E_{r|_v}}$ by Lemmas B.1 and B.4. (The same is true for the local model corresponding to $A_0$ at every $v \in \mathcal {V}_p$, by these lemmas and Remark B.2) The local model $M_{v_0}$ is smooth by [Reference GörtzGör01].

It remains to prove the last assertion. Let $S$ be a scheme over $E$, and let $(A_0, \iota _0, \lambda _0, A, \iota , \lambda , \overline \eta ^{p})$ be a point of $\mathcal {M}_{K_{\widetilde G}}(S)$. We want to associate to this a point of $M_{K_{\widetilde G}}(S)$, that is, we want to add the $p$-component of $\overline \eta$. The product of hermitian spaces $W \otimes _\mathbb{Q} \mathbb{Q}_p=\prod _{v \in \mathcal {V}_p} W_v$ contains the lattice $\prod _{v \in \mathcal {V}_p}\Lambda _v$, where $\Lambda _v$ is a vertex lattice in $W_v$. By assumption on the polarization $\lambda$, the product of hermitian spaces

\[ V_p(A_0, A) := \operatorname{Hom}_F(V_p(A_0), V_p(A)) = \prod_{v \in \mathcal{V}_p} V_v(A_0, A) \]

contains $\operatorname {Hom}_{O_F}(T_p(A_0), T_p(A))$ as a product of vertex lattices, where the factor at each $v$ is of the same type as $\Lambda _v$. Since $p\neq 2$ when there are non-split places in $\mathcal {V}_p$, it follows that, if there exists an isometry $\eta _p\colon V_p(A_0, A)\simeq -W \otimes _\mathbb{Q} \mathbb{Q}_p$ at all, then there also exists one that maps these two vertex lattices of identical type into one another, and the class modulo $K_{G, p}$ of such an isometry is then uniquely determined.

Hence we are reduced to showing that there exists an isometry $\eta _p$, that is, the equality of Hasse invariants ${{\mathrm {inv}}}_v(V_v(A_0, A))={{\mathrm {inv}}}_v(-W_v)$ for all $v \in \mathcal {V}_p$. By the Hasse principle and the product formula for hermitian spaces, it suffices to prove that for any $\mathbb{C}$-valued point $(A_0, \iota _0, \lambda _0, A, \iota , \lambda , \overline \eta ^{p})$ of $\mathcal {M}_{K_{\widetilde G}}$,

\[ {{\mathrm{inv}}}_v (V(A_0, A)_v)={{\mathrm{inv}}}_v(-W_v) \quad \text{for all}\ v\neq v_0 , \]

where $v$ runs through all places of $F_0$, including the archimedean ones. Here, as in the proof of Proposition 3.7, $V(A_0, A)=\operatorname {Hom}_F (\mathcal {H}_0, \mathcal {H})$, where $\mathcal {H}_0=H_1(A_0,\mathbb{Q})$ and $\mathcal {H}= H_1(A, \mathbb{Q})$. For the non-archimedean places not lying over $p$, this follows from the existence of the level structure; for the places $v\in \mathcal {V}_p\smallsetminus \{v_0\}$, this follows from the sign condition (4.4) at $v$; and finally, for the archimedean places, this follows from the fact that the signatures of $V(A_0, A)$ and $-W$ at all archimedean places are identical (cf. the proof of Proposition 3.7).

We analogously define the DM stacks $\mathcal {M}_{K_{\widetilde H}}(\widetilde H)$ and $\mathcal {M}_{K_{\widetilde {HG}}}(\widetilde {HG})$ over $\operatorname {Spec} O_{E,(\nu )}$ (cf. the end of § 3.2). Both are again smooth over $\operatorname {Spec} O_{E,(\nu )}$.

