Assumption 2.1. Let $\mathsf{X}_{{\mathcal{H}}}\rightarrow \mathsf{S}$ be a scheme over the spectrum of a discrete valuation ring $R_{0}$ of mixed characteristics $(0,p)$ , which is the pullback of one the following integral models in the literature. (The various notations $\mathsf{S}_{0}$ , $\vec{\mathsf{S}}_{0}$ , etc., below are those in the works we cited, which we shall freely use, but mostly only in proofs.)

(Sm)

A smooth integral model $\mathsf{M}_{{\mathcal{H}}^{\Box }}\rightarrow \mathsf{S}_{0}=\operatorname{Spec}({\mathcal{O}}_{F_{0},(\Box )})$ defined as a moduli problem of abelian schemes with PEL structures at a neat level ${\mathcal{H}}^{\Box }\subset \text{G}(\hat{\mathbb{Z}}^{\Box })$ , as in [Reference LanLan13, chs 1, 2, and 7], with $p\in \Box$ and ${\mathcal{H}}={\mathcal{H}}^{\Box }\times \prod _{q\in \Box }\text{G}(\mathbb{Z}_{q})$ . (When $\Box =\{p\}$ , it is shown in [Reference LanLan13, Proposition 1.4.3.4] that the definition in [Reference LanLan13, §1.4.1] by isomorphism classes agrees with the one in [Reference LanLan13, §1.4.2] by $\mathbb{Z}_{(p)}^{\times }$ -isogeny classes, the latter being Kottwitz’s definition in [Reference KottwitzKot92, §5].)

(Nm)

A flat integral model $\vec{\mathsf{M}}_{{\mathcal{H}}}\rightarrow \vec{\mathsf{S}}_{0}=\operatorname{Spec}({\mathcal{O}}_{F_{0},(p)})$ of a moduli problem $\mathsf{M}_{{\mathcal{H}}}\rightarrow \mathsf{S}_{0}=\operatorname{Spec}(F_{0})$ at a neat level ${\mathcal{H}}\subset \text{G}(\hat{\mathbb{Z}})$ (essentially the same as above, but with $\Box =\emptyset$ ) defined by taking normalizations over certain auxiliary good reduction models as in [Reference LanLan16, §6] (which allow bad reductions due to arbitrarily high levels, ramifications, polarization degrees, and collections of isogenies). (In this case, we also allow $F_{0}$ to be a finite extension of the reflex field, with $\mathsf{M}_{{\mathcal{H}}}$ etc., replaced with their pullbacks.) For simplicity, we shall assume that, in the choice of the collection $\{(g_{\text{j}},L_{\text{j}},\langle \,\cdot \,,\cdot \,\rangle _{\text{j}})\}_{\text{j}\in \text{J}}$ in [Reference LanLan16, §2], we have $g_{\text{j}}=1$ for all $\text{j}\in \text{J}$ and $(L_{\text{j}_{0}},\langle \,\cdot \,,\cdot \,\rangle _{\text{j}_{0}})=(p^{r_{0}}L,p^{-2r_{0}}\langle \,\cdot \,,\cdot \,\rangle )$ for some $\text{j}_{0}\in \text{J}$ and some $r_{0}\in \mathbb{Z}$ .

(Spl)

A flat integral model $\vec{\mathsf{M}}_{{\mathcal{H}}}^{\text{spl}}\rightarrow \operatorname{Spec}({\mathcal{O}}_{K})$ of $\mathsf{M}_{{\mathcal{H}}}\otimes _{F_{0}}K\rightarrow \operatorname{Spec}(K)$ defined by taking normalizations as in [Reference LanLan15b, §2.4] over the splitting models defined by Pappas–Rapoport as in [Reference Pappas and RapoportPR05, §15]. (By taking normalizations, we mean we also allow ${\mathcal{H}}$ to be arbitrarily higher levels, not just the same levels considered in [Reference Pappas and RapoportPR05, §15].) For simplicity, we shall assume that, in the choice of the collection $\{(g_{\text{j}},L_{\text{j}},\langle \,\cdot \,,\cdot \,\rangle _{\text{j}})\}_{\text{j}\in \text{J}}$ in [Reference LanLan15b, Choices 2.2.9], we have $g_{\text{j}}=1$ for all $\text{j}\in \text{J}$ and $(L_{\text{j}_{0}},\langle \,\cdot \,,\cdot \,\rangle _{\text{j}_{0}})=(p^{r_{0}}L,p^{-2r_{0}}\langle \,\cdot \,,\cdot \,\rangle )$ for some $\text{j}_{0}\in \text{J}$ and some $r_{0}\in \mathbb{Z}$ .

(Hdg)

A flat integral model $\mathscr{S}_{K}\rightarrow \operatorname{Spec}(\mathscr{O}_{E,(v)})$ in the notation of [Reference Madapusi PeraMad15, Introduction] at a neat level $K$ . For consistency with the notation in other cases, we shall denote $K$ , $E$ , and $\mathscr{S}_{K}$ as ${\mathcal{H}}$ , $F_{0}$ , and $\mathsf{M}_{{\mathcal{H}}}$ , respectively, in what follows. Essentially by construction, there exists some auxiliary good reduction Siegel moduli $\mathsf{M}_{{\mathcal{H}}_{\text{aux}}}\rightarrow \operatorname{Spec}(\mathbb{Z}_{(p)})$ in Case (Sm) above, with a finite morphism $\mathsf{M}_{{\mathcal{H}}}\rightarrow \mathsf{M}_{{\mathcal{H}}_{\text{aux}}}\otimes _{\mathbb{Z}_{(p)}}{\mathcal{O}}_{F_{0},(v)}$ extending a closed immersion $\mathsf{M}_{{\mathcal{H}}}\otimes _{\mathbb{Z}}\mathbb{Q}\rightarrow \mathsf{M}_{{\mathcal{H}}_{\text{aux}}}\otimes _{\mathbb{Z}}F_{0}$ .

In all cases, there is some group functor $\text{G}$ over $\operatorname{Spec}(\mathbb{Z})$ , and some reflex field $F_{0}$ .

• ∙ In Cases (Sm), (Nm), and (Spl), the integral models are defined by (among other data) an integral PEL datum $({\mathcal{O}},\star ,L,\langle \,\cdot \,,\cdot \,\rangle ,h_{0})$ (cf. [Reference LanLan13, Definition 1.2.1.3]), which defines the group functor $\text{G}$ as in [Reference LanLan13, Definition 1.2.1.6], and the reflex field $F_{0}$ as in [Reference LanLan13, Definition 1.2.5.4]. For technical reasons, we shall insist that [Reference LanLan13, Condition 1.4.3.10] is satisfied. In Cases (Nm) and (Spl), we allow the level ${\mathcal{H}}$ to be arbitrarily high at $p$ .

• ∙ In Case (Hdg), we still have an integral PEL datum defining the auxiliary good reduction Siegel moduli $\mathsf{M}_{{\mathcal{H}}_{\text{aux}}}$ , which we abusively denote as $({\mathcal{O}},\star ,L,\langle \,\cdot \,,\cdot \,\rangle ,h_{0})$ (with ${\mathcal{O}}=\mathbb{Z}$ , without ‘ $\text{aux}$ ’ in the notation), which also defines a group functor $\text{G}_{\text{aux}}$ with an injective homomorphism $\text{G}\rightarrow \text{G}_{\text{aux}}$ .

Proof. This is because the invertible sheaves $\unicode[STIX]{x1D6F9}(\ell )$ over $C$ are Zariski locally trivial, and because the semigroup $\unicode[STIX]{x1D70E}^{\vee }$ is finitely generated (see [Reference Kempf, Knudsen, Mumford and Saint-DonatKKMS73, ch. I, §1, Lemma 2]).◻

Proof. The first paragraph is a consequence of Proposition 2.2, especially the properties (7) and (9), and of Lemma 2.3. The second and third paragraphs are straightforward consequences of the first paragraph, and of the various definitions. ◻

Proof of Proposition 2.2 in Case (Sm).

In this case, $\mathsf{X}_{{\mathcal{H}}}\rightarrow \mathsf{S}$ is the pullback of some integral model $\mathsf{M}_{{\mathcal{H}}^{\Box }}\rightarrow \mathsf{S}_{0}$ defined in [Reference LanLan13, ch. 1], and we can take $\mathsf{X}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}\rightarrow \mathsf{S}$ and $\mathsf{X}_{{\mathcal{H}}}^{\min }\rightarrow \mathsf{S}$ to be the pullbacks of the toroidal and minimal compactifications $\mathsf{M}_{{\mathcal{H}}^{\Box },\unicode[STIX]{x1D6F4}}^{\text{tor}}\rightarrow \mathsf{S}_{0}$ and $\mathsf{M}_{{\mathcal{H}}^{\Box }}^{\min }\rightarrow \mathsf{S}_{0}$ in [Reference LanLan13, Theorems 6.4.1.1, 7.2.4.1, and 7.3.3.4], where the collections $\unicode[STIX]{x1D6F4}$ can be any smooth and projective ones as in [Reference LanLan13, Definitions 6.3.3.4 and 7.3.1.3] (satisfying [Reference LanLan13, Condition 6.2.5.25]). Then the properties (2)–(8) follow from the statements there (where the last requirement in the property (4) is satisfied by [Reference LanLan13, Lemma 6.2.5.27]), and the property (9) follows from the construction of $\mathsf{M}_{{\mathcal{H}}^{\Box },\unicode[STIX]{x1D6F4}}^{\text{tor}}$ (with its stratification) by gluing good algebraic models as in [Reference LanLan13, §6.3].◻

Proof of Proposition 2.2 in Case (Nm).

In this case, $\mathsf{X}_{{\mathcal{H}}}\rightarrow \mathsf{S}$ is the pullback of some integral model $\vec{\mathsf{M}}_{{\mathcal{H}}}\rightarrow \vec{\mathsf{S}}_{0}$ as in [Reference LanLan16, §6], and we can take $\mathsf{X}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}\rightarrow \mathsf{S}$ and $\mathsf{X}_{{\mathcal{H}}}^{\min }\rightarrow \mathsf{S}$ to be the pullbacks of the toroidal and minimal compactifications $\vec{\mathsf{M}}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}\rightarrow \vec{\mathsf{S}}_{0}$ and $\vec{\mathsf{M}}_{{\mathcal{H}}}^{\min }\rightarrow \vec{\mathsf{S}}_{0}$ in [Reference LanLan17b, Theorem 6.1] and [Reference LanLan16, Proposition 6.4], where $\unicode[STIX]{x1D6F4}$ can be any collections which are projective and smooth, and satisfy [Reference LanLan13, Condition 6.2.5.25]. Then the properties (2)–(8) follow from [Reference LanLan17b, Theorem 6.1] (and the references from there to various results in [Reference LanLan16]) and [Reference LanLan16, Theorems 12.1 and 12.6].

It remains to verify the property (9). Given the isomorphism $\mathfrak{X}_{\unicode[STIX]{x1D70E}}=(\unicode[STIX]{x1D6EF}(\unicode[STIX]{x1D70E}))_{\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D70E}}}^{\wedge }\cong (\mathsf{X}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}})_{\mathsf{Z}_{[\unicode[STIX]{x1D70E}]}}^{\wedge }$ in the property (8), which is the pullback of the isomorphism

$$\begin{eqnarray}\vec{\mathfrak{X}}_{\unicode[STIX]{x1D6F7}_{{\mathcal{H}}},\unicode[STIX]{x1D6FF}_{{\mathcal{H}}},\unicode[STIX]{x1D70E}}=(\vec{\unicode[STIX]{x1D6EF}}_{\unicode[STIX]{x1D6F7}_{{\mathcal{H}}},\unicode[STIX]{x1D6FF}_{{\mathcal{H}}}}(\unicode[STIX]{x1D70E}))_{\vec{\unicode[STIX]{x1D6EF}}_{\unicode[STIX]{x1D6F7}_{{\mathcal{H}}},\unicode[STIX]{x1D6FF}_{{\mathcal{H}}},\unicode[STIX]{x1D70E}}}^{\wedge }\stackrel{{\sim}}{\rightarrow }(\vec{\mathsf{M}}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}})_{\vec{\mathsf{Z}}_{[(\unicode[STIX]{x1D6F7}_{{\mathcal{H}}},\unicode[STIX]{x1D6FF}_{{\mathcal{H}}},\unicode[STIX]{x1D70E})]}}^{\wedge }\end{eqnarray}$$

given by [Reference LanLan17b, Theorem 6.1(4)] (see also [Reference LanLan16, (10.3) and Theorem 10.14]), since both $\vec{\unicode[STIX]{x1D6EF}}_{\unicode[STIX]{x1D6F7}_{{\mathcal{H}}},\unicode[STIX]{x1D6FF}_{{\mathcal{H}}}}(\unicode[STIX]{x1D70E})$ and $\vec{\mathsf{M}}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}$ are separated and of finite type over the excellent Dedekind base scheme $\vec{\mathsf{S}}_{0}=\operatorname{Spec}({\mathcal{O}}_{F_{0},(p)})$ , it follows from Artin’s approximation (see [Reference ArtinArt69, Theorem 1.12, and the proof of the corollaries in §2]) that, at each point $x$ of $\vec{\unicode[STIX]{x1D6EF}}_{\unicode[STIX]{x1D6F7}_{{\mathcal{H}}},\unicode[STIX]{x1D6FF}_{{\mathcal{H}}},\unicode[STIX]{x1D70E}}\cong \vec{\mathsf{Z}}_{[(\unicode[STIX]{x1D6F7}_{{\mathcal{H}}},\unicode[STIX]{x1D6FF}_{{\mathcal{H}}},\unicode[STIX]{x1D70E})]}$ , there exist an étale neighborhood $\overline{U}\rightarrow \vec{\unicode[STIX]{x1D6EF}}_{\unicode[STIX]{x1D6F7}_{{\mathcal{H}}},\unicode[STIX]{x1D6FF}_{{\mathcal{H}}}}(\unicode[STIX]{x1D70E})$ of $x$ and an étale morphism $\overline{U}\rightarrow \vec{\mathsf{M}}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}$ such that the pullbacks of these morphisms to $\vec{\unicode[STIX]{x1D6EF}}_{\unicode[STIX]{x1D6F7}_{{\mathcal{H}}},\unicode[STIX]{x1D6FF}_{{\mathcal{H}}},\unicode[STIX]{x1D70E}}$ and to $\vec{\mathsf{Z}}_{[(\unicode[STIX]{x1D6F7}_{{\mathcal{H}}},\unicode[STIX]{x1D6FF}_{{\mathcal{H}}},\unicode[STIX]{x1D70E})]}$ are both open immersions. By also approximating in characteristic zero the various additional structures as in the proof of [Reference LanLan13, Proposition 6.3.2.1] (cf. Remark 2.7), we can ensure in the above that the pullbacks of the stratifications of $\vec{\unicode[STIX]{x1D6EF}}_{\unicode[STIX]{x1D6F7}_{{\mathcal{H}}},\unicode[STIX]{x1D6FF}_{{\mathcal{H}}}}(\unicode[STIX]{x1D70E})$ and $\vec{\mathsf{M}}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}$ to $\overline{U}$ are compatible with each other in characteristic zero. Since the strata of these stratifications are flat over $\vec{\mathsf{S}}_{0}$ (see [Reference LanLan16, Corollary 10.15] and [Reference LanLan17b, Theorem 6.1(5)]), they are induced by their restrictions to the characteristic zero fiber, and hence they are also compatible with each other in mixed characteristics. Since the images of such $\overline{U}\rightarrow \vec{\mathsf{M}}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}$ cover $\vec{\mathsf{Z}}_{[(\unicode[STIX]{x1D6F7}_{{\mathcal{H}}},\unicode[STIX]{x1D6FF}_{{\mathcal{H}}},\unicode[STIX]{x1D70E})]}$ , by considering their pullbacks under $\mathsf{S}\rightarrow \mathsf{S}_{0}=\operatorname{Spec}({\mathcal{O}}_{F_{0},(p)})$ , the property (9) follows.◻

Proof of Proposition 2.2 in Case (Spl).

