3.1.1 Symplectic and unitary similitude groups

Let $\unicode[STIX]{x1D6F9}_{n}$ denote the $n\times n$ matrix with ones on the antidiagonal and zeros elsewhere. We set

$$\begin{eqnarray}J_{n}=\left(\begin{array}{@{}cc@{}}0 & \unicode[STIX]{x1D6F9}_{n}\\ -\unicode[STIX]{x1D6F9}_{n} & 0\end{array}\right).\end{eqnarray}$$

Consider the free ${\mathcal{O}}_{F}$-module $L:={\mathcal{O}}_{F}^{2n}$ of rank $2n$. Then $J_{n}$ determines a non-degenerate alternating pairing

$$\begin{eqnarray}\unicode[STIX]{x1D713}:L\times L\rightarrow \mathbb{Z},\quad \unicode[STIX]{x1D713}(x,y):=\text{Tr}_{{\mathcal{O}}_{F}/\mathbb{Z}}(x^{t}J_{n}y^{c}),\end{eqnarray}$$

which is perfect when localized at any rational prime which is unramified in $F$. We define $G$ to be the group scheme over $\mathbb{Z}$ given on $R$-points, for $R$ a ring, by

When $F$ is totally real, $G$ is a symplectic similitude group; when $F$ is imaginary CM, $G$ is a unitary similitude group (in both cases, we note that the similitude factor is required to lie in $\mathbb{G}_{m,\mathbb{Z}}$). We remark that in a pair $(g,r)\in G(R)$, $r$ is determined uniquely by $g$ and hence projection onto the first factor defines an injection $G\rightarrow \text{Res}_{{\mathcal{O}}_{F}/\mathbb{Z}}(\operatorname{GL}_{2n,{\mathcal{O}}_{F}})$. We will use this to represent elements of $G$ by $2\times 2$ block $n\times n$ matrices, as with $J_{n}$ above. The group $G$ is manifestly a subgroup scheme of the symplectic similitude group $\widetilde{G}/\mathbb{Z}$, defined on $R$-points by

Since $\unicode[STIX]{x1D713}$ is perfect when localized at primes $\ell$ unramified in $F$, we see that $\widetilde{G}_{\mathbb{Z}_{(\ell )}}\cong \text{GSp}_{2nf,\mathbb{Z}_{(\ell )}}$ for those $\ell$.

3.1.2 Shimura data

In this section, we recall the integral PEL data attached to $G$ and $\widetilde{G}$, starting with $G$. In addition to the ${\mathcal{O}}_{F}$-module $L$ and the pairing $\unicode[STIX]{x1D713}$, we need an $\mathbb{R}$-algebra homomorphism

$$\begin{eqnarray}h:\mathbb{C}\rightarrow \operatorname{End}_{{\mathcal{O}}_{F}\otimes _{\mathbb{Z}}\mathbb{R}}(L\otimes _{\mathbb{Z}}\mathbb{R})\end{eqnarray}$$

such that $\unicode[STIX]{x1D713}(h(z)x,y)=\unicode[STIX]{x1D713}(x,h(z^{c})y)$ and such that $(x,y):=\unicode[STIX]{x1D713}(x,h(i)y)$ is symmetric and positive definite. We set $h(i)=-J_{n}$; this uniquely determines the $\mathbb{R}$-algebra homomorphism $h$ and one checks by direct calculation that it has the required properties. The triple $(L,\unicode[STIX]{x1D713},h)$ is then an integral PEL datum for ${\mathcal{O}}_{F}$ according to [Reference LanLan13, Definition 1.2.13] (where this notion is called a PEL-type ${\mathcal{O}}_{F}$-lattice). Restricting $h$ to $\mathbb{C}^{\times }$ gives a homomorphism $h:\mathbb{S}\rightarrow G_{\mathbb{R}}$ of real algebraic groups, where $\mathbb{S}=\text{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m,\mathbb{C}}$ is the Deligne torus. The Shimura datum of $G$ is then $(G_{\mathbb{Q}},X)$, where $X$ is the $G(\mathbb{R})$-conjugacy class of $h$.

To define the integral PEL datum attached to $\widetilde{G}$, we simply define $\widetilde{h}:\mathbb{C}\rightarrow \operatorname{End}_{\mathbb{R}}(L\otimes _{\mathbb{Z}}\mathbb{R})$ by postcomposing $h$ with the natural map $\operatorname{End}_{{\mathcal{O}}_{F}\otimes _{\mathbb{Z}}\mathbb{R}}(L\otimes _{\mathbb{Z}}\mathbb{R})\rightarrow \operatorname{End}_{\mathbb{R}}(L\otimes _{\mathbb{Z}}\mathbb{R})$. The triple $(L,\unicode[STIX]{x1D713},\widetilde{h})$ is then an integral PEL datum for $\mathbb{Z}$, and the corresponding Shimura datum $(\widetilde{G}_{\mathbb{Q}},\widetilde{X})$ is a Siegel Shimura datum (here $\widetilde{X}$ is the $\widetilde{G}(\mathbb{R})$-conjugacy class of $\widetilde{h}$).