Let us now assume that the special vector $u \in W$ has norm $(u, u)\in O_{F_0,(p)}^{\times }$. Then we obtain a finite unramified morphism (respectively, a closed embedding), in analogy with (3.16) (respectively, (3.17)),

(4.13)\begin{equation} \mathcal{M}_{K_{\widetilde H}}(\widetilde H) \longrightarrow \mathcal{M}_{K_{\widetilde G}}(\widetilde G) \quad\text{and}\quad \mathcal{M}_{K_{\widetilde H}}(\widetilde H) {\hookrightarrow} \mathcal{M}_{K_{\widetilde{HG}}}(\widetilde{HG}). \end{equation}

For this we assume that $K_H^{p} \subset H(\mathbb{A}_{F_0,f}^{p}) \cap K_G^{p}$. Furthermore, we assume for each $v\in \mathcal {V}_p$ that the lattices in $W_v$ and $W_v^{\flat }$ satisfy the relation

(4.14)\begin{equation} \Lambda_v = \Lambda_v^{\flat} \oplus O_{F,v}u. \end{equation}

For the lattice in $W_v^{\flat } \oplus W_v$, we take the direct sum $\Lambda _v^{\flat } \oplus \Lambda _v$.

We end this subsection by defining Hecke correspondences attached to adelic elements prime to $p$. We first consider the case of $\mathcal {M}_{K_{\widetilde G}}(\widetilde G)$. Fix $g \in \widetilde G(\mathbb{A}_f^{p})$. Let $K_{\widetilde G}^{p} := Z^{\mathbb{Q}}(\widehat{\mathbb{Z}}^{p}) \times K_G^{p}$ and $K_{\widetilde G,p} := Z^{\mathbb{Q}}(\mathbb{Z}_p) \times K_{G,p}$, and set

\[ K_{\widetilde G}^{\prime p} := K_{\widetilde G}^{p} \cap g K_{\widetilde G}^{p} g{{}^{-1}} \quad\text{and}\quad K_{\widetilde G}' := K_{\widetilde G}^{\prime p} \times K_{\widetilde G,p}. \]

Then we obtain in the standard way a diagram of finite étale morphisms

(4.15)

which we view as a correspondence from $\mathcal {M}_{K_{\widetilde G}}(\widetilde G)$ to itself. Note that for a central element $g=z\in Z(G)(\mathbb{A}_{F_0,f}^{p})=\{z\in (\mathbb{A}^{p}_{F,f})^{\times }\mid \operatorname {Nm}_{F/F_0}(z)=1\}$, the diagram (

) collapses to a map

(4.16)\begin{equation} \mathcal{M}_{K_{\widetilde G}}(\widetilde G) {\mathop \to \limits^z} \mathcal{M}_{K_{\widetilde G}}(\widetilde G), \end{equation}

and this induces an action of $Z(G)(\mathbb{A}_{F_0,f}^{p})$ on $\mathcal {M}_{K_{\widetilde G}}(\widetilde G)$.

The cases of $\mathcal {M}_{K_{\widetilde H}}(\widetilde H)$ and $\mathcal {M}_{K_{\widetilde {HG}}}(\widetilde {HG})$ are completely analogous, simply replacing $\widetilde G$ everywhere by $\widetilde H$ and $\widetilde {HG}$, respectively. For later use, we record the diagram of finite étale morphisms we obtain for $\widetilde {HG}$:

(4.17)

4.2 Split level at $v_0$

We continue with the setup and assumptions of the previous subsection, except we now allow $v_0$ to be ramified over $p$. In addition, we assume that $v_0$ splits in $F$, say into $w_0$ and another place $\overline w_0$. We will define smooth integral models over $O_{E,(\nu )}$.

We define the moduli functor $\mathcal {M}_{K_{\widetilde G}}(\widetilde G)$ as follows. To each locally noetherian $O_{E,(\nu )}$-scheme $S$, we associate the groupoid of tuples $(A_0,\iota _0,\lambda _0,A,\iota ,\lambda ,\overline \eta ^{p})$ as in the previous subsection, except that we impose the further condition corresponding to $w_0$ that when $p$ is locally nilpotent on $S$, the $p$-divisible group $A[w_0^{\infty }]$ is a Lubin–Tate group of type $r|_{w_0}$, in the sense of [Reference Rapoport and ZinkRZ17, § 8] (note that this involves the Eisenstein condition of [Reference Rapoport and ZinkRZ17, § 8]). Here $r|_{w_0}$ is the restriction of the function $r$ on $\operatorname {Hom}_\mathbb{Q}(F, \overline{\mathbb{Q}})$ to $\operatorname {Hom}_{\mathbb{Q}_p}(F_{w_0}, \overline{\mathbb{Q}}_p)$, in the sense of the identification (4.8). We note that if $v_0$ is unramified over $p$, then this further Eisenstein condition is redundant, and the moduli functor $\mathcal {M}_{K_{\widetilde G}}(\widetilde G)$ is the same as the one defined in the previous subsection (cf. [Reference Rapoport and ZinkRZ17]).