In this case, $\mathsf{X}_{{\mathcal{H}}}\rightarrow \mathsf{S}$ is the pullback of some integral model $\vec{\mathsf{M}}_{{\mathcal{H}}}^{\text{spl}}\rightarrow \operatorname{Spec}({\mathcal{O}}_{K})$ as in [Reference LanLan15b, Definition 2.4.5], and we can take $\mathsf{X}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}\rightarrow \mathsf{S}$ and $\mathsf{X}_{{\mathcal{H}}}^{\min }\rightarrow \mathsf{S}$ to be the pullbacks of the toroidal and minimal compactifications $\vec{\mathsf{M}}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{spl},\text{tor}}\rightarrow \operatorname{Spec}({\mathcal{O}}_{K})$ and $\vec{\mathsf{M}}_{{\mathcal{H}}}^{\text{spl},\min }\rightarrow \operatorname{Spec}({\mathcal{O}}_{K})$ in [Reference LanLan15b, Theorems 3.4.1 and 4.3.1], where $\unicode[STIX]{x1D6F4}$ can be any compatible collections which are projective and smooth, and satisfy the mild [Reference LanLan13, Condition 6.2.5.25]. Then the properties (2)–(8) follow from the statements there, and the property (9) follows from the same argument as in the above proof of Proposition 2.2 in Case (Nm), using the analogous canonical isomorphism between formal completions in [Reference LanLan15b, Theorem 3.4.1(3)], the flatness of strata in [Reference LanLan15b, Theorem 3.4.1(2)], and Artin’s approximation. (Note that the stratification of $\vec{\mathsf{M}}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{spl},\text{tor}}$ is the pullback of the one of $\vec{\mathsf{M}}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}$ , by [Reference LanLan15b, Definition 3.1.8 and (3.1.9)], which is just a base change from $F_{0}$ to $K$ in characteristic zero, by [Reference LanLan15b, Proposition 2.3.10].)◻

Proof of Proposition 2.2 in Case (Hdg).

In this case, $\mathsf{X}_{{\mathcal{H}}}\rightarrow \mathsf{S}$ is the pullback of some integral model $\mathscr{S}_{K}\rightarrow \operatorname{Spec}(\mathscr{O}_{E,(v)})$ as in [Reference Madapusi PeraMad15, Introduction], and we can take $\mathsf{X}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}\rightarrow \mathsf{S}$ and $\mathsf{X}_{{\mathcal{H}}}^{\min }\rightarrow \mathsf{S}$ to be the pullbacks of the toroidal and minimal compactifications $\mathscr{S}_{K}^{\unicode[STIX]{x1D6F4}}\rightarrow \operatorname{Spec}(\mathscr{O}_{E,(v)})$ and $\mathscr{S}_{K}^{\text{min}}\rightarrow \operatorname{Spec}(\mathscr{O}_{E,(v)})$ in [Reference Madapusi PeraMad15, Theorems 4.1.5 and 5.2.11], where $\unicode[STIX]{x1D6F4}$ is induced by some auxiliary choice $\widetilde{\unicode[STIX]{x1D6F4}}$ for ${\mathcal{S}}_{{\mathcal{K}}}$ as in [Reference Madapusi PeraMad15, §4.1.4], or some refinement of it (see [Reference Madapusi PeraMad15, Remark 4.1.6]) which can be assumed to be projective and smooth. Hence, the properties (2)–(8) follow from the statements there. By the proof of [Reference Madapusi PeraMad15, Proposition 4.2.11], each stratum $\mathsf{Z}_{[\unicode[STIX]{x1D70E}]}$ of $\mathsf{X}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}$ is open and closed in the preimage of a stratum of an auxiliary good reduction toroidal compactification in Case (Sm), and the isomorphism in the property (8) is compatible with the pullback of the corresponding isomorphism in Case (Sm). Therefore, the property (9) follows from also approximating such open and closed subsets of the preimages in the argument in Case (Nm). ◻

Proof. It suffices to prove these in Cases (Sm), (Nm), and (Spl), because the construction in Case (Hdg) is the pullback of the analogous construction in Case (Sm). In all of these three cases, the abelian scheme $\mathsf{f}:\mathsf{A}\rightarrow \mathsf{X}_{{\mathcal{H}}}$ extends to some object $(\mathsf{A},\unicode[STIX]{x1D706}_{\mathsf{A}},i_{\mathsf{A}},\unicode[STIX]{x1D6FC}_{\mathsf{A},{\mathcal{H}}^{\Box }})$ , where $\unicode[STIX]{x1D706}_{\mathsf{A}}:\mathsf{A}\rightarrow \mathsf{A}^{\vee }$ is a polarization, $i_{\mathsf{A}}:{\mathcal{O}}\rightarrow \operatorname{End}_{\mathsf{X}_{{\mathcal{H}}}}(\mathsf{A})$ is an ${\mathcal{O}}$ -endomorphism structure for $(\mathsf{A},\unicode[STIX]{x1D706}_{\mathsf{A}})$ as in [Reference LanLan13, Definition 1.3.3.1], and where $\unicode[STIX]{x1D6FC}_{\mathsf{A},{\mathcal{H}}^{\Box }}$ is a level- ${\mathcal{H}}^{\Box }$ structure for $(\mathsf{A}\otimes _{\mathbb{Z}}\mathbb{Q},\unicode[STIX]{x1D706}_{\mathsf{A}}\otimes _{\mathbb{Z}}\mathbb{Q},i_{\mathsf{A}}\otimes _{\mathbb{Z}}\mathbb{Q})$ , with $\Box =\emptyset$ in Cases (Nm) and (Spl), such that $(\mathsf{A}\otimes _{\mathbb{Z}}\mathbb{Q},\unicode[STIX]{x1D706}_{\mathsf{A}}\otimes _{\mathbb{Z}}\mathbb{Q},i_{\mathsf{A}}\otimes _{\mathbb{Z}}\mathbb{Q},\unicode[STIX]{x1D6FC}_{\mathsf{A},{\mathcal{H}}^{\Box }})$ defines an object of $\mathsf{M}_{{\mathcal{H}}^{\Box }}$ over $\mathsf{X}_{{\mathcal{H}}}\,\otimes _{\mathbb{Z}}\,\mathbb{Q}$ as in [Reference LanLan13, Definition 1.4.1.4]. (In Case (Sm), this follows from the definition of $\mathsf{M}_{{\mathcal{H}}^{\Box }}$ in [Reference LanLan13, §1.4]. In Cases (Nm) and (Spl), this is because $\mathsf{X}_{{\mathcal{H}}}\otimes _{\mathbb{Z}}\mathbb{Q}$ is isomorphic to a base change of some $\mathsf{M}_{{\mathcal{H}}}$ defined in [Reference LanLan13, §1.4], and because $\mathsf{A}\rightarrow \mathsf{X}_{{\mathcal{H}}}$ is isomorphic to the pullback of $\vec{A}_{\text{j}_{0}}\rightarrow \vec{\mathsf{M}}_{{\mathcal{H}}}$ .) For any geometric point $\bar{s}\rightarrow \mathsf{X}_{{\mathcal{H}}}$ of residue characteristic zero, $\unicode[STIX]{x1D6FC}_{\mathsf{A},{\mathcal{H}}^{\Box }}$ induces, in particular, a $\unicode[STIX]{x1D70B}_{1}(\mathsf{X}_{{\mathcal{H}}}\otimes _{\mathbb{Z}}\mathbb{Q},\bar{s})$ -invariant ${\mathcal{H}}_{\ell }$ -orbit of isomorphisms $L\otimes _{\mathbb{Z}}\mathbb{Z}_{\ell }\stackrel{{\sim}}{\rightarrow }\operatorname{T}_{\ell }\mathsf{A}_{\bar{s}}$ matching the pairing $\langle \,\cdot \,,\cdot \,\rangle$ on $L$ with the $\unicode[STIX]{x1D706}_{\mathsf{A}}$ -Weil pairing on $\operatorname{T}_{\ell }\mathsf{A}_{\bar{s}}$ up to scalar multiples, compatible with the ${\mathcal{O}}$ -module structures of $L$ and of $\operatorname{T}_{\ell }\mathsf{A}_{\bar{s}}$ given by $i_{\mathsf{A}}$ . If $\ell \neq p$ , then this isomorphism also extends to similar isomorphisms at geometric points $\bar{s}\rightarrow \mathsf{X}_{{\mathcal{H}}}$ of characteristic $p$ . Hence, the lemma follows from the very definitions of the various objects.◻

Proof. By [Reference LanLan13, Definition 1.2.1.6] and the assumption that $\text{G}\otimes _{\mathbb{Z}}\mathbb{Q}$ is a subgroup of $\text{G}_{\text{aux}}\otimes _{\mathbb{Z}}\mathbb{Q}$ in Case (Hdg), $L\otimes _{\mathbb{Z}}\mathbb{Q}$ is a faithful representation of the reductive group $\text{G}\otimes _{\mathbb{Z}}\mathbb{Q}$ . By [Reference DeligneDel82, Proposition 3.1(a)], the irreducible representation $V_{\unicode[STIX]{x1D709}}$ is a direct summand of $(L\otimes _{\mathbb{Z}}\bar{\mathbb{Q}}_{\ell })^{\otimes a_{\unicode[STIX]{x1D709}}}\otimes (L^{\vee }\otimes _{\mathbb{Z}}\bar{\mathbb{Q}}_{\ell })^{\otimes b_{\unicode[STIX]{x1D709}}}$ , for some integers $a_{\unicode[STIX]{x1D709}},b_{\unicode[STIX]{x1D709}}\geqslant 0$ , where the tensor products are over $\bar{\mathbb{Q}}_{\ell }$ , and where $L^{\vee }:=\operatorname{Hom}_{\mathbb{Z}}(L,\mathbb{Z})$ . Since $\langle \,\cdot \,,\cdot \,\rangle$ induces a perfect pairing $(L\otimes _{\mathbb{Z}}\mathbb{Q}_{\ell })\times (L\otimes _{\mathbb{Z}}\mathbb{Q}_{\ell })\rightarrow \mathbb{Q}_{\ell }(1)$ (where the Tate twist is formal, by tensor product with $\mathbb{Z}(1):=\ker (\exp :\mathbb{C}\rightarrow \mathbb{C}^{\times })$ ), which is matched via the level structure (in the proof of Lemma 3.1) with the $\unicode[STIX]{x1D706}$ -Weil pairing $(\operatorname{V}_{\ell }\mathsf{A}_{\bar{s}})\times (\operatorname{V}_{\ell }\mathsf{A}_{\bar{s}})\rightarrow \operatorname{V}_{\ell }\mathbf{G}_{\text{m},\bar{s}}$ up to scalar multiples at each geometric point $\bar{s}\rightarrow \mathsf{X}_{{\mathcal{H}}}$ of residue characteristic zero, we obtain an isomorphism $L\otimes _{\mathbb{Z}}\mathbb{Q}_{\ell }\cong \operatorname{Hom}_{\mathbb{Z}}(L,\mathbb{Q}_{\ell })(1)$ , which is matched with the isomorphism $\text{}\underline{\operatorname{Hom}}_{\mathbb{Q}_{\ell }}(R^{1}(\mathsf{f}\otimes _{\mathbb{Z}}\mathbb{Q})_{\ast }\mathbb{Q}_{\ell },\mathbb{Q}_{\ell })\cong R^{1}(\mathsf{f}\otimes _{\mathbb{Z}}\mathbb{Q})_{\ast }\mathbb{Q}_{\ell }(1)$ between étale sheaves. Hence, the proposition follows from Lemma 3.1, from [Reference DeligneDel80, 6.2.5(b) and 6.2.6], from the fact that $(R^{1}(\mathsf{f}\otimes _{\mathbb{Z}}\mathbb{Q})_{\ast }\bar{\mathbb{Q}}_{\ell })^{\otimes n_{\unicode[STIX]{x1D709}}}$ is a direct summand of $R^{n_{\unicode[STIX]{x1D709}}}(\mathsf{f}^{\times n_{\unicode[STIX]{x1D709}}}\otimes _{\mathbb{Z}}\mathbb{Q})_{\ast }\bar{\mathbb{Q}}_{\ell }\cong \wedge ^{n_{\unicode[STIX]{x1D709}}}((R^{1}(\mathsf{f}\otimes _{\mathbb{Z}}\mathbb{Q})_{\ast }\bar{\mathbb{Q}}_{\ell })^{\oplus n_{\unicode[STIX]{x1D709}}})$ , and from the fact that $R^{n_{\unicode[STIX]{x1D709}}}(\mathsf{f}^{\times n_{\unicode[STIX]{x1D709}}}\otimes _{\mathbb{Z}}\mathbb{Q})_{\ast }\bar{\mathbb{Q}}_{\ell }$ is a direct summand of $R(\mathsf{f}^{\times n_{\unicode[STIX]{x1D709}}}\otimes _{\mathbb{Z}}\mathbb{Q})_{\ast }\bar{\mathbb{Q}}_{\ell }[n_{\unicode[STIX]{x1D709}}]$ thanks to Lieberman’s trick (cf. [Reference Lan and SuhLS12, §3.2]).◻