We end with a short discussion of the Hodge cocharacter. Recall that the Hodge cocharacter for $G$ is given by the composition

$$\begin{eqnarray}\unicode[STIX]{x1D707}^{\prime }:\mathbb{G}_{m,\mathbb{C}}\xrightarrow[{}]{z\mapsto (z,1)}\mathbb{G}_{m,\mathbb{C}}\times \mathbb{G}_{m,\mathbb{C}}\simeq \mathbb{S}_{\mathbb{C}}\xrightarrow[{}]{h_{\mathbb{C}}}G_{\mathbb{C}}.\end{eqnarray}$$

Using the definition, one checks (by a calculation entirely analogous to the classical case of $\operatorname{GL}_{2,\mathbb{Q}}$) that $\unicode[STIX]{x1D707}^{\prime }$ is conjugate over $\mathbb{C}$ to the cocharacter $\unicode[STIX]{x1D707}:\mathbb{G}_{m,\mathbb{Z}}\rightarrow G$ defined by

$$\begin{eqnarray}\unicode[STIX]{x1D707}(z)=\left(\begin{array}{@{}cc@{}}zI_{n} & 0\\ 0 & I_{n}\end{array}\right),\end{eqnarray}$$

where $I_{n}$ denotes the $n\times n$ identity matrix. In particular, the reflex field of $(G_{\mathbb{Q}},X)$ is $\mathbb{Q}$. By composing $\unicode[STIX]{x1D707}$ with $G\rightarrow \widetilde{G}$, we get a Hodge cocharacter $\widetilde{\unicode[STIX]{x1D707}}$ for $\widetilde{G}$.

3.1.3 Shimura varieties and canonical models

Given a neat open compact subgroup $K$ of $G(\mathbb{A}_{f})$, the Shimura variety for $G$ with level $K$ is the double quotient

$$\begin{eqnarray}G(\mathbb{Q})\backslash X\times G(\mathbb{A}_{f})/K,\end{eqnarray}$$

which has the structure of a complex manifold. We will only consider neat $K$. As is well known, the above double quotient may be viewed as the complex points of a moduli space parametrizing abelian varieties with certain extra structures. We briefly recall these moduli spaces over $\mathbb{Q}$ (for general $K$) and $\mathbb{Z}_{(p)}$ (when $K=K^{p}G(\mathbb{Z}_{p})$ with $K^{p}\subseteq G(\mathbb{A}_{f}^{p})$), following Deligne and Kottwitz. We refer to [Reference KottwitzKot92, §5] and [Reference LanLan13, Chaps 1 and 2] for more detailed information and proofs.

Over $\mathbb{Q}$, initially consider as test objects the category of connected, locally noetherian $\mathbb{Q}$-schemes. For any such test object $S$, one defines a functor $S\mapsto X_{K}(S)$ by letting $X_{K}(S)$ be the set of equivalence classes of quadruples $(A,\unicode[STIX]{x1D706},\unicode[STIX]{x1D704},\unicode[STIX]{x1D702}K)$, where:

• ∙ $A$ is an abelian scheme over $S$;

• ∙ $\unicode[STIX]{x1D706}:A\rightarrow A^{\vee }$ is a polarization of $A$ (i.e. an isogeny inducing a polarization over each geometric fiber);

• ∙ $\unicode[STIX]{x1D704}:F{\hookrightarrow}\operatorname{End}^{0}(A):=\operatorname{End}(A)\,\otimes _{\mathbb{Z}}\,Z\mathbb{Q}$ is a $\mathbb{Q}$-algebra homomorphism satisfying $\unicode[STIX]{x1D706}\circ \unicode[STIX]{x1D704}(x^{c})=\unicode[STIX]{x1D704}(x)^{\vee }\circ \unicode[STIX]{x1D706}$ for all $x\in F$, where $-^{\vee }$ denotes the dual quasi-isogeny;

• ∙ for any geometric point $\bar{s}$ of $S$, $\unicode[STIX]{x1D702}K$ is a $\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(S,\bar{s})$-invariant $K$-orbit of $F\,\otimes _{\mathbb{Q}}\,\mathbb{A}_{f}$-isomorphisms

  $$\begin{eqnarray}\unicode[STIX]{x1D702}:L\,\otimes _{\mathbb{Z}}\,\mathbb{A}_{f}\stackrel{{\sim}}{\rightarrow }V_{f}(A_{\bar{s}})\end{eqnarray}$$
  identifying the pairing $\unicode[STIX]{x1D713}\,\otimes _{\mathbb{Z}}\,\mathbb{A}_{f}$ with the $\unicode[STIX]{x1D706}_{\bar{s}}$-Weil pairing up to an $\mathbb{A}_{f}^{\times }$-multiple, where $V_{f}(A_{\bar{s}})$ is the rational adelic Tate module of the geometric fiber $A_{\bar{s}}$ at $\bar{s}$.