We analogously define the DM stacks $\mathcal {M}_{K_{\widetilde H}}(\widetilde H)$ and $\mathcal {M}_{K_{\widetilde {HG}}}(\widetilde {HG})$ over $\operatorname {Spec} O_{E,(\nu )}$ (cf. the end of § 3.2). Both are again smooth over $\operatorname {Spec} O_{E,(\nu )}$. We then obtain a finite unramified morphism (respectively, a closed embedding),

(4.18)\begin{equation} \mathcal{M}_{K_{\widetilde H}}(\widetilde H) \longrightarrow \mathcal{M}_{K_{\widetilde G}}(\widetilde G) \quad\text{and}\quad \mathcal{M}_{K_{\widetilde H}}(\widetilde H) {\hookrightarrow} \mathcal{M}_{K_{\widetilde{HG}}}(\widetilde{HG}), \end{equation}

and Hecke correspondences exactly as in the previous subsection.

4.3 Drinfeld level at $v_0$

We continue with the setup and assumptions of § 4.2, with $v_0$ split in $F$ and possibly ramified over $p$. In this subsection we will define integral models over $O_{E,(\nu )}$ where we impose a Drinfeld level structure at $v_0$. To do this, we require that the matching condition between the CM type $\Phi$ and the chosen place $\nu$ of $E$ is satisfied, which demands that

(4.19)\begin{equation} \{\varphi\in \operatorname{Hom}(F, \overline{\mathbb{Q}}) \,|\, w_\varphi=w_0\}\subset \Phi, \end{equation}

where $w_\varphi$ is the place of $F$ induced by $\widetilde \nu \circ \varphi$, as in (4.8). We note that this condition only depends on the place $\nu$ of $E$ induced by $\widetilde \nu$. When condition (4.19) is satisfied, we also say that the CM type $\Phi$ and the place $\nu$ of $E$ are matched. Here are some examples in which the matching condition is guaranteed to hold.

Now let $m$ be a non-negative integer. We define the level subgroup $K_G^{m} \subset G(\mathbb{A}_{F_0,f})$ in exactly the same way as $K_G$ in § 4.1 (subject to the choice of certain vertex lattices $\Lambda _v$ for $v \in \mathcal {V}_p$), except that in the $v_0$-factor where $G(F_{0, v_0})\simeq \mathrm {GL}_n(F_{0, v_0})$, we take $K^{m}_{G, v_0}$ to be the principal congruence subgroup modulo $\mathfrak {p}_{v_0}^{m}$ inside $K_{G,v_0}$. In particular, $K_G$ coincides with $K^{m}_G$ for $m=0$. We define $K_{\widetilde G}^{m} := K_{Z^{\mathbb{Q}}} \times K_G^{m}$ as in (3.6).

We now define the moduli functor $\mathcal {M}_{K^{m}_{\widetilde G}}(\widetilde G)$ over $\mathcal {M}_{K_{\widetilde G}}(\widetilde G)$. Let $(A_0,\iota _0,\lambda _0,A,\iota ,\lambda , \overline \eta ^{p})\in \mathcal {M}_{K_{\widetilde G}}(\widetilde G)(S)$. Consider the factors occurring in the decomposition (4.5) of the $p$-divisible group $A[p^{\infty }]$,

(4.20)\begin{equation} A[v_0^{\infty}] = A[w_0^{\infty}] \times A[\overline w_0^{\infty}]. \end{equation}