Proof. By [Reference DeligneDel82, Proposition 3.1(a)], there exist $a_{W},b_{W}\in \mathbb{Z}_{{\geqslant}0}$ such that $W=W_{0}\otimes _{{\mathcal{O}}_{K_{0}}}K_{0}$ is a direct summand of $(L\otimes _{\mathbb{Z}}K_{0})^{\otimes a_{W}}\otimes (L^{\vee }\otimes _{\mathbb{Z}}K_{0})^{\otimes b_{W}}$ . For all sufficiently large $\ell$ , and for each $K$ as in the statement of the proposition, the weights of $W\otimes _{K_{0}}K$ and $(L\otimes _{\mathbb{Z}}K)^{\otimes a_{W}}\otimes (L^{\vee }\otimes _{\mathbb{Z}}K)^{\otimes b_{W}}$ are all $\ell$ -small, and there are no nontrivial extensions between admissible $\text{G}\otimes _{\mathbb{Z}}{\mathcal{O}}_{K}$ -lattices in irreducible representations of $\text{G}\otimes _{\mathbb{Z}}K$ of such $\ell$ -small weights (see, for example, [Reference Polo and TilouinePT02, §1] and the references there, which are applicable because $\text{G}\otimes _{\mathbb{Z}}K$ is split). Hence, there exists some integer $c_{W}\geqslant 0$ (as in the statement of the proposition) such that the induced pairing $\langle \,\cdot \,,\cdot \,\rangle \otimes _{\mathbb{Z}}\mathbb{Z}_{\ell }:(L\otimes _{\mathbb{Z}}\mathbb{Z}_{\ell })\times (L\otimes _{\mathbb{Z}}\mathbb{Z}_{\ell })\rightarrow \mathbb{Z}_{\ell }(1)$ is perfect, and such that $W_{0}\otimes _{{\mathcal{O}}_{K_{0}}}{\mathcal{O}}_{K}$ is a direct summand of $(L\otimes _{\mathbb{Z}}{\mathcal{O}}_{K})^{\otimes a_{W}}\otimes (L^{\vee }\otimes _{\mathbb{Z}}{\mathcal{O}}_{K})^{\otimes b_{W}}$ , whenever $\ell >c_{W}$ and $K$ is as above; and we can finish as in the proof of Proposition 3.2. In Cases (Sm), (Nm), or (Spl), since we are in PEL-type cases with residue characteristics prime to $\ell$ , we can explicitly determine $t_{W}$ , $n_{W}$ , and $c_{W}$ using only the PEL datum $({\mathcal{O}},\star ,L,\langle \,\cdot \,,\cdot \,\rangle ,h_{0})$ and the weights of $W$ , as in [Reference Lan and SuhLS12, §§2–4].◻

Proof of Proposition 4.1 in Case (Sm).

Let $({\mathcal{O}},\star ,L,\langle \,\cdot \,,\cdot \,\rangle ,h_{0})$ be the integral PEL datum defining $\mathsf{M}_{{\mathcal{H}}}$ . By assumption, there exists a maximal order ${\mathcal{O}}^{\prime }$ in ${\mathcal{O}}\otimes _{\mathbb{Z}}\mathbb{Q}$ such that the action of ${\mathcal{O}}$ on $L$ extends to an action of ${\mathcal{O}}^{\prime }$ on $L$ . Consider

$$\begin{eqnarray}\displaystyle Q & := & \displaystyle {\mathcal{O}}^{\oplus n},\nonumber\\ \displaystyle Q_{-2} & := & \displaystyle (\operatorname{Diff}_{{\mathcal{O}}^{\prime }/\mathbb{Z}}^{-1}(1))^{\oplus n}\cong \operatorname{Hom}_{{\mathcal{O}}}(Q,\operatorname{Diff}_{{\mathcal{O}}^{\prime }/\mathbb{Z}}^{-1}(1))\subset Q^{\vee }\otimes _{\mathbb{Z}}\mathbb{Q}(1),\nonumber\\ \displaystyle Q_{0} & := & \displaystyle ({\mathcal{O}}^{\prime })^{\oplus n}\cong {\mathcal{O}}^{\prime }\cdot Q\subset Q\otimes _{\mathbb{Z}}\mathbb{Q},\nonumber\end{eqnarray}$$

where the ${\mathcal{O}}$ -actions on

$$\begin{eqnarray}Q^{\vee }:=\operatorname{Hom}_{\mathbb{ Z}}(Q,\mathbb{Z})\cong \operatorname{Hom}_{{\mathcal{O}}}(Q,\operatorname{Diff}_{{\mathcal{O}}/\mathbb{Z}}^{-1})\end{eqnarray}$$

and similar dual modules are induced by the right action of ${\mathcal{O}}^{\text{op}}$ (and the anti-isomorphism $\star :{\mathcal{O}}\stackrel{{\sim}}{\rightarrow }{\mathcal{O}}^{\text{op}}$ ), and consider the canonical pairing

$$\begin{eqnarray}\langle \,\cdot \,,\cdot \,\rangle _{Q}:Q_{-2}\times Q_{0}\rightarrow \mathbb{Z}(1)\end{eqnarray}$$

given by

$$\begin{eqnarray}\langle (x_{-2},x_{0}),(y_{-2},y_{0})\rangle _{Q}=\operatorname{Tr}_{({\mathcal{O}}\otimes _{\mathbb{Z}}\mathbb{Q})/\mathbb{Q}}(\text{}^{t}x_{-2}\,y_{0}-\text{}^{t}x_{0}\,y_{-2}).\end{eqnarray}$$

Consider the integral PEL datum $({\mathcal{O}},\star ,\widetilde{L},\langle \,\cdot \,,\cdot \,\rangle ^{\widetilde{~}},\widetilde{h})$ defined as follows:

• ∙ $\widetilde{L}:=Q_{-2}\oplus L\oplus Q_{0}$ ;

• ∙ $\langle \,\cdot \,,\cdot \,\rangle ^{\widetilde{~}}:\widetilde{L}\times \widetilde{L}\rightarrow \mathbb{Z}(1)$ is defined (symbolically) by the matrix

  $$\begin{eqnarray}\langle x,y\rangle ^{\widetilde{~}}:=\text{}^{t}\left(\begin{array}{@{}c@{}}x_{-2}\\ x_{-1}\\ x_{0}\end{array}\right)\left(\begin{array}{@{}ccc@{}} & & \langle \,\cdot \,,\cdot \,\rangle _{Q}\\ & \langle \,\cdot \,,\cdot \,\rangle & \\ -\text{}^{t}\langle \,\cdot \,,\cdot \,\rangle _{Q}\end{array}\right)\left(\begin{array}{@{}c@{}}y_{-2}\\ y_{-1}\\ y_{0}\end{array}\right);\end{eqnarray}$$

• ∙ $\widetilde{h}:\mathbb{C}\rightarrow \operatorname{End}_{{\mathcal{O}}\otimes _{\mathbb{Z}}\mathbb{R}}(\widetilde{L}\otimes _{\mathbb{Z}}\mathbb{R})$ is defined by

  $$\begin{eqnarray}\displaystyle z & = & \displaystyle z_{1}+\sqrt{-1}\;z_{2}\nonumber\\ \displaystyle & \mapsto & \displaystyle \widetilde{h}(z):=\left(\begin{array}{@{}ccc@{}}z_{1}\operatorname{Id}_{Q_{-2}\otimes _{\mathbb{Z}}\mathbb{R}} & & -z_{2}((2\unicode[STIX]{x1D70B}\sqrt{-1})\circ j_{Q}^{-1})\\ & h(z) & \\ z_{2}(j_{Q}\circ (2\unicode[STIX]{x1D70B}\sqrt{-1})^{-1}) & & z_{1}\operatorname{Id}_{Q_{0}\otimes _{\mathbb{Z}}\mathbb{R}}\end{array}\right),\nonumber\end{eqnarray}$$
  where $2\unicode[STIX]{x1D70B}\sqrt{-1}:\mathbb{Z}\stackrel{{\sim}}{\rightarrow }\mathbb{Z}(1)$ and $(2\unicode[STIX]{x1D70B}\sqrt{-1})^{-1}:\mathbb{Z}(1)\stackrel{{\sim}}{\rightarrow }\mathbb{Z}$ stand for the isomorphisms defined by the choice of $\sqrt{-1}$ in $\mathbb{C}$ , and where
  $$\begin{eqnarray}j_{Q}:(\operatorname{Diff}_{{\mathcal{O}}^{\prime }/\mathbb{Z}}^{-1})^{\oplus n}\otimes _{\mathbb{ Z}}\mathbb{R}\cong Q^{\vee }\otimes _{\mathbb{ Z}}\mathbb{R}\stackrel{{\sim}}{\rightarrow }Q\otimes _{\mathbb{Z}}\mathbb{R}\cong ({\mathcal{O}}^{\prime })^{\oplus n}\otimes _{\mathbb{ Z}}\mathbb{R}\end{eqnarray}$$
  can be identified with the identity morphism on ${\mathcal{O}}^{\oplus n}\otimes _{\mathbb{Z}}\mathbb{R}$ .

These define a group functor $\widetilde{\text{G}}$ as in [Reference LanLan13, Definition 1.2.1.6] (and the same reflex field $F_{0}$ ). By construction, there is a fully symplectic admissible filtration $\widetilde{\mathtt{Z}}$ on $\widetilde{L}\otimes _{\mathbb{Z}}\hat{\mathbb{Z}}^{\Box }$ (see [Reference LanLan13, Definition 5.2.7.3]) induced by

$$\begin{eqnarray}\displaystyle \widetilde{\mathtt{Z}}_{-3}:=0 & \subset & \displaystyle \widetilde{\mathtt{Z}}_{-2}:=Q_{-2}\otimes _{\mathbb{Z}}\hat{\mathbb{Z}}^{\Box }\nonumber\\ \displaystyle & \subset & \displaystyle \widetilde{\mathtt{Z}}_{-1}:=(Q_{-2}\oplus L)\otimes _{\mathbb{Z}}\hat{\mathbb{Z}}^{\Box }\nonumber\\ \displaystyle & \subset & \displaystyle \widetilde{\mathtt{Z}}_{0}:=(Q_{-2}\oplus L\oplus Q_{0})\otimes _{\mathbb{Z}}\hat{\mathbb{Z}}^{\Box }=\widetilde{L}\otimes _{\mathbb{Z}}\hat{\mathbb{Z}}^{\Box },\nonumber\end{eqnarray}$$

so that there are canonical isomorphisms $\operatorname{Gr}_{-2}^{\widetilde{\mathtt{Z}}}\cong Q_{-2}\,\otimes _{\mathbb{Z}}\,\hat{\mathbb{Z}}^{\Box }$ , $\operatorname{Gr}_{-1}^{\widetilde{\mathtt{Z}}}\cong L\,\otimes _{\mathbb{Z}}\,\hat{\mathbb{Z}}^{\Box }$ , and $\operatorname{Gr}_{0}^{\widetilde{\mathtt{Z}}}\cong Q_{0}\,\otimes _{\mathbb{Z}}\,\hat{\mathbb{Z}}^{\Box }$ matching the pairings $\operatorname{Gr}_{-2}^{\widetilde{\mathtt{Z}}}\times \operatorname{Gr}_{0}^{\widetilde{\mathtt{Z}}}\rightarrow \hat{\mathbb{Z}}^{\Box }(1)$ and $\operatorname{Gr}_{-1}^{\widetilde{\mathtt{Z}}}\times \operatorname{Gr}_{-1}^{\widetilde{\mathtt{Z}}}\rightarrow \hat{\mathbb{Z}}^{\Box }(1)$ induced by $\langle \,\cdot \,,\cdot \,\rangle ^{\widetilde{~}}$ with $\langle \,\cdot \,,\cdot \,\rangle _{Q}$ and $\langle \,\cdot \,,\cdot \,\rangle$ , respectively. Let

$$\begin{eqnarray}\widetilde{X}:=\operatorname{Hom}_{{\mathcal{O}}}(Q_{-2},\operatorname{Diff}_{{\mathcal{O}}/\mathbb{Z}}^{-1}(1))\cong ({\mathcal{O}}^{\prime })^{\oplus n}\end{eqnarray}$$

and

$$\begin{eqnarray}\widetilde{Y}:=Q_{0}=({\mathcal{O}}^{\prime })^{\oplus n}.\end{eqnarray}$$

Then the pairing $\langle \,\cdot \,,\cdot \,\rangle _{Q}:Q_{-2}\times Q_{0}\rightarrow \mathbb{Z}(1)$ induces a canonical embedding $\widetilde{\unicode[STIX]{x1D719}}:\widetilde{Y}{\hookrightarrow}\widetilde{X}$ , which can be identified with the identity morphism on $({\mathcal{O}}^{\prime })^{\oplus n}$ in this case, and there are canonical isomorphisms $\widetilde{\unicode[STIX]{x1D711}}_{-2}:\operatorname{Gr}_{-2}^{\widetilde{\mathtt{Z}}}\stackrel{{\sim}}{\rightarrow }\operatorname{Hom}_{\hat{\mathbb{Z}}^{\Box }}(\widetilde{X}\otimes _{\mathbb{Z}}\hat{\mathbb{Z}}^{\Box },\hat{\mathbb{Z}}^{\Box }(1))$ and $\widetilde{\unicode[STIX]{x1D711}}_{0}:\operatorname{Gr}_{0}^{\widetilde{\mathtt{Z}}}\stackrel{{\sim}}{\rightarrow }\widetilde{Y}\otimes _{\mathbb{Z}}\hat{\mathbb{Z}}^{\Box }$ (of $\hat{\mathbb{Z}}^{\Box }$ -modules). These data define a torus argument

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6F7}}:=(\widetilde{X},\widetilde{Y},\widetilde{\unicode[STIX]{x1D719}},\widetilde{\unicode[STIX]{x1D711}}_{-2},\widetilde{\unicode[STIX]{x1D711}}_{0})\end{eqnarray}$$

for $\widetilde{\mathtt{Z}}$ as in [Reference LanLan13, Definition 5.4.1.3]. Let

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6FF}}:\operatorname{Gr}^{\widetilde{\mathtt{Z}}}=\operatorname{Gr}_{-2}^{\widetilde{\mathtt{Z}}}\oplus \operatorname{Gr}_{-1}^{\widetilde{\mathtt{Z}}}\oplus \operatorname{Gr}_{0}^{\widetilde{\mathtt{Z}}}\stackrel{{\sim}}{\rightarrow }L\otimes _{\mathbb{ Z}}\hat{\mathbb{Z}}^{\Box }\end{eqnarray}$$

be the obvious splitting of $\widetilde{\mathtt{Z}}$ induced by the equality $Q_{-2}\oplus L\oplus Q_{0}=\widetilde{L}$ .