The quadruples $(A,\unicode[STIX]{x1D706},\unicode[STIX]{x1D704},\unicode[STIX]{x1D702}\widetilde{K})$ in this moduli description are further required to satisfy Kottwitz’s determinant condition; we refer to [Reference KottwitzKot92, §5] for the precise formulation of this. Two quadruples $(A,\unicode[STIX]{x1D706},\unicode[STIX]{x1D704},\unicode[STIX]{x1D702}K)$ and $(A^{\prime },\unicode[STIX]{x1D706}^{\prime },\unicode[STIX]{x1D704}^{\prime },\unicode[STIX]{x1D702}^{\prime }K)$ are equivalent if there is a quasi-isogeny between $A$ and $A^{\prime }$, which is compatible with the polarizations $\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D706}^{\prime }$ up to a $\mathbb{Q}^{\times }$-multiple, respects the $F$-actions induced by $\unicode[STIX]{x1D704}$ and $\unicode[STIX]{x1D704}^{\prime }$, and takes the $K$-orbit $\unicode[STIX]{x1D702}K$ to $\unicode[STIX]{x1D702}^{\prime }K$.

This moduli problem can be extended to locally noetherian $\mathbb{Q}$-schemes $S$ which are not necessarily connected by taking a disjoint union over the sets corresponding to the connected components of $S$, and is then representable by a smooth quasi-projective scheme $X_{K}$ over $\mathbb{Q}$, whose $\mathbb{C}$-points $X_{K}(\mathbb{C})$ can be canonically identified with the double quotient $G(\mathbb{Q})\backslash X\times G(\mathbb{A}_{f})/K$.Footnote ³ We recall that there is a right action of $\widetilde{G}(\mathbb{A}_{f})$ on the tower $(X_{K})_{K}$ by precomposition $\unicode[STIX]{x1D702}\mapsto \unicode[STIX]{x1D702}\circ g$ on the level structures, which we will make much use of later. We also note in passing that our moduli problem uses a ‘definition by isogeny classes’ [Reference LanLan13, §1.4.2] whereas, for example, [Reference ScholzeSch15] (which we will compare with later) uses a ‘definition by isomorphism classes’ [Reference LanLan13, §1.4.1] when $K\subseteq G(\widehat{\mathbb{Z}})$. These two formulations are equivalent, as demonstrated in [Reference LanLan13, §1.4.3].

When $K=K^{p}G(\mathbb{Z}_{p})$ with $K^{p}\subseteq G(\mathbb{A}_{f}^{p})$, Kottwitz has constructed a smooth quasi-projective (integral, canonical) model of $X_{K}$ over $\mathbb{Z}_{(p)}$, which represents an extension of the above moduli problem (suitably modified) to locally noetherian $\mathbb{Z}_{(p)}$-schemes. We refer to [Reference KottwitzKot92, §5] for the details. By abuse of notation, we will also denote this model by $X_{K}$. Finally, the whole discussion above holds with $G$ replaced by $\widetilde{G}$; in this case the models were constructed by Mumford. We will use the notation $\widetilde{X}_{\widetilde{K}}$ for the (models of the) Shimura varieties for $\widetilde{G}$, for $\widetilde{K}\subseteq \widetilde{G}(\mathbb{A}_{f})$ a neat open compact subgroup. Whenever $K\subseteq \widetilde{K}$, we have a natural finite map $X_{K}\rightarrow \widetilde{X}_{\widetilde{K}}$.

3.1.4 Compactifications of canonical models

In this paper we will need to consider two types of compactifications of our Shimura varieties. The first is the minimal (Satake–Baily–Borel) compactification. We will denote all minimal compactifications by a superscript $-^{\ast }$. For the varieties $X_{K}$ and $\widetilde{X}_{\widetilde{K}}$ over $\mathbb{Q}$, the minimal compactifications $X_{K}^{\ast }$ and $\widetilde{X}_{\widetilde{K}}^{\ast }$ were constructed by Pink [Reference PinkPin90]. Over $\mathbb{Z}_{(p)}$, the minimal compactifications were constructed by Faltings–Chai [Reference Faltings and ChaiFC90] and Lan [Reference LanLan13]. They are normal, and projective over the base. The right action of $G(\mathbb{A}_{f})$ and $\widetilde{G}(\mathbb{A}_{f})$ extends to the tower of minimal compactifications $(X_{K}^{\ast })_{K}$ and $(\widetilde{X}_{\widetilde{K}}^{\ast })_{\widetilde{K}}$, respectively, and, when $K\subseteq \widetilde{K}$, the finite map $X_{K}\rightarrow \widetilde{X}_{\widetilde{K}}$ extends to a finite map $X_{K}^{\ast }\rightarrow \widetilde{X}_{\widetilde{K}}^{\ast }$.

For the Shimura varieties over $G$, we will mainly use compactifications that are different from the minimal compactifications, following [Reference ScholzeSch15, §4.1]. In the following discussion, fix $K$ and assume that $K=\widetilde{K}\cap G(\mathbb{A}_{f})$. For sufficiently small such $\widetilde{K}$, the map $X_{K}\rightarrow \widetilde{X}_{\widetilde{K}}$ is a closed immersion by [Reference DeligneDel71, Proposition 1.15], but this does not need to hold for the maps $X_{K}^{\ast }\rightarrow \widetilde{X}_{\widetilde{K}}^{\ast }$. Following Scholze, we define the ‘ad hoc’-compactification $\overline{X}_{K}$ of $X_{K}$ to be the universal finite map $X_{K}^{\ast }\rightarrow \overline{X}_{K}$ over which all the $X_{K}^{\ast }\rightarrow \widetilde{X}_{\widetilde{K}}^{\ast }$ vanish; as noted by Scholze, $\overline{X}_{K}$ is the scheme-theoretic image of $X_{K}^{\ast }\rightarrow \widetilde{X}_{\widetilde{K}}^{\ast }$ for sufficiently small $\widetilde{K}$. The right action of $G(\mathbb{A}_{f})$ extends to the tower of ad hoc compactifications $(\overline{X}_{K})_{K}$.