When $p$ is locally nilpotent on $S$, the $p$-divisible group $A[w_0^{\infty }]$ satisfies the Kottwitz condition of type $r|_{w_0}$ for the action of $O_{F,w_0}$ on its Lie algebra, in the sense of the previous subsection. Thus, by the formulation of the Kottwitz condition in Remark 3.6(i) and the matching condition (4.19),

\[ \operatorname{char}(\iota(a)\mid\operatorname{Lie} A[w_0^{\infty}]) = \prod_{\substack{\varphi \colon \! F \rightarrow \mathbb{Q}\\ w_\varphi=w_0}} (T-\varphi(a))^{r_\varphi} = T - \varphi_0(a) \quad\text{for all}\ a \in O_{F,w_0}. \]

Hence $A[w_0^{\infty }]$ is a one-dimensional formal $O_{F,w_0}$-module. Similarly, $A[\overline w_0^{\infty }]$ has dimension $n \cdot [F_{w_0}:\mathbb{Q}_p]-1$. Since $\operatorname {Ros}_\lambda$ induces the conjugation automorphism on $O_F$, $\lambda$ furthermore identifies $A[w_0^{\infty }]$ and $A[\overline w_0^{\infty }]$ with the (absolute) duals of each other, and hence both have (absolute) height $n \cdot [F_{w_0}:\mathbb{Q}_p]$. Analogously,

\[ A_0[v_0^{\infty}] = A_0[w_0^{\infty}] \times A_0[\overline w_0^{\infty}] , \]

where, by (4.19) and the Kottwitz condition on $A_0$, the $p$-divisible group $A_0[w_0^{\infty }]$ with $O_{F,w_0}$-action is étale of height $[F_{w_0}:\mathbb{Q}_p]$, whereas $A_0[\overline w_0^{\infty }]$ is identified with the dual of $A_0[w_0^{\infty }]$.

In analogy with the prime-to-$p$ theory, we introduce the finite flat group scheme over $S$,

\[ T_{w_0}(A_0,A)[w_0^{m}] := \operatorname{\underline{Hom}}_{O_{F,w_0}}(A_0[w_0^{m}],A[w_0^{m}]). \]

Note that as $m$ varies, the right-hand side is naturally an inverse system under restriction of homomorphisms, and making it into a directed system depends on the choice of uniformizer $\pi _{w_0}$ of $F_{0, w_0}$. The colimit $T_{w_0}(A_0,A) := \varinjlim _{m} T_{w_0}(A_0,A)[w_0^{m}]$ is a one-dimensional formal $O_{F,w_0}$-module since $A[w_0^{\infty }]$ is.

For the moduli problem $\mathcal {M}_{K^{m}_{\widetilde G}}(\widetilde G)$, we equip the object $(A_0,\iota _0,\lambda _0,A,\iota ,\lambda , \overline \eta ^{p})\in \mathcal {M}_{K_{\widetilde G}}(\widetilde G)$ with the following additional datum. Let $\Lambda _{v_0} = \Lambda _{w_0} \oplus \Lambda _{\overline w_0}$ denote the natural decomposition of the lattice $\Lambda _{v_0}$ attached to the split place $v_0$. The additional datum is

• • an $O_{F,w_0}$-linear homomorphism of finite flat group schemes,

  (4.21)\begin{equation} \varphi\colon \pi_{w_0}^{-m}\Lambda_{w_0}/\Lambda_{w_0} \longrightarrow \operatorname{\underline{Hom}}_{O_{F,w_0}}(A_0[w_0^{m}],A[w_0^{m}]), \end{equation}
  which is a Drinfeld $w_0^{m}$-structure on the target (cf. [Reference Harris and TaylorHT01, § II.2]).