For any $\hat{\mathbb{Z}}^{\Box }$ -algebra $R$ , let $\widetilde{\text{P}}_{\widetilde{\mathtt{Z}}}(R)$ denote the subgroup of $\widetilde{\text{G}}(R)$ consisting of elements $g$ such that $g(\widetilde{\mathtt{Z}}_{-2}\otimes _{\hat{\mathbb{Z}}^{\Box }}R)=\widetilde{\mathtt{Z}}_{-2}\otimes _{\hat{\mathbb{Z}}^{\Box }}R$ and $g(\widetilde{\mathtt{Z}}_{-1}\otimes _{\hat{\mathbb{Z}}^{\Box }}R)=\widetilde{\mathtt{Z}}_{-1}\otimes _{\hat{\mathbb{Z}}^{\Box }}R$ . Then we have a homomorphism $\operatorname{Gr}_{-1}^{\widetilde{\mathtt{Z}}}:\widetilde{\text{P}}_{\widetilde{\mathtt{Z}}}(\hat{\mathbb{Z}}^{\Box })\rightarrow \text{G}(\hat{\mathbb{Z}}^{\Box })$ defined by taking graded pieces. Let $\widetilde{\text{P}}_{\widetilde{\mathtt{Z}}}^{\prime }(\hat{\mathbb{Z}}^{\Box })$ be the kernel of $\operatorname{Gr}_{-2}^{\widetilde{\mathtt{Z}}}\times \operatorname{Gr}_{0}^{\widetilde{\mathtt{Z}}}$ , where the homomorphisms $\operatorname{Gr}_{-2}^{\widetilde{\mathtt{Z}}}$ and $\operatorname{Gr}_{0}^{\widetilde{\mathtt{Z}}}$ are defined analogously. Let $\widetilde{{\mathcal{H}}}^{\Box }$ be any neat open compact subgroup of $\widetilde{\text{G}}(\hat{\mathbb{Z}}^{\Box })$ satisfying the following conditions.

1. (1) We have equalities $\operatorname{Gr}_{-1}^{\widetilde{\mathtt{Z}}}(\widetilde{{\mathcal{H}}}^{\Box }\cap \widetilde{\text{P}}_{\widetilde{\mathtt{Z}}}^{\prime }(\hat{\mathbb{Z}}^{\Box }))=\operatorname{Gr}_{-1}^{\widetilde{\mathtt{Z}}}(\widetilde{{\mathcal{H}}}^{\Box }\cap \widetilde{\text{P}}_{\widetilde{\mathtt{Z}}}(\hat{\mathbb{Z}}^{\Box }))={\mathcal{H}}^{\Box }$ . (Both equalities are conditions. Then ${\mathcal{H}}^{\Box }$ is a direct factor of $\operatorname{Gr}^{\widetilde{\mathtt{Z}}}(\widetilde{{\mathcal{H}}}^{\Box }\cap \widetilde{\text{P}}_{\widetilde{\mathtt{Z}}}(\hat{\mathbb{Z}}^{\Box }))$ .)

2. (2) The splitting $\widetilde{\unicode[STIX]{x1D6FF}}$ defines a (group-theoretic) splitting of the surjection $\widetilde{{\mathcal{H}}}^{\Box }\cap \widetilde{\text{P}}_{\widetilde{\mathtt{Z}}}^{\prime }(\hat{\mathbb{Z}}^{\Box }){\twoheadrightarrow}{\mathcal{H}}^{\Box }$ induced by $\operatorname{Gr}_{-1}^{\widetilde{\mathtt{Z}}}$ .

(Such an $\widetilde{{\mathcal{H}}}^{\Box }$ exists because the pairing $\langle \,\cdot \,,\cdot \,\rangle ^{\widetilde{~}}$ is the direct sum of the pairings on $Q_{-2}\oplus Q_{0}$ and on $L$ .) The data of ${\mathcal{O}}$ , $(\widetilde{L},\langle \,\cdot \,,\cdot \,\rangle ^{\widetilde{~}},\widetilde{h})$ , $\Box$ , and $\widetilde{{\mathcal{H}}}^{\Box }\subset \widetilde{\text{G}}(\hat{\mathbb{Z}}^{\Box })$ define a moduli problem $\widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}}}$ as in [Reference LanLan13, Definition 1.4.1.4]. Let $\widetilde{{\mathcal{H}}}:=\widetilde{{\mathcal{H}}}^{\Box }\times \prod _{q\in \Box }\widetilde{\text{G}}(\mathbb{Z}_{q})$ .

Take any projective smooth $\widetilde{\unicode[STIX]{x1D6F4}}$ for $\widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}}^{\Box }}$ , which defines a toroidal compactification $\widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}}^{\Box },\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{tor}}$ which is projective and smooth over $\mathsf{S}_{0}$ by [Reference LanLan13, Theorem 7.3.3.4]. Let $(\widetilde{\mathtt{Z}},\widetilde{\unicode[STIX]{x1D6F7}},\widetilde{\unicode[STIX]{x1D6FF}})$ be as above, and let $(\widetilde{\mathtt{Z}}_{\widetilde{{\mathcal{H}}}^{\Box }},\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}^{\Box }}=(\widetilde{X},\widetilde{Y},\widetilde{\unicode[STIX]{x1D719}},\widetilde{\unicode[STIX]{x1D711}}_{-2,\widetilde{{\mathcal{H}}}^{\Box }},\widetilde{\unicode[STIX]{x1D711}}_{0,\widetilde{{\mathcal{H}}}^{\Box }}),\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}^{\Box }})$ be the induced triple at level $\widetilde{{\mathcal{H}}}^{\Box }$ , inducing a cusp label $[(\widetilde{\mathtt{Z}}_{\widetilde{{\mathcal{H}}}^{\Box }},\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}^{\Box }},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}^{\Box }})]$ at level $\widetilde{{\mathcal{H}}}^{\Box }$ . Let $\widetilde{\unicode[STIX]{x1D70E}}\subset \mathbf{P}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}^{\Box }}}^{+}$ be any top-dimensional nondegenerate rational polyhedral cone in the cone decomposition $\widetilde{\unicode[STIX]{x1D6F4}}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}^{\Box }}}$ in $\widetilde{\unicode[STIX]{x1D6F4}}$ . By [Reference LanLan13, Theorems 6.4.1.1(2) and 7.2.4.1(5); see also the errata], the stratum $\widetilde{\mathsf{Z}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}^{\Box }},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}^{\Box }},\widetilde{\unicode[STIX]{x1D70E}})]}$ of $\widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}}^{\Box }}^{\text{tor}}$ is a zero-dimensional torus bundle over the abelian scheme $\widetilde{C}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}^{\Box }},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}^{\Box }}}$ over $\mathsf{M}_{{\mathcal{H}}^{\Box }}$ . (We have canonical isomorphisms $\widetilde{\mathsf{Z}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}^{\Box }},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}^{\Box }})]}\cong \widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}}^{\Box }}^{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}^{\Box }}}\cong \widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}}^{\Box }}^{\widetilde{\mathtt{Z}}_{\widetilde{{\mathcal{H}}}^{\Box }}}\cong \mathsf{M}_{{\mathcal{H}}^{\Box }}$ because of the condition (1) above on the choice of $\widetilde{{\mathcal{H}}}^{\Box }$ . The abelian scheme torsor $\widetilde{C}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}^{\Box }},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}^{\Box }}}\rightarrow \widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}}^{\Box }}^{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}^{\Box }}}$ is an abelian scheme because of the condition (2) above on the choice of $\widetilde{{\mathcal{H}}}^{\Box }$ .) In other words, $\widetilde{\mathsf{Z}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}^{\Box }},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}^{\Box }},\widetilde{\unicode[STIX]{x1D70E}})]}\cong \widetilde{C}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}^{\Box }},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}^{\Box }}}$ . By the construction of $\widetilde{C}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}^{\Box }},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}^{\Box }}}$ as in [Reference LanLan13, §§6.2.3–6.2.4], and by the same arguments as in the proofs of [Reference LanLan17a, Lemmas 8.1.3.2 and 8.1.3.5], it is $\mathbb{Z}_{(\Box )}^{\times }$ -isogenous to the abelian scheme $\text{}\underline{\operatorname{Hom}}_{{\mathcal{O}}}(Q,A)^{\circ }\cong A^{\times n}$ over $\mathsf{M}_{{\mathcal{H}}^{\Box }}$ . Since $\ell \not \in \Box$ , up to modifying the choice of $\widetilde{{\mathcal{H}}}^{\Box }$ , this $\mathbb{Z}_{(\Box )}^{\times }$ -isogeny can be assumed to be also a $\mathbb{Z}_{(\ell )}^{\times }$ -isogeny.

To finish, it suffices to take $\widetilde{\mathsf{X}}_{\widetilde{{\mathcal{H}}}}\rightarrow \mathsf{S}$ , $\widetilde{\mathsf{X}}_{\widetilde{{\mathcal{H}}}}^{\min }\rightarrow \mathsf{S}$ , $\widetilde{\mathsf{X}}_{\widetilde{{\mathcal{H}}},\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{tor}}\rightarrow \mathsf{S}$ , and $\widetilde{\mathsf{Z}}_{[\widetilde{\unicode[STIX]{x1D70E}}]}\rightarrow \widetilde{\mathsf{Z}}$ to be the respective pullbacks (under $\mathsf{S}\rightarrow \operatorname{Spec}({\mathcal{O}}_{F_{0},(\Box )})$ ) of $\widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}}^{\Box }}\rightarrow \operatorname{Spec}({\mathcal{O}}_{F_{0},(\Box )})$ , $\widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}}^{\Box }}^{\min }\rightarrow \operatorname{Spec}({\mathcal{O}}_{F_{0},(\Box )})$ , $\widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}}^{\Box },\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{tor}}\rightarrow \operatorname{Spec}({\mathcal{O}}_{F_{0},(\Box )})$ , and $\widetilde{\mathsf{Z}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}^{\Box }},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}^{\Box }},\widetilde{\unicode[STIX]{x1D70E}})]}\rightarrow \widetilde{\mathsf{Z}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}^{\Box }},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}^{\Box }})]}$ .◻

Proof of Proposition 4.2 in Case (Sm).

In this case, the proposition follows [Reference LanLan12, §3B, the paragraph following Condition 3.8]. ◻

Proof of Proposition 4.1 in Case (Nm).

Let us proceed with the same choices made in the proof of Proposition 4.1 in Case (Sm), with $\Box =\emptyset$ . (We shall henceforth omit all the superscripts with $\Box$ .) In this case, $\vec{\mathsf{M}}_{{\mathcal{H}}}\rightarrow \vec{\mathsf{S}}_{0}=\operatorname{Spec}({\mathcal{O}}_{F_{0},(p)})$ is defined with some choice of $\{(L_{\text{j}},\langle \,\cdot \,,\cdot \,\rangle _{\text{j}})\}_{\text{j}\in \text{J}}$ , with $(L_{\text{j}_{0}},\langle \,\cdot \,,\cdot \,\rangle _{\text{j}_{0}})=(p^{r_{0}}L,p^{-2r_{0}}\langle \,\cdot \,,\cdot \,\rangle )$ for some $\text{j}_{0}\in \text{J}$ and some $r_{0}\in \mathbb{Z}$ , which allows us to take $\mathsf{f}:\mathsf{A}\rightarrow \mathsf{X}_{{\mathcal{H}}}$ to be $\vec{A}_{\text{j}_{0}}\rightarrow \vec{\mathsf{M}}_{{\mathcal{H}}}$ .

For each $\text{j}\in \text{J}$ , let $(\widetilde{L}_{\text{j}},\langle \,\cdot \,,\cdot \,\rangle _{\text{j}}^{\widetilde{~}})$ be defined as in the case of $(\widetilde{L},\langle \,\cdot \,,\cdot \,\rangle ^{\widetilde{~}})$ , but with $\langle \,\cdot \,,\cdot \,\rangle _{Q}$ and $(L,\langle \,\cdot \,,\cdot \,\rangle )$ replaced with $r_{\text{j}}\langle \,\cdot \,,\cdot \,\rangle _{Q}$ and $(L_{\text{j}},r_{\text{j}}^{\prime }\langle \,\cdot \,,\cdot \,\rangle _{\text{j}})$ for some sufficiently large integers $r_{\text{j}},r_{\text{j}}^{\prime }\geqslant 1$ such that $\langle \,\cdot \,,\cdot \,\rangle _{\text{j}}^{\widetilde{~}}$ is induced by the restriction of $r_{\text{j}}\langle \,\cdot \,,\cdot \,\rangle ^{\widetilde{~}}\otimes _{\mathbb{Z}}\mathbb{Q}$ .