3.1.5 Parabolic and level subgroups

We finish this subsection by defining some subgroups which we will need throughout the paper. Recall that we can represent elements of $G$ by $2\times 2$ block $n\times n$ matrices. We define closed subgroup schemes $P_{\unicode[STIX]{x1D707}}\subseteq G$ and $N\subseteq G$ on $R$-points ($R$ a ring) by

where $\ast$ means an arbitrary $n\times n$ matrix and we recall that $I_{n}$ is the $n\times n$ identity matrix. Equivalently, we may define $P_{\unicode[STIX]{x1D707}}\subseteq G$ as the stabilizer of ${\mathcal{O}}_{F}^{n}\oplus 0\subseteq L$, and $N\subseteq G$ as the subgroup which acts trivially on ${\mathcal{O}}_{F}^{n}\oplus 0$ and on the quotient $L/({\mathcal{O}}_{F}^{n}\oplus 0)$. One may also describe $P_{\unicode[STIX]{x1D707}}(\mathbb{C})$ as

In particular, $P_{\unicode[STIX]{x1D707},\mathbb{C}}$ is the parabolic denoted by $P_{\unicode[STIX]{x1D707}}$ (for our choice of $\unicode[STIX]{x1D707}$) in [Reference Caraiani and ScholzeCS17, §2.1]. When $\ell$ is a prime unramified in $F$, $G_{\mathbb{Z}_{(\ell )}}$ is reductive and $P_{\unicode[STIX]{x1D707},\mathbb{Z}_{(\ell )}}$ is a parabolic subgroup of $G$, with unipotent radical $N_{\mathbb{Z}_{(\ell )}}$. We then define, for any integer $m\geqslant 0$, some open subgroups of $G(\mathbb{Z}_{p})$ by

Finally, we make similar definitions for $\widetilde{G}$ and the analogous statements hold; we define $\widetilde{P}_{\unicode[STIX]{x1D707}}\subseteq \widetilde{G}$ as the stabilizer of ${\mathcal{O}}_{F}^{n}\oplus 0\subseteq L$, and $\widetilde{N}\subseteq \widetilde{G}$ as the subgroup which acts trivially on ${\mathcal{O}}_{F}^{n}\oplus 0$ and on the quotient $L/({\mathcal{O}}_{F}^{n}\oplus 0)$ (we have elected to use the notation $\widetilde{P}_{\unicode[STIX]{x1D707}}$ instead of $\widetilde{P}_{\widetilde{\unicode[STIX]{x1D707}}}$ for simplicity). We then put

$$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m}) & := & \displaystyle \{(g,r)\in \widetilde{G}(\mathbb{Z}_{p})\mid (g\hspace{0.2em}{\rm mod}\hspace{0.2em}p^{m})\in \widetilde{P}_{\unicode[STIX]{x1D707}}(\mathbb{Z}/p^{m})\};\nonumber\\ \displaystyle \widetilde{\unicode[STIX]{x1D6E4}}_{1}(p^{m}) & := & \displaystyle \{(g,r)\in \widetilde{G}(\mathbb{Z}_{p})\mid (g\hspace{0.2em}{\rm mod}\hspace{0.2em}p^{m})\in \widetilde{N}(\mathbb{Z}/p^{m})\};\nonumber\\ \displaystyle \widetilde{\unicode[STIX]{x1D6E4}}(p^{m}) & := & \displaystyle \{(g,r)\in \widetilde{G}(\mathbb{Z}_{p})\mid g\equiv I_{2n}\hspace{0.2em}{\rm mod}\hspace{0.2em}p^{m}\}.\nonumber\end{eqnarray}$$

3.2.1 Formal and adic models

Recall that we have fixed a complete non-archimedean algebraically closed extension $C$ of $\mathbb{Q}_{p}$. Ultimately, we will work over $C$ in this paper, so to keep the notation uniform throughout we will stick to working over $C$ in the remainder of this section, but we remark that we could have chosen to work over any perfectoid extension of $\mathbb{Q}_{p}^{\text{cycl}}$, the completion of $\mathbb{Q}_{p}(\unicode[STIX]{x1D701}_{p^{\infty }})$.Footnote ⁴ We set $X^{\ast }:=X_{G(\mathbb{Z}_{p}),\mathbb{Z}_{p}}^{\ast }$ and we let $\mathfrak{X}_{\mathbb{Z}_{p}}^{\ast }$ be the formal completion of $X^{\ast }$ along $p=0$. This is a formal scheme over $\operatorname{Spf}\mathbb{Z}_{p}$. We let $\mathfrak{X}^{\ast }$ denote the base change of $\mathfrak{X}_{\mathbb{Z}_{p}}^{\ast }$ to $\operatorname{Spf}{\mathcal{O}}_{C}$ and ${\mathcal{X}}^{\ast }$ denote the generic fiber of $\mathfrak{X}^{\ast }$, viewed as an adic space. This is a proper adic space over $\operatorname{Spa}(C,{\mathcal{O}}_{C})$. We define $\widetilde{X}^{\ast }$, $\widetilde{\mathfrak{X}}_{\mathbb{Z}_{p}}^{\ast }$, $\widetilde{\mathfrak{X}}^{\ast }$ and $\widetilde{{\mathcal{X}}}^{\ast }$ analogously for $\widetilde{G}$.