We analogously define the DM stack $\mathcal {M}_{K^{m}_{\widetilde H}}(\widetilde H)$ over $\operatorname {Spec} O_{E,(\nu )}$, and obtain for it the analog of Theorem 4.5. We also define $\mathcal {M}_{K^{m}_{\widetilde {HG}}}(\widetilde {HG}) := \mathcal {M}_{K_{\widetilde H}^{m}}(\widetilde H) \times _{\mathcal {M}^{\mathfrak {a},\xi }_0} \mathcal {M}_{K_{\widetilde G}^{m}}(\widetilde G)$, but we note that this stack is not regular for $m > 0$. We then obtain a finite unramified morphism (respectively, a closed embedding)

\[ \mathcal{M}_{K^{m}_{\widetilde H}}(\widetilde H) \longrightarrow \mathcal{M}_{K^{m}_{\widetilde G}}(\widetilde G) \quad\text{and}\quad \mathcal{M}_{K^{m}_{\widetilde H}}(\widetilde H) {\hookrightarrow} \mathcal{M}_{K^{m}_{\widetilde{HG}}}(\widetilde{HG}), \]

provided that $K_H^{p} \subset H(\mathbb{A}_{F_0,f}^{p}) \cap K_G^{p}$ and that the lattices $\Lambda _v$ and $\Lambda _v^{\flat }$ are as in (4.14). Indeed, the Drinfeld level structure $\varphi ^{\flat } \colon \pi _{w_0}^{-m}\Lambda _{w_0}^{\flat }/\Lambda _{w_0}^{\flat } \to \operatorname {\underline {Hom}}_{O_{F,w_0}}(A_0[w_0^{m}],A^{\flat }[w_0^{m}])$ induces a Drinfeld level structure

\[ \varphi\colon \pi_{w_0}^{-m}\Lambda_{w_0}/\Lambda_{w_0} \longrightarrow \operatorname{\underline{Hom}}_{O_{F,w_0}}(A_0[w_0^{m}], (A^{\flat} \times A_0)[w_0^{m}]) \]

by taking the direct sum of $\varphi ^{\flat }$ and the $O_{F,w_0}$-linear homomorphism

(4.22)

with respect to the natural decompositions in the source and target of $\varphi$.

In the present situation there are more Hecke correspondences than those defined for adelic elements prime to $p$ in the previous two subsections (cf. [Reference Harris and TaylorHT01, § III.4]). We again first treat the case for $\widetilde G$. With respect to the decomposition obtained from (2.1),

\[ \widetilde G(\mathbb{Q}_p) = Z^{\mathbb{Q}}(\mathbb{Q}_p) \times \prod_{v \in \mathcal{V}_p} G(F_{0,v}), \]

let $g \in G(F_{0,v_0})$, considered as an element on the left-hand side. In the special case where $g \in K_{G,v_0}$, since $K_{G,v_0}^{m}$ is a normal subgroup of $K_G$, we get a diagram of finite flat morphisms

analogously to (4.15). For general $g \in G(F_{0,v_0})$, choose $\mu$ large enough that

\[ p^{\mu}\Lambda_{v_0} \subset g\Lambda_{v_0} \subset \varpi_{v_0}^{-m} g \Lambda_{v_0} \subset p^{-\mu} \Lambda_{v_0}. \]

Then $K_{G,v_0}^{2\mu e} \subset K_{G,v_0}^{m} \cap gK_{G,v_0}^{m} g{{}^{-1}}$, where $e = e_{v_0}$ is the ramification index of $v_0$ over $p$. Hence we obtain a diagram of finite flat morphisms

(4.23)

as before. In terms of the moduli descriptions above, these morphisms are given as follows. Consider a point $(A_0,\iota _0,\lambda _0,A,\iota ,\lambda , \overline \eta ^{p},\varphi )$ of $\mathcal {M}_{K_{\widetilde G}^{2\mu e}}(\widetilde G)$. Then $\operatorname {nat}_1$ sends this point to the point represented by $(A_0,\iota _0,\lambda _0,A,\iota ,\lambda , \overline \eta ^{p},\varphi |_{(\pi _{w_0}^{-m}\Lambda _{w_0}/\Lambda _{w_0})})$. To describe $\operatorname {nat}_g$, let $C_{w_0} \subset A[w_0^{2\mu e}]$ be the unique closed subgroup scheme for which the set of $\varphi (x)$, $x \in p^{-\mu } g\Lambda _{w_0}/\Lambda _{w_0} \subset p^{-2\mu }\Lambda _{w_0}/\Lambda _{w_0}$, is a complete set of sections; cf. [Reference Katz and MazurKM85, Cor. 1.10.3] (note that $\operatorname {\underline {Hom}}_{O_{F,w_0}}(A_0[w_0^{2\mu e}],A[w_0^{2\mu e}])$ is étale-locally isomorphic to $A[w_0^{2\mu e}]$). Since $\lambda$ induces a principal polarization on the $p$-divisible group $A[v_0^{\infty }]$, the $\lambda$-Weil pairing on $A[p^{2\mu }]$ induces a perfect duality between $A[w_0^{2\mu e}]$ and $A[\overline w_0^{2\mu e}]$; in this way, let $C_{\overline w_0} \subset A[\overline w_0^{2\mu e}]$ be the annihilator of $C_{w_0}$. Let $C$ be the (totally isotropic) subgroup of $A[p^{2\mu }]$,