For any $\hat{\mathbb{Z}}$ -algebra $R$ , let $\widetilde{\text{U}}_{\widetilde{\mathtt{Z}}}(R)$ denote the subgroup of $\widetilde{\text{P}}(R)$ consisting of elements $g$ such that $g(\widetilde{\mathtt{Z}}_{-i}\,\otimes _{\hat{\mathbb{Z}}^{\Box }}\,R)\subset \widetilde{\mathtt{Z}}_{-i-1}\otimes _{\hat{\mathbb{Z}}^{\Box }}R$ , for all $i$ ; let $\widetilde{\text{U}}_{\widetilde{\mathtt{Z}},-2}(R)$ denote the subgroup of $\widetilde{\text{U}}_{\widetilde{\mathtt{Z}}}(R)$ consisting of elements $g$ such that $g(\widetilde{\mathtt{Z}}_{-i}\otimes _{\hat{\mathbb{Z}}^{\Box }}R)\subset \widetilde{\mathtt{Z}}_{-i-2}\otimes _{\hat{\mathbb{Z}}^{\Box }}R$ , for all $i$ ; and let $\widetilde{\text{U}}_{\widetilde{\mathtt{Z}},-1}(R):=\widetilde{\text{U}}_{\widetilde{\mathtt{Z}}}(R)/\widetilde{\text{U}}_{\widetilde{\mathtt{Z}},-2}(R)$ .

Each element $g\in \widetilde{\text{U}}_{\widetilde{\mathtt{Z}},-1}(\hat{\mathbb{Z}})$ is determined by its induced morphisms

$$\begin{eqnarray}g_{12}:\operatorname{Gr}_{-1}^{\widetilde{\mathtt{Z}}}=L\otimes _{\mathbb{ Z}}\hat{\mathbb{Z}}\rightarrow \operatorname{Gr}_{-2}^{\widetilde{\mathtt{Z}}}=Q_{-2}\otimes _{\mathbb{Z}}\hat{\mathbb{Z}}=(\operatorname{Diff}_{{\mathcal{O}}^{\prime }/\mathbb{Z}}^{-1})^{\oplus n}\end{eqnarray}$$

and

$$\begin{eqnarray}g_{01}:\operatorname{Gr}_{0}^{\widetilde{\mathtt{Z}}}=({\mathcal{O}}^{\prime })^{\oplus n}\rightarrow \operatorname{Gr}_{-1}^{\widetilde{\mathtt{Z}}}=L\otimes _{\mathbb{ Z}}\hat{\mathbb{Z}}\end{eqnarray}$$

on the graded pieces, and $g^{m}$ induces $mg_{12}$ and $mg_{01}$ for each integer $m$ , which satisfy

$$\begin{eqnarray}mg_{12}(L_{\text{j}}\otimes _{\mathbb{Z}}\hat{\mathbb{Z}})\subset (\operatorname{Diff}_{{\mathcal{O}}^{\prime }/\mathbb{Z}}^{-1})^{\oplus n}\end{eqnarray}$$

and

$$\begin{eqnarray}mg_{01}(({\mathcal{O}}^{\prime })^{\oplus n})\subset L_{\text{j}}\otimes _{\mathbb{Z}}\hat{\mathbb{Z}}\end{eqnarray}$$

for all $\text{j}\in \text{J}$ and all sufficiently large $m$ . Consequently, up to replacing $\widetilde{{\mathcal{H}}}$ with an open compact subgroup, we may and we shall assume that it also satisfies the following additional requirements.

(3)

The action of $\widetilde{{\mathcal{H}}}$ stabilizes all the lattices $\widetilde{L}_{\text{j}}\otimes _{\mathbb{Z}}\hat{\mathbb{Z}}$ .

(4)

The subgroup $(\widetilde{{\mathcal{H}}}\cap \widetilde{\text{U}}_{\widetilde{\mathtt{Z}}}(\hat{\mathbb{Z}}))/(\widetilde{{\mathcal{H}}}\cap \widetilde{\text{U}}_{\widetilde{\mathtt{Z}},-2}(\hat{\mathbb{Z}}))$ of $\widetilde{\text{U}}_{\widetilde{\mathtt{Z}},-1}(\hat{\mathbb{Z}})$ is of the form $(\widetilde{\text{U}}_{\widetilde{\mathtt{Z}},-1}(\hat{\mathbb{Z}}))^{m}$ for some integer $m\geqslant 1$ satisfying $mL\subset L_{\text{j}}$ for all $\text{j}\in \text{J}$ .

These choices of $\{(\widetilde{L}_{\text{j}},\langle \,\cdot \,,\cdot \,\rangle _{\text{j}}^{\widetilde{~}})\}_{\text{j}\in \text{J}}$ and $\widetilde{{\mathcal{H}}}$ (and, in particular, the condition (3) above) allow us to define a moduli problem $\widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}}}\rightarrow \operatorname{Spec}(F_{0})$ as in [Reference LanLan13, Definition 1.4.1.4] and an integral model $\vec{\widetilde{\mathsf{M}}}_{\widetilde{{\mathcal{H}}}}\rightarrow \operatorname{Spec}({\mathcal{O}}_{F_{0},(p)})$ as in [Reference LanLan16, Proposition 6.1], and its minimal compactification $\widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}}}^{\min }\rightarrow \operatorname{Spec}(F_{0})$ has an integral model $\vec{\widetilde{\mathsf{M}}}_{\widetilde{{\mathcal{H}}}}^{\min }\rightarrow \operatorname{Spec}({\mathcal{O}}_{F_{0},(p)})$ as in [Reference LanLan16, Proposition 6.4, and Theorems 12.1 and 12.16]. Moreover, for any projective smooth $\widetilde{\unicode[STIX]{x1D6F4}}$ for $\widetilde{\mathsf{M}}$ , the toroidal compactification $\widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}},\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{tor}}\rightarrow \operatorname{Spec}(F_{0})$ as in [Reference LanLan13, Theorems 6.4.1.1 and 7.3.3.4] extends to an integral model $\vec{\widetilde{\mathsf{M}}}_{\widetilde{{\mathcal{H}}},\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{tor}}\rightarrow \operatorname{Spec}({\mathcal{O}}_{F_{0},(p)})$ as in [Reference LanLan17b, Theorem 6.1]. Then the pullbacks of $\vec{\widetilde{\mathsf{M}}}_{\widetilde{{\mathcal{H}}}}\rightarrow \operatorname{Spec}({\mathcal{O}}_{F_{0},(p)})$ , $\vec{\widetilde{\mathsf{M}}}_{\widetilde{{\mathcal{H}}}}^{\min }\rightarrow \operatorname{Spec}({\mathcal{O}}_{F_{0},(p)})$ , and $\vec{\widetilde{\mathsf{M}}}_{\widetilde{{\mathcal{H}}},\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{tor}}\rightarrow \operatorname{Spec}({\mathcal{O}}_{F_{0},(p)})$ , for all projective smooth $\widetilde{\unicode[STIX]{x1D6F4}}$ , define the models $\widetilde{\mathsf{X}}_{\widetilde{{\mathcal{H}}}}\rightarrow \mathsf{S}$ , $\widetilde{\mathsf{X}}_{\widetilde{{\mathcal{H}}}}^{\min }\rightarrow \mathsf{S}$ , and $\widetilde{\mathsf{X}}_{\widetilde{{\mathcal{H}}},\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{tor}}\rightarrow \mathsf{S}$ , which satisfy the analogue of Proposition 2.2 for $\widetilde{\mathsf{X}}_{\widetilde{{\mathcal{H}}}}$ etc.

We claim that the pullback $\widetilde{\mathsf{Z}}_{[\widetilde{\unicode[STIX]{x1D70E}}]}\rightarrow \widetilde{\mathsf{Z}}$ of $\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}\rightarrow \vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}})]}$ under $\mathsf{S}\rightarrow \operatorname{Spec}({\mathcal{O}}_{F_{0},(p)})$ is isomorphic to the abelian scheme $\mathsf{f}^{\times n}:\mathsf{A}^{\times n}\rightarrow \mathsf{X}_{{\mathcal{H}}}$ we want.

Firstly, by the constructions of $\vec{\widetilde{\mathsf{M}}}_{\widetilde{{\mathcal{H}}}}^{\widetilde{\mathtt{Z}}_{\widetilde{{\mathcal{H}}}}}$ and $\vec{\widetilde{\mathsf{M}}}_{\widetilde{{\mathcal{H}}}}^{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}}}$ by taking normalizations (see [Reference LanLan16, Propositions 7.4 and 8.1]), the canonical isomorphisms $\widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}}}^{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}}}\cong \widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}}}^{\widetilde{\mathtt{Z}}_{\widetilde{{\mathcal{H}}}}}\cong \mathsf{M}_{{\mathcal{H}}}$ over $\operatorname{Spec}(F_{0})$ (because of the condition (1) on the choice of $\widetilde{{\mathcal{H}}}$ in the proof in Case (Sm)) induce canonical isomorphisms $\vec{\widetilde{\mathsf{M}}}_{\widetilde{{\mathcal{H}}}}^{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}}}\cong \vec{\widetilde{\mathsf{M}}}_{\widetilde{{\mathcal{H}}}}^{\widetilde{\mathtt{Z}}_{\widetilde{{\mathcal{H}}}}}\cong \vec{\mathsf{M}}_{{\mathcal{H}}}$ over $\operatorname{Spec}({\mathcal{O}}_{F_{0},(p)})$ . Secondly, the condition (2) above on the choice of $\widetilde{{\mathcal{H}}}$ implies that the abelian scheme torsor $\widetilde{C}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}}}\rightarrow \widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}}}^{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}}}$ over $\operatorname{Spec}(F_{0})$ is an abelian scheme. Moreover, the condition (4) above on the choice of $\widetilde{{\mathcal{H}}}$ implies that the canonical morphism $\widetilde{C}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}}}\rightarrow \text{}\underline{\operatorname{Hom}}_{{\mathcal{O}}}(Q,A)^{\circ }\cong A^{\times n}$ over $\widetilde{\mathsf{Z}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}})]}\cong \widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}}}^{\widetilde{\mathtt{Z}}_{\widetilde{{\mathcal{H}}}}}\cong \mathsf{M}_{{\mathcal{H}}}$ (where $A\rightarrow \mathsf{M}_{{\mathcal{H}}}$ is the tautological abelian scheme) can be identified with $[m]:A^{\times n}\rightarrow A^{\times n}$ (the multiplication by $m$ over $\mathsf{M}_{{\mathcal{H}}}$ ). Since $p^{r_{0}}L=L_{\text{j}_{0}}$ and $mL\subset L_{\text{j}}$ , for all $\text{j}\in \text{J}$ , there exists some integer $r_{1}\geqslant 0$ (depending on $r_{0}$ ) such that $[mp^{r_{1}}]:A^{\times n}\rightarrow A^{\times n}$ extends to isogenies $\vec{A}_{\text{j}_{0}}^{\times n}\rightarrow \vec{A}_{\text{j}}^{\times n}$ over $\vec{\mathsf{M}}_{{\mathcal{H}}}$ for all $\text{j}\in \text{J}$ , by [Reference LanLan13, Corollary 1.3.5.4, Proposition 1.4.3.4, and Proposition 3.3.1.5] and the normality of $\vec{\mathsf{M}}_{{\mathcal{H}}}$ . By the construction of $\vec{\widetilde{C}}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}}}$ by taking normalization over the product of the auxiliary models indexed by $\text{j}\in \text{J}$ (see [Reference LanLan16, Proposition 8.4]), and by Zariski’s main theorem (see [Reference Grothendieck and DieudonnéEGA, III-1, 4.4.3, 4.4.11]), the canonical finite morphism $\vec{A}_{\text{j}_{0}}^{\times n}\rightarrow \prod _{\text{j}\in \text{J}}\vec{A}_{\text{j}}^{\times n}$ (defined by the above isogenies) induces the composition of $[p^{r_{1}}]:\vec{A}_{\text{j}_{0}}^{\times n}\rightarrow \vec{A}_{\text{j}_{0}}^{\times n}$ and an isomorphism $\vec{A}_{\text{j}_{0}}^{\times n}\stackrel{{\sim}}{\rightarrow }\vec{\widetilde{C}}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}}}$ . Finally, since $\widetilde{\unicode[STIX]{x1D70E}}$ is top-dimensional, we have $\widetilde{\mathsf{Z}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}\cong \widetilde{C}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}}}$ and $\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}\cong \vec{\widetilde{C}}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}}}$ , which are compatible with their structural morphisms to $\widetilde{\mathsf{Z}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}})]}\cong \mathsf{M}_{{\mathcal{H}}}$ and $\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}})]}\cong \vec{\mathsf{M}}_{{\mathcal{H}}}$ , respectively, by [Reference LanLan16, Lemma 8.20 and Theorem 12.16] and [Reference LanLan17b, Theorem 6.1(5)]. Thus, by pulling back everything under $S\rightarrow \operatorname{Spec}({\mathcal{O}}_{F_{0},(p)})$ , the above claim follows.◻

Proof of Proposition 4.2 in Case (Nm).

Let us proceed with the same setting of the proof of Proposition 4.1 in Case (Nm).

Since $\widetilde{\unicode[STIX]{x1D6F4}}$ satisfies [Reference LanLan13, Condition 6.2.5.25] by assumption, by the same argument as in the proof of [Reference LanLan13, Lemma 6.2.5.27], the closure of each stratum of $\vec{\widetilde{\mathsf{M}}}_{\widetilde{{\mathcal{H}}},\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{tor}}$ has no self-intersection. Therefore, by the étale local description spelled out in the property (9) of Proposition 2.2 and in Corollary 2.4, and by the geometric normality of affine toroidal embeddings over their base schemes as explained in the proof of [Reference LanLan16, Proposition 8.14], the closure of each stratum of $\vec{\widetilde{\mathsf{M}}}_{\widetilde{{\mathcal{H}}},\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{tor}}$ is noetherian and normal. In particular, this is the case for the closure $\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{tor}}$ of  $\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}$ .

Following the same strategy as in [Reference LanLan12, §3B], we would like to show that, for suitable choices of $\widetilde{\unicode[STIX]{x1D6F4}}$ and $\widetilde{\unicode[STIX]{x1D70E}}$ , the noetherian normal scheme $\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{tor}}$ carries some object parameterized by $\vec{\mathsf{M}}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}$ , which satisfies the condition in [Reference LanLan17b, Theorem 6.1(6)]. Compared with the condition in [Reference LanLan13, Theorem 6.4.1.1(6)], the main difference is that the condition in [Reference LanLan17b, Theorem 6.1(6)] also requires a collection of semi-abelian degenerations parameterized by $\text{j}\in \text{J}$ , whose pullback to the generic point in characteristic zero is $\mathbb{Q}^{\times }$ -isogenous to the tautological object parameterized by $\mathsf{M}_{{\mathcal{H}}}$ . Without repeating (essentially verbatim) all the steps in [Reference LanLan12, §3B], we shall at least explain how to create such a collection.