For any level $K_{p}\subset G(\mathbb{Z}_{p})$, we let $\overline{{\mathcal{X}}}_{K_{p}}$ and ${\mathcal{X}}_{K_{p}}^{\ast }$ be the adic spaces corresponding to the base change to $C$ of the $\mathbb{Q}$-schemes $\overline{X}_{K_{p}}$ and $X_{K_{p}}^{\ast }$, respectively.Footnote ⁵ Note that ${\mathcal{X}}^{\ast }={\mathcal{X}}_{G(\mathbb{Z}_{p})}^{\ast }$. We define ${\mathcal{X}}_{K_{p}}$ similarly, as the adic space corresponding to the base change to $C$ of the $\mathbb{Q}$-scheme $X_{K_{p}}$. This is not quasi-compact, but it is open in $\overline{{\mathcal{X}}}_{K_{p}}$. The closed complement is the boundary ${\mathcal{Z}}_{K_{p}}:=\overline{{\mathcal{X}}}_{K_{p}}\setminus {\mathcal{X}}_{K_{p}}$. Again, we define $\widetilde{{\mathcal{X}}}_{\widetilde{K}_{p}}^{\ast }$, $\widetilde{{\mathcal{X}}}_{\widetilde{K}_{p}}$ and $\widetilde{{\mathcal{Z}}}_{\widetilde{K}_{p}}=\widetilde{{\mathcal{X}}}_{\widetilde{K}_{p}}^{\ast }\setminus \widetilde{X}_{\widetilde{K}_{p}}$ analogously for $\widetilde{G}$. Although we will only need the following result for $\widetilde{G}$, we state it in general. For the analogue for schemes, see [Reference LanLan13, Corollary 7.2.5.2].

3.2.6 Notation for ‘infinite-level’ Shimura varieties

Before proceeding, we will set out the notation for infinite-level Shimura varieties. We mostly discuss the case of $G$; the case of $\widetilde{G}$ is entirely analogous. Recall that we have fixed a prime-to-$p$-level. Our finite-level Shimura varieties ${\mathcal{X}}_{K}$ are then indexed by the open subgroups $K\subseteq G(\mathbb{Z}_{p})$. Our infinite-level Shimura varieties will arise as limits of towers $({\mathcal{X}}_{K})_{K}$, where the $K$ run through a cofiltered inverse system of open subgroups. Thus, the resulting limit (if it exists in a suitable sense) will only depend on the intersection $H=\bigcap K$, which is a closed subgroup of $G(\mathbb{Z}_{p})$. It therefore makes sense to make the following general definition.

Definition 3.2.7. Let $H\subseteq G(\mathbb{Z}_{p})$ be a closed subgroup. We define a locally spatial diamond ${\mathcal{X}}_{H}$ by

$$\begin{eqnarray}{\mathcal{X}}_{H}:=\mathop{\varprojlim }\nolimits_{H\subseteq K}{\mathcal{X}}_{K}^{\diamondsuit },\end{eqnarray}$$

where the limit ranges over the open subgroups $K$ with $H\subseteq K\subseteq G(\mathbb{Z}_{p})$. Note that this exists by Proposition 2.1.1. We define $\overline{{\mathcal{X}}}_{H}$ and ${\mathcal{X}}_{H}^{\ast }$ similarly; these are spatial diamonds (again by Proposition 2.1.1). We also define $\widetilde{{\mathcal{X}}}_{\widetilde{H}}$ and $\widetilde{{\mathcal{X}}}_{\widetilde{H}}^{\ast }$ analogously for $\widetilde{G}$, when $\widetilde{H}\subseteq \widetilde{G}(\mathbb{Z}_{p})$ is a closed subgroup.

We have a natural identification $|{\mathcal{X}}_{H}|=\mathop{\varprojlim }\nolimits_{H\subseteq K}|{\mathcal{X}}_{K}|$ by Proposition 2.1.1, and similarly for the other infinite-level Shimura varieties.

With this definition, we may extend the towers $({\mathcal{X}}_{K})_{K}$ and $({\mathcal{X}}_{K}^{\ast })_{K}$, for open subgroups $K$, to towers $({\mathcal{X}}_{H})_{H}$ and $({\mathcal{X}}_{H}^{\ast })_{H}$, where we index over all closed subgroups. The right actions of $G(\mathbb{Q}_{p})$ on $({\mathcal{X}}_{K})_{K}$, $(\overline{{\mathcal{X}}}_{K})_{K}$ and $({\mathcal{X}}_{K}^{\ast })_{K}$ extend naturally to right actions on $({\mathcal{X}}_{H})_{H}$, $(\overline{{\mathcal{X}}}_{K})_{K}$ and $({\mathcal{X}}_{H}^{\ast })_{H}$. If $H\subseteq G(\mathbb{Z}_{p})$ is normal, then $G(\mathbb{Z}_{p})/H$ acts on ${\mathcal{X}}_{H}$, $\overline{{\mathcal{X}}}_{H}$ and ${\mathcal{X}}_{H}^{\ast }$. Note that, for arbitrary $H$, if $g\in G(\mathbb{Q}_{p})$ and $H,g^{-1}Hg\subseteq G(\mathbb{Z}_{p})$, then right multiplication of $g$ induces an isomorphism ${\mathcal{X}}_{H}\xrightarrow[{}]{{\sim}}{\mathcal{X}}_{g^{-1}Hg}$ of diamonds, and similarly for the compactifications.