\[ C := C_{w_0} \times C_{\overline w_0} \times \prod_{v \in \mathcal{V}_p \smallsetminus \{v_0\}} A[v^{\mu e_v}], \]

where $e_v$ denotes the ramification index of $v$ over $p$. Let

\[ A' := A/C. \]

Let $\iota '$ denote the induced $O_F \otimes \mathbb{Z}_{(p)}$-action on $A'$, and let $\alpha \colon A \to A'$ be the corresponding $O_F$-linear isogeny. Then there is a unique polarization $\lambda '$ on $A'$ such that the diagram

commutes, and $\lambda '$ is a polarization of the type in the moduli problem for $\mathcal {M}_{K_{\widetilde G}}(\widetilde G)$. Furthermore, the level structure $\overline \eta ^{p}$ induces, via the quasi-isogeny $p^{-\mu }\alpha$, a level structure $\overline \eta ^{\prime p}$ for the pair $(A_0,A')$. We obtain a Drinfeld level $w_0^{m}$-structure $\varphi '$ on $\operatorname {\underline {Hom}}_{O_{F,w_0}}(A_0[w_0^{m}],A'[w_0^{m}])$ via the following commutative diagram:

Here the dashed arrows are induced by $\varphi$. Then the image of $(A_0,\iota _0,\lambda _0,A,\iota ,\lambda , \overline \eta ^{p},\varphi )$ under $\operatorname {nat}_g$ is $(A_0,\iota _0,\lambda _0,A',\iota ',\lambda ',\overline \eta ^{\prime p}, \varphi ')$. We remark that the construction of the latter tuple is, up to canonical isomorphism in the moduli problem for $\mathcal {M}_{K_{\widetilde G}^{m}}(\widetilde G)$, independent of the choice of $\mu$.

Again, as in (4.16), this defines an action of the center of $G(F_{0, v_0})$ on $\mathcal {M}_{K_{\widetilde G}^{m}}(\widetilde G)$.

The above construction carries over in the obvious way with $\widetilde H$ in place of $\widetilde G$. Taking the product over $\mathcal {M}^{\mathfrak {a},\xi }_0$ of the diagram (4.23) with the one attached to $\widetilde H$ and an element $h \in H(F_{0,v_0})$, we obtain, for $\mu$ sufficiently large, an analogous diagram of finite flat morphisms for $\widetilde {HG}$. We record this as the following diagram, where $g\in (H\times G)(F_{0,v_0})$:

(4.24)

4.4 AT parahoric level at $v_0$

In this subsection we assume that $p \neq 2$ and that the place $v_0$ is unramified over $p$. We again choose a vertex lattice $\Lambda _v \subset W_v$ for each $v \in \mathcal {V}_p$, as in § 4.1. We require that the pair $(v_0,\Lambda _{v_0})$ is of one of the following four types, which we call AT types.

1. (1) $v_0$ is inert in $F$ and $\Lambda _{v_0}$ is almost self-dual as an $O_{F,v_0}$-lattice.

2. (2) $v_0$ ramifies in $F$, $n$ is even, and $\Lambda _{v_0}$ is $\pi _{v_0}$-modular.

3. (3) $v_0$ ramifies in $F$, $n$ is odd, and $\Lambda _{v_0}$ is almost $\pi _{v_0}$-modular.

4. (4) $v_0$ ramifies in $F$, $n = 2$, and $\Lambda _{v_0}$ is self-dual.