Consider the tautological semi-abelian objects $(\vec{\widetilde{G}}_{\text{j}},\vec{\widetilde{\unicode[STIX]{x1D706}}}_{\text{j}},\vec{\widetilde{i}}_{\text{j}},\vec{\widetilde{\unicode[STIX]{x1D6FC}}}_{\widetilde{{\mathcal{H}}}})\rightarrow \vec{\widetilde{\mathsf{M}}}_{\widetilde{{\mathcal{H}}},\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{tor}}$ . By construction of $\vec{\widetilde{C}}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}}}$ , the pullbacks $\vec{G}_{\text{j},\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}}$ of $\vec{G}_{\text{j}}$ to $\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}$ is an extension of the pullback of the abelian scheme $\vec{A}_{\text{j}}\rightarrow \mathsf{M}_{{\mathcal{H}}}$ by a split torus $\vec{\widetilde{T}}_{\text{j},\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}}$ . Since $\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{tor}}$ is noetherian and normal, by [Reference LanLan13, Proposition 3.3.1.7], for each $\text{j}\in \text{J}$ , the subtorus $\vec{\widetilde{T}}_{\text{j},\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}}$ of $\vec{\widetilde{G}}_{\text{j},\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}}$ extends to a subtorus $\vec{\widetilde{T}}_{\text{j},\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{tor}}}$ of the pullback $\vec{\widetilde{G}}_{\text{j},\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{tor}}}$ of $\vec{\widetilde{G}}_{\text{j}}$ to $\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{tor}}$ , which allows us to define a quotient semi-abelian scheme $\vec{\overline{G}}_{\text{j}}:=\vec{\widetilde{G}}_{\text{j},\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{tor}}}/\vec{\widetilde{T}}_{\text{j},\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{tor}}}$ over $\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{tor}}$ . On the other hand, in characteristic zero, the pullback of the tautological semi-abelian object $(\widetilde{G},\widetilde{\unicode[STIX]{x1D706}},\widetilde{i},\widetilde{\unicode[STIX]{x1D6FC}}_{\widetilde{{\mathcal{H}}}})\rightarrow \widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}},\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{tor}}$ to the closure $\widetilde{\mathsf{Z}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{tor}}$ of $\widetilde{\mathsf{Z}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}$ in $\widetilde{\mathsf{M}}_{\widetilde{{\mathcal{H}}},\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{tor}}$ defines the semi-abelian scheme $\overline{G}:=\widetilde{G}_{\widetilde{\mathsf{Z}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{tor}}}/\widetilde{T}_{\widetilde{\mathsf{Z}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{tor}}}$ over $\widetilde{\mathsf{Z}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{tor}}$ , exactly as in [Reference LanLan12, §3B]. By construction, these quotient semi-abelian schemes $\overline{G}$ and $\{\vec{\overline{G}}_{\text{j}}\}_{\text{j}\in \text{J}}$ are $\mathbb{Q}^{\times }$ -isogenous to each other over $\widetilde{\mathsf{Z}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{tor}}\cong \vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{tor}}\otimes _{\mathbb{Z}}\mathbb{Q}$ . By the same argument as in [Reference LanLan12, §3B], they also carry the additional PEL structures, providing the collection we need, and satisfying the condition in [Reference LanLan17b, Theorem 6.1(6)] as long as the same combinatorial [Reference LanLan12, Condition 3.8] is satisfied. Thus the proposition follows. ◻

Proof of Proposition 4.1 in Case (Spl).

Let us proceed with the setting of the proof of Proposition 4.1 in Case (Nm), so that $\vec{\widetilde{C}}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}}}\rightarrow \vec{\mathsf{M}}_{{\mathcal{H}}}$ is isomorphic to the abelian scheme $\vec{A}_{\text{j}_{0}}^{\times n}\rightarrow \vec{\mathsf{M}}_{{\mathcal{H}}}$ , which is smooth. Since $\vec{\widetilde{C}}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}}}^{\text{spl}}$ is the normalization of $\vec{\widetilde{C}}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}}}\times _{\vec{\mathsf{M}}_{{\mathcal{H}}}}\vec{\mathsf{M}}_{{\mathcal{H}}}^{\text{spl}}$ (see [Reference LanLan15b, Definition 3.2.3 and Lemma 3.2.4]), and since this fiber product is already normal (by the smoothness of $\vec{\widetilde{C}}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}}}\rightarrow \vec{\mathsf{M}}_{{\mathcal{H}}}$ and the normality of $\vec{\mathsf{M}}_{{\mathcal{H}}}^{\text{spl}}$ ), the induced morphism $\vec{\widetilde{C}}_{\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}}}^{\text{spl}}\rightarrow \vec{\mathsf{M}}_{{\mathcal{H}}}^{\text{spl}}$ is isomorphic to the pullback of $\vec{A}_{\text{j}_{0}}^{\times n}\rightarrow \vec{\mathsf{M}}_{{\mathcal{H}}}$ under $\vec{\mathsf{M}}_{{\mathcal{H}}}^{\text{spl}}\rightarrow \vec{\mathsf{M}}_{{\mathcal{H}}}$ . Again, since $\widetilde{\unicode[STIX]{x1D70E}}$ is top-dimensional, by [Reference LanLan15b, Proposition 3.2.11, Theorems 3.4.1(2) and 4.3.1(5)], the pullback $\widetilde{\mathsf{Z}}_{[\widetilde{\unicode[STIX]{x1D70E}}]}\rightarrow \widetilde{\mathsf{Z}}$ of $\vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{spl}}\rightarrow \vec{\widetilde{\mathsf{Z}}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}})]}^{\text{spl}}$ under $\mathsf{S}\rightarrow \operatorname{Spec}({\mathcal{O}}_{K})$ is isomorphic to $\mathsf{f}^{\times n}:\mathsf{A}^{\times n}\rightarrow \mathsf{X}_{{\mathcal{H}}}$ , as desired. ◻

Proof of Proposition 4.2 in Case (Spl).

In this case, by the same argument as in the proof of Proposition 4.2 in Case (Nm), the pullbacks of the semi-abelian objects over $\vec{\widetilde{\mathsf{M}}}_{\widetilde{{\mathcal{H}}},\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{spl},\text{tor}}$ to the closure $\vec{\mathsf{Z}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{spl},\text{tor}}$ of $\vec{\mathsf{Z}}_{[(\widetilde{\unicode[STIX]{x1D6F7}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D6FF}}_{\widetilde{{\mathcal{H}}}},\widetilde{\unicode[STIX]{x1D70E}})]}^{\text{spl}}$ in $\vec{\widetilde{\mathsf{M}}}_{\widetilde{{\mathcal{H}}},\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{spl},\text{tor}}$ define quotient objects which also carry the splitting structures given by the analogue of the recipe [Reference LanLan15b, (3.3.12)] (for $\vec{\widetilde{\mathsf{M}}}_{\widetilde{{\mathcal{H}}},\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{spl},\text{tor}}$ instead of $\vec{\mathsf{M}}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{spl},\text{tor}}$ ). Hence, the condition in [Reference LanLan15b, Theorem 3.4.1(4)] applies to these quotient objects, which is satisfied under the same combinatorial [Reference LanLan12, Condition 3.8], by the same argument as in [Reference LanLan12, §3B] (again).◻

Proof. Without loss of generality, we may assume that $\mathsf{S}=\operatorname{Spec}({\mathcal{O}}_{F_{0},(v)})$ . Suppose that $x$ is mapped to the stratum $\mathsf{Z}_{[\breve{\unicode[STIX]{x1D70F}}]}$ of $\widetilde{\mathsf{X}}_{\widetilde{{\mathcal{H}}},\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{tor}}$ , which is above some stratum $\breve{\mathsf{Z}}$ (in the closure of $\widetilde{\mathsf{Z}}\cong \mathsf{X}_{{\mathcal{H}}_{\text{aux}}}$ ) of $\widetilde{\mathsf{X}}_{\widetilde{{\mathcal{H}}}}^{\min }$ . Suppose that $x$ is mapped to the stratum $\mathsf{Z}_{[\unicode[STIX]{x1D70F}]}$ of $\mathsf{X}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}$ , which is above some stratum $\mathsf{Z}$ of $\mathsf{X}_{{\mathcal{H}}}^{\min }$ . Suppose that $x$ is mapped to the stratum $\mathsf{Z}_{[\unicode[STIX]{x1D70F}_{\text{aux}}]}$ of $\mathsf{X}_{{\mathcal{H}}_{\text{aux}},\unicode[STIX]{x1D6F4}_{\text{aux}}}^{\text{tor}}$ , which is above some stratum $\mathsf{Z}_{\text{aux}}$ of $\mathsf{X}_{{\mathcal{H}}_{\text{aux}}}^{\min }$ .

By the property (9) of Proposition 2.2 (see also the second paragraph of Corollary 2.4), étale locally at each point of $\mathsf{Z}_{[\widetilde{\unicode[STIX]{x1D70F}}]}$ , for each representative $\widetilde{\unicode[STIX]{x1D70F}}$ of $[\widetilde{\unicode[STIX]{x1D70F}}]$ , and for some representative $\widetilde{\unicode[STIX]{x1D70F}}^{\prime }$ of $[\widetilde{\unicode[STIX]{x1D70F}}^{\prime }]$ , where $[\breve{\unicode[STIX]{x1D70E}}]$ determines $[\widetilde{\unicode[STIX]{x1D70E}}]$ as in the case of $[\unicode[STIX]{x1D70F}]$ determining $[\unicode[STIX]{x1D70F}^{\prime }]$ in the property (9) of Proposition 2.2, the canonical open immersion $Y_{\text{aux}}=\mathsf{Z}_{[\widetilde{\unicode[STIX]{x1D70E}}]}{\hookrightarrow}\overline{Y}_{\text{aux}}=\mathsf{Z}_{[\widetilde{\unicode[STIX]{x1D70E}}]}^{\text{tor}}$ is isomorphic to the affine toroidal embedding $\breve{\unicode[STIX]{x1D6EF}}_{\breve{\unicode[STIX]{x1D70E}}}{\hookrightarrow}\breve{\unicode[STIX]{x1D6EF}}_{\breve{\unicode[STIX]{x1D70E}}}(\breve{\unicode[STIX]{x1D70F}})$ over the $\breve{C}$ over $\breve{\mathsf{Z}}$ , where $\breve{\unicode[STIX]{x1D6EF}}_{\breve{\unicode[STIX]{x1D70E}}}(\breve{\unicode[STIX]{x1D70F}})$ is the closure of $\breve{\unicode[STIX]{x1D6EF}}_{\breve{\unicode[STIX]{x1D70E}}}$ in $\breve{\unicode[STIX]{x1D6EF}}(\breve{\unicode[STIX]{x1D70F}})$ . (Such notation with $\breve{~}$ instead of $\widetilde{~}$ is to make it clear that $\breve{\unicode[STIX]{x1D70F}}\in \unicode[STIX]{x1D6F4}_{\breve{\mathsf{Z}}}^{+}$ and $\widetilde{\unicode[STIX]{x1D70E}}\in \unicode[STIX]{x1D6F4}_{\widetilde{\mathsf{Z}}}^{+}$ cannot be directly compared; cf. [Reference LanLan12, §2D].) Similarly, étale locally at each point of $\mathsf{Z}_{[\unicode[STIX]{x1D70F}_{\text{aux}}]}$ , for each representative $\unicode[STIX]{x1D70F}_{\text{aux}}$ of $[\unicode[STIX]{x1D70F}_{\text{aux}}]$ , the canonical open immersion $\mathsf{X}_{{\mathcal{H}}_{\text{aux}}}{\hookrightarrow}\mathsf{X}_{{\mathcal{H}}_{\text{aux}},\unicode[STIX]{x1D6F4}_{\text{aux}}}^{\text{tor}}$ is isomorphic to the affine toroidal embedding $\unicode[STIX]{x1D6EF}_{\text{aux}}{\hookrightarrow}\unicode[STIX]{x1D6EF}_{\text{aux}}(\unicode[STIX]{x1D70F}_{\text{aux}})$ over the $C_{\text{aux}}$ over $\mathsf{Z}_{\text{aux}}$ . As explained in [Reference Madapusi PeraMad15, the proof of Theorem 1, after Remark 4.1.6], étale locally at each point of $\mathsf{Z}_{[\unicode[STIX]{x1D70F}]}$ , for each representative $\unicode[STIX]{x1D70F}$ , the canonical open immersion $\mathsf{X}_{{\mathcal{H}}}{\hookrightarrow}\mathsf{X}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}$ is isomorphic to the affine toroidal embedding $\unicode[STIX]{x1D6EF}{\hookrightarrow}\unicode[STIX]{x1D6EF}(\unicode[STIX]{x1D70F})$ over the $C$ over $\mathsf{Z}$ .

As explained in [Reference LanLan12, §3B], we have $\breve{\mathsf{Z}}\cong \mathsf{Z}_{\text{aux}}$ , together with canonical morphisms $\breve{C}\rightarrow C_{\text{aux}}$ , $\breve{\unicode[STIX]{x1D6EF}}_{\breve{\unicode[STIX]{x1D70E}}}\rightarrow \unicode[STIX]{x1D6EF}_{\text{aux}}$ , and $\breve{\unicode[STIX]{x1D6EF}}_{\breve{\unicode[STIX]{x1D70E}}}(\breve{\unicode[STIX]{x1D70F}})\rightarrow \unicode[STIX]{x1D6EF}_{\text{aux}}(\unicode[STIX]{x1D70F}_{\text{aux}})$ . By [Reference LanLan12, Lemma 4.9], $\breve{C}\rightarrow C_{\text{aux}}$ is an abelian scheme torsor, and hence so is the pullback $C_{x}:=C\times _{C_{\text{aux}}}\breve{C}\rightarrow C$ . Let $\unicode[STIX]{x1D6EF}_{x}:=\unicode[STIX]{x1D6EF}\times _{\unicode[STIX]{x1D6EF}_{\text{aux}}}\breve{\unicode[STIX]{x1D6EF}}_{\breve{\unicode[STIX]{x1D70E}}}$ , which is a torsor over $C_{x}$ under the subtorus $E_{x}:=E\times _{E_{\text{aux}}}\breve{E}_{\breve{\unicode[STIX]{x1D70E}}}$ of $\breve{E}_{\breve{\unicode[STIX]{x1D70E}}}$ . By [Reference HarrisHar89, Lemma 3.2], the normalization of $\breve{\unicode[STIX]{x1D6EF}}_{\breve{\unicode[STIX]{x1D70E}}}(\breve{\unicode[STIX]{x1D70F}})$ under the canonical morphism $\unicode[STIX]{x1D6EF}_{x}\rightarrow \breve{\unicode[STIX]{x1D6EF}}_{\breve{\unicode[STIX]{x1D70E}}}(\breve{\unicode[STIX]{x1D70F}})$ is an affine toroidal embedding over $C_{x}$ , which we can write as $\unicode[STIX]{x1D6EF}_{x}{\hookrightarrow}\unicode[STIX]{x1D6EF}_{x}(\unicode[STIX]{x1D70F}_{x})$ for some cone $\unicode[STIX]{x1D70F}_{x}$ induced by $\breve{\unicode[STIX]{x1D70F}}$ .