Next, we establish notation for some closed subgroups that will occur frequently. Recall that $P_{\unicode[STIX]{x1D707}}$ and $N$ were defined in §3.1.5.

Definition 3.2.9. We let $\mathbf{1}\subseteq G(\mathbb{Z}_{p})$ denote the trivial subgroup and define

$$\begin{eqnarray}P_{\unicode[STIX]{x1D707},0}:=P_{\unicode[STIX]{x1D707}}(\mathbb{Z}_{p})=\mathop{\bigcap }_{m\geqslant 0}\unicode[STIX]{x1D6E4}_{0}(p^{m}),\quad N_{0}:=N(\mathbb{Z}_{p})=\mathop{\bigcap }_{m\geqslant 0}\unicode[STIX]{x1D6E4}_{1}(p^{m}).\end{eqnarray}$$

We define $\widetilde{\mathbf{1}}$, $\widetilde{P}_{\unicode[STIX]{x1D707},0}$ and $\widetilde{N}_{0}$ analogously for $\widetilde{G}$.

Note that $\mathbf{1}=\bigcap _{m\geqslant 0}\unicode[STIX]{x1D6E4}(p^{m})$. We finish by recording some results about group actions. We begin with a result for open Shimura varieties.

Proof. Set $K_{i,m}=H_{i}\unicode[STIX]{x1D6E4}(p^{m})$ for $i=1,2$ and $m\geqslant 0$. Then $K_{1,m}$ is normal in $K_{2,m}$ and hence ${\mathcal{X}}_{K_{1,m}}\rightarrow {\mathcal{X}}_{K_{2,m}}$ is a $K_{2,m}/K_{1,m}$-torsor compatibly in $m$, i.e. we have compatible isomorphisms

$$\begin{eqnarray}{\mathcal{X}}_{K_{2,m}}\times \text{}\underline{K_{2,m}/K_{1,m}}\xrightarrow[{}]{{\sim}}{\mathcal{X}}_{K_{1,m}}\times _{{\mathcal{X}}_{K_{2,m}}}{\mathcal{X}}_{K_{1,m}}\end{eqnarray}$$

(this holds for rigid spaces and follows formally for diamonds, since fiber products are preserved). Taking the inverse limit over $m$ gives us the isomorphism ${\mathcal{X}}_{H_{2}}\times \text{}\underline{H_{2}/H_{1}}\xrightarrow[{}]{{\sim}}{\mathcal{X}}_{H_{1}}\times _{{\mathcal{X}}_{H_{2}}}{\mathcal{X}}_{H_{1}}$, so ${\mathcal{X}}_{H_{1}}\rightarrow {\mathcal{X}}_{H_{2}}$ is a $H_{2}/H_{1}$-torsor, as desired.

We now move on to compactifications, where we will content ourselves with proving statements at the level of topological space. We will only need the first lemma for $\widetilde{G}$, but we state it in general.

Our task is now to prove the corresponding statement for ad hoc compactifications. Because of the way the ad hoc compactifications are defined, it is easiest not to work with a fixed $K^{p}$, so we drop this from now on until the beginning of § 3.2.16; we make definitions analogous to the above for closed subgroups $H\subseteq G(\mathbb{A}_{f})$. We will break up the proof into a sequence of short results.

Next, we state a general lemma on diamonds of rigid spaces. If $Z$ is a topological space, there is a v-sheaf $\text{}\underline{Z}$ defined just before [Reference ScholzeSch17, Definition 10.12].

We may then give a convenient description of the diamond of the ad hoc compactification.

Proof. Let $\widetilde{K}\subseteq \widetilde{G}(\mathbb{A}_{f})$ be a sufficiently small open compact subgroup throughout this proof, in the sense of Proposition 3.2.12. Proposition 3.2.12 and Lemma 3.2.13 show that, if $S_{\widetilde{K}}$ is the closure of $|{\mathcal{X}}_{K}|$ in $|\widetilde{{\mathcal{X}}}_{\widetilde{K}}^{\ast }|$, then $\overline{{\mathcal{X}}}_{K}^{\diamondsuit }=S_{\widetilde{K}}\times _{\text{}\underline{|\widetilde{{\mathcal{X}}}_{\widetilde{K}}^{\ast }|}}\widetilde{{\mathcal{X}}}_{\widetilde{K}}^{\ast ,\diamondsuit }$. Taking inverse limits over $\widetilde{K}$, we get

$$\begin{eqnarray}\overline{{\mathcal{X}}}_{K}^{\diamondsuit }=S_{K}\times _{\text{}\underline{|\widetilde{{\mathcal{X}}}_{K}^{\ast }|}}\widetilde{{\mathcal{X}}}_{K}^{\ast },\end{eqnarray}$$