Since the canonical morphism $\overline{Y}_{\text{aux}}\rightarrow \mathsf{X}_{{\mathcal{H}}_{\text{aux}},\unicode[STIX]{x1D6F4}_{\text{aux}}}^{\text{tor}}$ is induced by [Reference LanLan13, Theorem 6.4.1.1(6)] (see [Reference LanLan12, §3B]), whose proof is based on the properties of good algebraic models (constructed by Artin’s approximation), on which the proof of (9) of Proposition 2.2 in Case (Sm) (in § 2.3) is also based, it follows that, étale locally at each point of $\mathsf{Z}_{[\widetilde{\unicode[STIX]{x1D70F}}]}$ , the canonical morphisms $Y_{\text{aux}}{\hookrightarrow}\overline{Y}_{\text{aux}}$ and $\mathsf{X}_{{\mathcal{H}}_{\text{aux}}}{\hookrightarrow}\mathsf{X}_{{\mathcal{H}}_{\text{aux}},\unicode[STIX]{x1D6F4}_{\text{aux}}}^{\text{tor}}$ are compatibly isomorphic to $\breve{\unicode[STIX]{x1D6EF}}_{\breve{\unicode[STIX]{x1D70E}}}{\hookrightarrow}\breve{\unicode[STIX]{x1D6EF}}_{\breve{\unicode[STIX]{x1D70E}}}(\breve{\unicode[STIX]{x1D70F}})$ and $\unicode[STIX]{x1D6EF}_{\text{aux}}{\hookrightarrow}\unicode[STIX]{x1D6EF}_{\text{aux}}(\unicode[STIX]{x1D70E}_{\text{aux}})$ , respectively. By using the nested approximation in [Reference TeissierTei95, 2.9] and [Reference SpivakovskySpi99, Theorems 11.4 and 11.5] instead of Artin’s approximation, where the hypothesis (in the notation there) that $A_{i}\otimes _{A_{i-1}}B_{i-1}$ is noetherian is satisfied because the canonical isomorphism $\mathfrak{X}_{\unicode[STIX]{x1D70F}}=(\unicode[STIX]{x1D6EF}(\unicode[STIX]{x1D70F}))_{\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D70F}}}^{\wedge }\stackrel{{\sim}}{\rightarrow }(\mathsf{X}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}})_{\mathsf{Z}_{[\unicode[STIX]{x1D70F}]}}^{\wedge }$ in [Reference Madapusi PeraMad15, Theorem 4.1.5(5); see the proof of Proposition 4.2.11] is induced by the canonical isomorphism $\mathfrak{X}_{\unicode[STIX]{x1D70F}_{\text{aux}}}=(\unicode[STIX]{x1D6EF}_{\text{aux}}(\unicode[STIX]{x1D70F}_{\text{aux}}))_{\unicode[STIX]{x1D6EF}_{\text{aux},\unicode[STIX]{x1D70F}_{\text{aux}}}}^{\wedge }\stackrel{{\sim}}{\rightarrow }(\mathsf{X}_{{\mathcal{H}}_{\text{aux}},\unicode[STIX]{x1D6F4}_{\text{aux}}}^{\text{tor}})_{\mathsf{Z}_{[\unicode[STIX]{x1D70F}_{\text{aux}}]}}^{\wedge }$ , it follows that $Y{\hookrightarrow}\overline{Y}$ and $\mathsf{X}_{{\mathcal{H}}}{\hookrightarrow}\mathsf{X}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}$ are compatibly isomorphic to $\unicode[STIX]{x1D6EF}_{x}{\hookrightarrow}\unicode[STIX]{x1D6EF}_{x}(\unicode[STIX]{x1D70F}_{x})$ and $\unicode[STIX]{x1D6EF}{\hookrightarrow}\unicode[STIX]{x1D6EF}(\unicode[STIX]{x1D70F})$ , respectively, étale locally at $x$ . Hence, the proposition follows.◻

Proof. Without loss of generality, by enlarging the collection $\unicode[STIX]{x1D6F4}$ if necessary, we may assume that $\overline{E}$ is proper over $\mathsf{S}$ . Note that we have not assumed that $\unicode[STIX]{x1D6F4}$ is smooth. Nevertheless, there exists a smooth refinement $\unicode[STIX]{x1D6F4}^{\prime }$ of $\unicode[STIX]{x1D6F4}$ , so that the corresponding toroidal embedding

$$\begin{eqnarray}J_{\overline{E}^{\prime }}:E{\hookrightarrow}\overline{E}^{\prime }=\mathop{\bigcup }_{\unicode[STIX]{x1D70E}^{\prime }\in \unicode[STIX]{x1D6F4}^{\prime }}E(\unicode[STIX]{x1D70E}^{\prime })\end{eqnarray}$$

is between smooth schemes over $\operatorname{Spec}(\mathbb{Z})$ , and so that $\overline{E}^{\prime }-E$ (with its reduced structure) is a simple normal crossings divisor flat over $\mathsf{S}$ . By [Reference Kempf, Knudsen, Mumford and Saint-DonatKKMS73, ch. I, §2, Theorem 8], $\overline{E}$ and $\overline{E}^{\prime }$ are proper over $\mathsf{S}$ , and there is a canonical proper morphism

$$\begin{eqnarray}\unicode[STIX]{x1D711}:\overline{E}^{\prime }\rightarrow \overline{E}\end{eqnarray}$$

satisfying

$$\begin{eqnarray}J_{\overline{E}}=\unicode[STIX]{x1D711}\circ J_{\overline{E}^{\prime }}.\end{eqnarray}$$

Therefore, by [Reference Deligne and KatzSGA7, XIII, 2.1.9], the adjunction morphisms

(5.8) $$\begin{eqnarray}R\unicode[STIX]{x1D6F9}_{\overline{E}^{\prime }}\,RJ_{\overline{E}_{\unicode[STIX]{x1D702}}^{\prime },\ast }(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}})\rightarrow RJ_{\overline{E}_{\bar{s}}^{\prime },\ast }\,R\unicode[STIX]{x1D6F9}_{E}(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}})\end{eqnarray}$$

and

(5.9) $$\begin{eqnarray}J_{\overline{E}_{\bar{s}}^{\prime },!}\,R\unicode[STIX]{x1D6F9}_{E}(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}})\rightarrow R\unicode[STIX]{x1D6F9}_{\overline{E}^{\prime }}\,J_{\overline{E}_{\unicode[STIX]{x1D702}}^{\prime },!}(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}})\end{eqnarray}$$

are isomorphisms. The proper morphism $\unicode[STIX]{x1D711}$ then induces canonical isomorphisms

$$\begin{eqnarray}R\unicode[STIX]{x1D6F9}_{\overline{E}}\,RJ_{\overline{E}_{\unicode[STIX]{x1D702}},\ast }(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}})\cong R\unicode[STIX]{x1D6F9}_{\overline{E}}\,R\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D702},\ast }\,RJ_{\overline{E}_{\unicode[STIX]{x1D702}}^{\prime },\ast }(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}})\stackrel{{\sim}}{\rightarrow }R\unicode[STIX]{x1D711}_{\bar{s},\ast }\,R\unicode[STIX]{x1D6F9}_{\overline{E}^{\prime }}\,RJ_{\overline{E}_{\unicode[STIX]{x1D702}}^{\prime },\ast }(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}})\end{eqnarray}$$

and

$$\begin{eqnarray}RJ_{\overline{E}_{\bar{s}},\ast }\,R\unicode[STIX]{x1D6F9}_{E}(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}})\cong R\unicode[STIX]{x1D711}_{\bar{s},\ast }\,RJ_{\overline{E}_{\bar{s}}^{\prime },\ast }\,R\unicode[STIX]{x1D6F9}_{E}(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}}),\end{eqnarray}$$

which are compatible with each other under the adjunction morphisms (5.6) and (5.8). It also induces canonical isomorphisms

$$\begin{eqnarray}J_{\overline{E}_{\bar{s}},!}\,R\unicode[STIX]{x1D6F9}_{E}(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}})\cong R\unicode[STIX]{x1D711}_{\bar{s},\ast }\,RJ_{\overline{E}_{\bar{s}}^{\prime },!}\,R\unicode[STIX]{x1D6F9}_{E}(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}})\end{eqnarray}$$

and

$$\begin{eqnarray}R\unicode[STIX]{x1D6F9}_{\overline{E}}\,J_{\overline{E}_{\unicode[STIX]{x1D702}},!}(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}})\cong R\unicode[STIX]{x1D6F9}_{\overline{E}}\,R\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D702},\ast }\,J_{\overline{E}_{\unicode[STIX]{x1D702}}^{\prime },!}(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}})\stackrel{{\sim}}{\rightarrow }R\unicode[STIX]{x1D711}_{\bar{s},\ast }\,R\unicode[STIX]{x1D6F9}_{\overline{E}^{\prime }}\,J_{\overline{E}_{\unicode[STIX]{x1D702}}^{\prime },!}(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}}),\end{eqnarray}$$

which are compatible with each other under the adjunction morphisms (5.7) and (5.9). Thus we can finish because (5.8) and (5.9) are isomorphisms. ◻

Proof. Note that $Y_{\unicode[STIX]{x1D702}}\rightarrow \unicode[STIX]{x1D702}$ might not be smooth. Since the assertion of the lemma can be verified étale locally over $\overline{Y}$ , it suffices to show that the adjunction morphisms

(5.13) $$\begin{eqnarray}R\unicode[STIX]{x1D6F9}_{\overline{E}\times _{\mathsf{S}}C}\,RJ_{(\overline{E}\times _{\mathsf{S}}C)_{\unicode[STIX]{x1D702}},\ast }(\unicode[STIX]{x1D6EC}_{(E\times _{\mathsf{S}}C)_{\unicode[STIX]{x1D702}}})\rightarrow RJ_{(\overline{E}\times _{\mathsf{S}}C)_{\bar{s}},\ast }\,R\unicode[STIX]{x1D6F9}_{E\times _{\mathsf{S}}C}(\unicode[STIX]{x1D6EC}_{(E\times _{\mathsf{S}}C)_{\unicode[STIX]{x1D702}}})\end{eqnarray}$$

and

(5.14) $$\begin{eqnarray}J_{(\overline{E}\times _{\mathsf{S}}C)_{\bar{s}},!}\,R\unicode[STIX]{x1D6F9}_{E\times _{\mathsf{S}}C}(\unicode[STIX]{x1D6EC}_{(E\times _{\mathsf{S}}C)_{\unicode[STIX]{x1D702}}})\rightarrow R\unicode[STIX]{x1D6F9}_{\overline{E}\times _{\mathsf{S}}C}\,J_{(\overline{E}\times _{\mathsf{S}}C)_{\unicode[STIX]{x1D702}},!}(\unicode[STIX]{x1D6EC}_{(E\times _{\mathsf{S}}C)_{\unicode[STIX]{x1D702}}})\end{eqnarray}$$

are isomorphisms, where

$$\begin{eqnarray}J_{\overline{E}\times _{\mathsf{S}}C}=J_{\overline{E}}\times _{\mathsf{S}}\operatorname{Id}_{C}:E\times _{\mathsf{S}}C{\hookrightarrow}\overline{E}\times _{\mathsf{S}}C\end{eqnarray}$$

is the canonical open immersion. By the Künneth isomorphisms as in [Reference Artin, Grothendieck and VerdierSGA4, XVII, 5.4.3] and [Reference Beilinson, Bernstein and DeligneBBD82, 4.2.7], and by Gabber’s theorem (see [Reference IllusieIll94, 4.7]) on nearby cycles over products of schemes of finite type over $\mathsf{S}$ , we have canonical isomorphisms