where $S_{K}=\mathop{\varprojlim }\nolimits_{\widetilde{K}}S_{\widetilde{K}}$, so it suffices to prove that $S_{K}$ is the closure of $|{\mathcal{X}}_{K}|$ in $|\widetilde{{\mathcal{X}}}_{K}^{\ast }|$. By the definition of the inverse limit topology, a point $x\in |\widetilde{{\mathcal{X}}}_{K}^{\ast }|$ is in the closure of $|{\mathcal{X}}_{K}|$ if and only if for every $\widetilde{K}$, every open neighborhood of the image of $x$ in $|\widetilde{{\mathcal{X}}}_{\widetilde{K}}^{\ast }|$ intersects $|{\mathcal{X}}_{K}|$, and this is equivalent to the image of $x$ in each $|\widetilde{{\mathcal{X}}}_{\widetilde{K}}^{\ast }|$ being in the closure of $|{\mathcal{X}}_{K}|$. This finishes the proof.

We now get to the analogue of Lemma 3.2.11.

3.2.16 The anticanonical tower

In this section, we adapt the construction of the anticanonical tower over a neighborhood of the ordinary locus for the Shimura varieties of $\widetilde{G}$ as in [Reference ScholzeSch15, § 3] to the Shimura varieties of $G$. Our strategy is to deduce results for $G$ from those for $\widetilde{G}$, but we will also need some refinements of the results of [Reference ScholzeSch15, § 3] for $\widetilde{G}$.

The special fiber of the integral model $X^{\ast }$ admits a Newton stratification: see, for example, [Reference Lan and StrohLS18, §3.3]. The reflex field of $G$ is $\mathbb{Q}$, so [Reference WedhornWed99, Theorem 1.6.3] (together with [Reference Lan and StrohLS18, §3.3] for the extension to the boundary) implies that there exists an open dense ordinary stratum in $X_{\overline{\mathbb{F}}_{p}}^{\ast }$. As usual, one can also recover the ordinary stratum as the complement of the vanishing locus of the Hasse invariant $\text{Ha}$. Over $X^{\ast }$, we have the Hodge line bundle $\unicode[STIX]{x1D714}$, and the Hasse invariant $\text{Ha}$ is a section of $\unicode[STIX]{x1D714}_{\overline{\mathbb{F}}_{p}}^{\otimes (p-1)}$. The above discussion also holds for $\widetilde{G}$, and we note that $\text{Ha}$ on $X_{\overline{\mathbb{F}}_{p}}^{\ast }$ is the pullback of the Hasse invariant on $\widetilde{X}_{\overline{\mathbb{F}}_{p}}^{\ast }$.

Let us now recall the results from [Reference ScholzeSch15, §3] for $\widetilde{G}$ that we need (partially to set up notation), and prove an additional result that will be crucial for us in this paper (Proposition 3.2.19). We start by recalling the anticanonical tower for $\widetilde{G}$. In this discussion, we will work over $\mathbb{Q}_{p}$ instead of $C$ until further notice to simplify referencing to [Reference ScholzeSch15], so all Shimura varieties for $\widetilde{G}$ (of finite or infinite level) are considered to be defined over $\mathbb{Q}_{p}$; we will sometimes add a subscript $\mathbb{Q}_{p}$ when we wish to emphasize this. Let $0\leqslant \unicode[STIX]{x1D716}<1/2$. The anticanonical locus $\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p)}^{\ast }(\unicode[STIX]{x1D716})_{a}$ of level $\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p)$ and radius of overconvergence $\unicode[STIX]{x1D716}$ is (essentially) defined in [Reference ScholzeSch15, Theorem 3.2.15(iii)]. This is an open subset of $\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p)}^{\ast }$, defined as the image of the map

$$\begin{eqnarray}\widetilde{{\mathcal{X}}}^{\ast }(\unicode[STIX]{x1D716})\rightarrow \widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p)}^{\ast },\end{eqnarray}$$

where $\widetilde{{\mathcal{X}}}^{\ast }(\unicode[STIX]{x1D716})$ is the locus where $|Ha|\geqslant p^{\unicode[STIX]{x1D716}}$ (here and elsewhere, this condition is defined in terms of local (integral) lifts as usual, and independent of the choices of lifts) and the map sends a principally polarized abelian variety $A$ (with tame level structure) to $(A/\text{Can},A[p]/\text{Can})$ (with the induced tame level structure), where $\text{Can}\subseteq A[p]$ is the canonical subgroup. For any closed subgroup $\widetilde{H}\subseteq \widetilde{\unicode[STIX]{x1D6E4}}_{0}(p)$, we define

When $\widetilde{H}$ is open, we primarily view this as a rigid space. By [Reference ScholzeSch15, Corollary 3.2.19] (see remark below), there is a unique affinoid perfectoid space $\widetilde{{\mathcal{X}}}_{\widetilde{P}_{\unicode[STIX]{x1D707},0}\cap \widetilde{G}^{\text{der}}(\mathbb{Z}_{p}),\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a}$ such that

where $\widetilde{G}^{\text{der}}\subseteq \widetilde{G}$ is the subgroup where the similitude factor is $1$, and $\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})^{\prime }$ is the subgroup of $\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})$ given by imposing the condition that the similitude factor is congruent to $1$ modulo $p^{m}$. Moreover, the boundary

is strongly Zariski closed, in the sense of [Reference ScholzeSch15, Definition 2.2.6].Footnote ⁶