$$\begin{eqnarray}\displaystyle (R\unicode[STIX]{x1D6F9}_{\overline{E}}\,RJ_{\overline{E}_{\unicode[STIX]{x1D702}},\ast }(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}}))\boxtimes _{\bar{s}}^{L}(R\unicode[STIX]{x1D6F9}_{C}(\unicode[STIX]{x1D6EC}_{C_{\unicode[STIX]{x1D702}}})) & \stackrel{{\sim}}{\rightarrow } & \displaystyle R\unicode[STIX]{x1D6F9}_{\overline{E}\times _{\mathsf{S}}C}((RJ_{\overline{E}_{\unicode[STIX]{x1D702}},\ast }(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}}))\boxtimes _{\unicode[STIX]{x1D702}}^{L}\unicode[STIX]{x1D6EC}_{C_{\unicode[STIX]{x1D702}}})\nonumber\\ \displaystyle & \stackrel{{\sim}}{\rightarrow } & \displaystyle R\unicode[STIX]{x1D6F9}_{\overline{E}\times _{\mathsf{S}}C}\,RJ_{(\overline{E}\times _{\mathsf{S}}C)_{\unicode[STIX]{x1D702}},\ast }(\unicode[STIX]{x1D6EC}_{(E\times _{\mathsf{S}}C)_{\unicode[STIX]{x1D702}}}),\nonumber\\ \displaystyle (RJ_{\overline{E}_{\bar{s}},\ast }\,R\unicode[STIX]{x1D6F9}_{E}(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}}))\boxtimes _{\bar{s}}^{L}(R\unicode[STIX]{x1D6F9}_{C}(\unicode[STIX]{x1D6EC}_{C_{\unicode[STIX]{x1D702}}})) & \stackrel{{\sim}}{\rightarrow } & \displaystyle RJ_{(\overline{E}\times _{\mathsf{S}}C)_{\bar{s}},\ast }((R\unicode[STIX]{x1D6F9}_{E}(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}}))\boxtimes _{\bar{s}}^{L}(R\unicode[STIX]{x1D6F9}_{C}(\unicode[STIX]{x1D6EC}_{C_{\unicode[STIX]{x1D702}}})))\nonumber\\ \displaystyle & \stackrel{{\sim}}{\rightarrow } & \displaystyle RJ_{(\overline{E}\times _{\mathsf{S}}C)_{\bar{s}},\ast }\,R\unicode[STIX]{x1D6F9}_{E\times _{\mathsf{S}}C}(\unicode[STIX]{x1D6EC}_{(E\times _{\mathsf{S}}C)_{\unicode[STIX]{x1D702}}}),\nonumber\\ \displaystyle (J_{\overline{E}_{\bar{s}},!}\,R\unicode[STIX]{x1D6F9}_{E}(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}}))\boxtimes _{\bar{s}}^{L}(R\unicode[STIX]{x1D6F9}_{C}(\unicode[STIX]{x1D6EC}_{C_{\unicode[STIX]{x1D702}}})) & \stackrel{{\sim}}{\rightarrow } & \displaystyle J_{(\overline{E}\times _{\mathsf{S}}C)_{\bar{s}},!}((R\unicode[STIX]{x1D6F9}_{E}(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}}))\boxtimes _{\bar{s}}^{L}(R\unicode[STIX]{x1D6F9}_{C}(\unicode[STIX]{x1D6EC}_{C_{\unicode[STIX]{x1D702}}})))\nonumber\\ \displaystyle & \stackrel{{\sim}}{\rightarrow } & \displaystyle J_{(\overline{E}\times _{\mathsf{S}}C)_{\bar{s}},!}\,R\unicode[STIX]{x1D6F9}_{E\times _{\mathsf{S}}C}(\unicode[STIX]{x1D6EC}_{(E\times _{\mathsf{S}}C)_{\unicode[STIX]{x1D702}}}),\nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle (R\unicode[STIX]{x1D6F9}_{\overline{E}}\,J_{\overline{E},!}(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}}))\boxtimes _{\bar{s}}^{L}(R\unicode[STIX]{x1D6F9}_{C}(\unicode[STIX]{x1D6EC}_{C_{\unicode[STIX]{x1D702}}})) & \stackrel{{\sim}}{\rightarrow } & \displaystyle R\unicode[STIX]{x1D6F9}_{\overline{E}\times _{\mathsf{S}}C}((J_{\overline{E}_{\unicode[STIX]{x1D702}},!}(\unicode[STIX]{x1D6EC}_{E_{\unicode[STIX]{x1D702}}}))\boxtimes _{\unicode[STIX]{x1D702}}^{L}\unicode[STIX]{x1D6EC}_{C_{\unicode[STIX]{x1D702}}})\nonumber\\ \displaystyle & \stackrel{{\sim}}{\rightarrow } & \displaystyle R\unicode[STIX]{x1D6F9}_{\overline{E}\times _{\mathsf{S}}C}\,J_{(\overline{E}\times _{\mathsf{S}}C)_{\unicode[STIX]{x1D702}},!}(\unicode[STIX]{x1D6EC}_{(E\times _{\mathsf{S}}C)_{\unicode[STIX]{x1D702}}}),\nonumber\end{eqnarray}$$

where the former (respectively latter) two isomorphisms are compatible with each other under the adjunction morphisms (5.6) and (5.13) (respectively (5.7) and (5.14)) (and the identity morphism on $R\unicode[STIX]{x1D6F9}_{C}(\unicode[STIX]{x1D6EC}_{C_{\unicode[STIX]{x1D702}}})$ ). Hence, (5.13) and (5.14) are isomorphisms because (5.6) and (5.7) are, by Lemma 5.5.◻

Proof. Let $t=t_{\unicode[STIX]{x1D709}}$ and $n=n_{\unicode[STIX]{x1D709}}$ if ${\mathcal{V}}={\mathcal{V}}_{\unicode[STIX]{x1D709}}$ as in Proposition 3.2, in which case we take $\unicode[STIX]{x1D6EC}=\bar{\mathbb{Q}}_{\ell }$ ; or let $t=t_{W}$ and $n=n_{W}$ if ${\mathcal{V}}={\mathcal{W}}_{0,M}$ as in Proposition 3.4 for some ring $M$ that is a finite free module over either $\mathbb{Z}_{\ell }$ or $\mathbb{Z}/\ell ^{m}\mathbb{Z}$ (for some integer $m\geqslant 1$ ), for some $\ell >c_{W}$ there, in which case we take $\unicode[STIX]{x1D6EC}=M$ .

In Cases (Sm), (Nm), or (Spl), let $\widetilde{\mathsf{Z}}_{[\widetilde{\unicode[STIX]{x1D70E}}]}\rightarrow \widetilde{\mathsf{Z}}$ be as in Proposition 4.1, which realizes the abelian scheme $\mathsf{f}^{\times n}:\mathsf{A}^{\times n}\rightarrow \mathsf{X}_{{\mathcal{H}}}$ up to $\mathbb{Z}_{(\ell )}^{\times }$ -isogeny. Let $Y:=\widetilde{\mathsf{Z}}_{[\widetilde{\unicode[STIX]{x1D70E}}]}$ ; and let $\overline{Y}:=\widetilde{\mathsf{Z}}_{[\widetilde{\unicode[STIX]{x1D70E}}]}^{\text{tor}}$ , the closure of $\widetilde{\mathsf{Z}}_{[\widetilde{\unicode[STIX]{x1D70E}}]}$ in $\widetilde{\mathsf{X}}_{\widetilde{{\mathcal{H}}},\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{tor}}$ . Let $J_{\overline{Y}}:Y{\hookrightarrow}\overline{Y}$ denote the canonical open immersion. By Proposition 4.2, up to modifying the choices of $\widetilde{\unicode[STIX]{x1D6F4}}$ and $\widetilde{\unicode[STIX]{x1D70E}}$ , we may assume that the induced morphism $h:Y\rightarrow \mathsf{X}_{{\mathcal{H}}}$ extends to a (necessarily proper surjective) morphism $\overline{h}:\overline{Y}\rightarrow \mathsf{X}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}$ . In Case (Hdg), we simply take $Y{\hookrightarrow}\overline{Y}$ as in the paragraph preceding Proposition 4.3, with canonical morphisms $h:Y\rightarrow \mathsf{X}_{{\mathcal{H}}}$ and $\overline{h}:\overline{Y}\rightarrow \mathsf{X}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}$ . In all cases, we know that ${\mathcal{V}}$ is a direct summand of $Rh_{\ast }(\unicode[STIX]{x1D6EC}_{Y})(-t)[n]$ , and therefore $RJ_{(\mathsf{X}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}})_{\unicode[STIX]{x1D702}},\ast }({\mathcal{V}})$ and $J_{(\mathsf{X}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}})_{\unicode[STIX]{x1D702}},!}({\mathcal{V}})$ are direct summands of $R\overline{h}_{\unicode[STIX]{x1D702},\ast }RJ_{\overline{Y}_{\unicode[STIX]{x1D702}},\ast }(\unicode[STIX]{x1D6EC}_{Y_{\unicode[STIX]{x1D702}}})(-t)[n]$ and $R\overline{h}_{\unicode[STIX]{x1D702},\ast }J_{\overline{Y}_{\unicode[STIX]{x1D702}},!}(\unicode[STIX]{x1D6EC}_{Y_{\unicode[STIX]{x1D702}}})(-t)[n]$ , respectively.

By the proper base change theorem (applied to $\overline{h}$ ), and by shifting and Tate twisting, it suffices to note that the adjunction morphisms

(5.18) $$\begin{eqnarray}R\unicode[STIX]{x1D6F9}_{\overline{Y}}\,RJ_{\overline{Y}_{\unicode[STIX]{x1D702}},\ast }(\unicode[STIX]{x1D6EC}_{Y_{\unicode[STIX]{x1D702}}})\rightarrow RJ_{\overline{Y}_{\bar{s}},\ast }\,R\unicode[STIX]{x1D6F9}_{Y}(\unicode[STIX]{x1D6EC}_{Y_{\unicode[STIX]{x1D702}}})\end{eqnarray}$$

and

(5.19) $$\begin{eqnarray}J_{\overline{Y}_{\bar{s}},!}\,R\unicode[STIX]{x1D6F9}_{Y}(\unicode[STIX]{x1D6EC}_{Y_{\unicode[STIX]{x1D702}}})\rightarrow R\unicode[STIX]{x1D6F9}_{\overline{Y}}\,J_{\overline{Y}_{\unicode[STIX]{x1D702}},!}(\unicode[STIX]{x1D6EC}_{Y_{\unicode[STIX]{x1D702}}})\end{eqnarray}$$

are isomorphisms. In Cases (Sm), (Nm), and (Spl), this follows from Lemma 5.10 and the second paragraph of Corollary 2.4 (applied to the strata of $\widetilde{\mathsf{X}}_{\widetilde{{\mathcal{H}}},\widetilde{\unicode[STIX]{x1D6F4}}}^{\text{tor}}$ ). In Case (Hdg), this follows from Lemma 5.10 and Proposition 4.3.◻

Proof. Since $\mathsf{X}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}\rightarrow \mathsf{S}$ is proper, by the proper base change theorem, these follow from the two canonical isomorphisms (5.16) and (5.17) in $D_{c}^{b}((\mathsf{X}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}})_{\bar{s}}\times \bar{\unicode[STIX]{x1D702}},\unicode[STIX]{x1D6EC})$ .◻

Proof. Since the structural morphism $\oint _{{\mathcal{H}},\unicode[STIX]{x1D6F4}}:\mathsf{X}_{{\mathcal{H}},\unicode[STIX]{x1D6F4}}^{\text{tor}}\rightarrow \mathsf{X}_{{\mathcal{H}}}^{\min }$ is proper (for any choice of $\unicode[STIX]{x1D6F4}$ ), this follows from Theorem 5.15 and the proper base change theorem.◻

Proof. Let us denote by $\text{}^{p}{\mathcal{H}}^{0}$ the zeroth perverse cohomology of a $\bar{\mathbb{Q}}_{\ell }$ -sheaf, which is a perverse sheaf. By definition, we have

$$\begin{eqnarray}J_{(\mathsf{X}_{{\mathcal{H}}}^{\min })_{?},!\ast }=\operatorname{Im}(\text{}^{p}{\mathcal{H}}^{0}\circ J_{(\mathsf{X}_{{\mathcal{H}}}^{\min })_{?},!}\rightarrow \text{}^{p}{\mathcal{H}}^{0}\circ RJ_{(\mathsf{X}_{{\mathcal{H}}}^{\min })_{?},\ast }),\end{eqnarray}$$

for $?=\unicode[STIX]{x1D702}$ or $\bar{s}$ , where the image is taken in the abelian category of perverse sheaves. Therefore, it suffices to show that there are canonical isomorphisms

(5.28) $$\begin{eqnarray}\text{}^{p}{\mathcal{H}}^{0}(J_{(\mathsf{X}_{{\mathcal{H}}}^{\min })_{\bar{s}},!}\,R\unicode[STIX]{x1D6F9}_{\mathsf{X}_{{\mathcal{H}}}}({\mathcal{V}}_{\unicode[STIX]{x1D709}}[d]))\stackrel{{\sim}}{\rightarrow }R\unicode[STIX]{x1D6F9}_{\mathsf{X}_{{\mathcal{H}}}^{\min }}\,\text{}^{p}{\mathcal{H}}^{0}(J_{(\mathsf{X}_{{\mathcal{H}}}^{\min })_{\unicode[STIX]{x1D702}},!}({\mathcal{V}}_{\unicode[STIX]{x1D709}}[d]))\end{eqnarray}$$

and

(5.29) $$\begin{eqnarray}R\unicode[STIX]{x1D6F9}_{\mathsf{X}_{{\mathcal{H}}}^{\min }}\,\text{}^{p}{\mathcal{H}}^{0}(RJ_{(\mathsf{X}_{{\mathcal{H}}}^{\min })_{\unicode[STIX]{x1D702}},\ast }({\mathcal{V}}_{\unicode[STIX]{x1D709}}[d]))\stackrel{{\sim}}{\rightarrow }\text{}^{p}{\mathcal{H}}^{0}(RJ_{(\mathsf{X}_{{\mathcal{H}}}^{\min })_{\bar{s}},\ast }\,R\unicode[STIX]{x1D6F9}_{\mathsf{X}_{{\mathcal{H}}}}({\mathcal{V}}_{\unicode[STIX]{x1D709}}[d]))\end{eqnarray}$$

which are compatible with the canonical morphisms induced by the canonical morphisms

$$\begin{eqnarray}J_{(\mathsf{X}_{{\mathcal{H}}}^{\min })_{?},!}\rightarrow RJ_{(\mathsf{X}_{{\mathcal{H}}}^{\min })_{?},\ast }\end{eqnarray}$$

of functors, for $?=\unicode[STIX]{x1D702}$ and $\bar{s}$ . By [Reference IllusieIll94, 4.5] again, we have a canonical isomorphism

$$\begin{eqnarray}\text{}^{p}{\mathcal{H}}^{0}\circ R\unicode[STIX]{x1D6F9}\cong R\unicode[STIX]{x1D6F9}\circ \text{}^{p}{\mathcal{H}}^{0}\end{eqnarray}$$

of functors, because the functor $R\unicode[STIX]{x1D6F9}$ is $t$ -exact for the middle perversity. Therefore, by applying the functor $\text{}^{p}{\mathcal{H}}^{0}$ to (5.25) and (5.24), we obtain the desired isomorphisms (5.28) and (5.29).◻

Proof. Since $\mathsf{X}_{{\mathcal{H}}}^{\min }\rightarrow \mathsf{S}$ is proper, by the proper base change theorem, these follow from the canonical isomorphism (5.27) in the category of perverse sheaves over $(\mathsf{X}_{{\mathcal{H}}}^{\min })_{\bar{s}}$ with compatible continuous $\operatorname{Gal}(\bar{K}/K)$ -actions.◻