Going further up, by [Reference ScholzeSch15, Proposition 3.2.34 and Theorem 3.2.36] there are unique affinoid perfectoid spaces $\widetilde{{\mathcal{X}}}_{\widetilde{N}_{0},\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a}$ and $\widetilde{{\mathcal{X}}}_{\widetilde{\mathbf{1}},\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a}$ such that

$$\begin{eqnarray}\widetilde{{\mathcal{X}}}_{\widetilde{N}_{0},\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a}\sim \mathop{\varprojlim }\nolimits_{m}\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{1}(p^{m}),\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a};\end{eqnarray}$$

$$\begin{eqnarray}\widetilde{{\mathcal{X}}}_{\widetilde{\mathbf{1}},\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a}\sim \mathop{\varprojlim }\nolimits_{m}\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}(p^{m}),\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a}.\end{eqnarray}$$

Moreover, the boundaries $\widetilde{{\mathcal{Z}}}_{\widetilde{N}_{0},\mathbb{Q}_{p}}(\unicode[STIX]{x1D716})_{a}\subseteq \widetilde{{\mathcal{X}}}_{\widetilde{N}_{0},\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a}$ and $\widetilde{{\mathcal{Z}}}_{\widetilde{\mathbf{1}},\mathbb{Q}_{p}}(\unicode[STIX]{x1D716})_{a}\subseteq \widetilde{{\mathcal{X}}}_{\widetilde{\mathbf{1}},\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a}$, respectively, are strongly Zariski closed, and functions on $\widetilde{{\mathcal{X}}}_{\widetilde{\mathbf{1}},\mathbb{Q}_{p}}(\unicode[STIX]{x1D716})_{a}$ extend uniquely to $\widetilde{{\mathcal{X}}}_{\widetilde{\mathbf{1}},\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a}$. It will be essential for the arguments of this paper to have a slightly more general result. Before we state and prove this generalization, we recall [Reference ScholzeSch15, Lemma 3.2.24(iii)]Footnote ⁷ which will be used in the proof. For all levels $\widetilde{H}$, we use $\widetilde{{\mathcal{X}}}_{\widetilde{H}}^{\text{gd}}(\unicode[STIX]{x1D716})_{a}$ and $\widetilde{{\mathcal{X}}}_{\widetilde{H}}^{\text{gd}}$ to denote the good reduction loci in $\widetilde{{\mathcal{X}}}_{\widetilde{H}}(\unicode[STIX]{x1D716})_{a}$ and $\widetilde{{\mathcal{X}}}_{\widetilde{H}}$, respectively; these are quasi-compact open subspaces (or diamonds).

Lemma 3.2.18 [Reference ScholzeSch15, Lemma 3.2.24(iii)].

Let ${\mathcal{Y}}_{m}^{\ast }\rightarrow \widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})^{\prime },\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a}$ be finite, étale away from the boundary and assume that ${\mathcal{Y}}_{m}^{\ast }$ is normal and that no irreducible component of ${\mathcal{Y}}_{m}^{\ast }$ maps to the boundary of $\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})^{\prime },\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a}$. In particular,

$$\begin{eqnarray}{\mathcal{Y}}_{m}^{\text{gd}}:=\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})^{\prime },\mathbb{Q}_{p}}^{\text{gd}}(\unicode[STIX]{x1D716})_{a}\times _{\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})^{\prime },\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a}}{\mathcal{Y}}_{m}^{\ast }\end{eqnarray}$$

is finite étale over ${\mathcal{X}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})^{\prime },\mathbb{Q}_{p}}^{\text{gd}}(\unicode[STIX]{x1D716})_{a}$. For any $m^{\prime }\geqslant m$, let ${\mathcal{Y}}_{m^{\prime }}^{\ast }$ be the normalization of $\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m^{\prime }})^{\prime },\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a}\times _{\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})^{\prime },\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a}}{\mathcal{Y}}_{m}^{\ast }$ and let ${\mathcal{Y}}_{m^{\prime }}^{\text{gd}}$ be the preimage of ${\mathcal{Y}}_{m}^{\text{gd}}$. Let

$$\begin{eqnarray}{\mathcal{Y}}_{\infty }^{\text{gd}}={\mathcal{Y}}_{m}^{\text{gd}}\times _{\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})^{\prime },\mathbb{Q}_{p}}^{\text{gd}}(\unicode[STIX]{x1D716})_{a}}\widetilde{{\mathcal{X}}}_{\widetilde{P}_{\unicode[STIX]{x1D707},0}\cap \widetilde{G}^{\text{der}}(\mathbb{Z}_{p}),\mathbb{Q}_{p}}^{\text{gd}}(\unicode[STIX]{x1D716})_{a};\end{eqnarray}$$

this exists as a perfectoid space, since ${\mathcal{Y}}_{m}^{\text{gd}}\rightarrow \widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})^{\prime },\mathbb{Q}_{p}}^{\text{gd}}(\unicode[STIX]{x1D716})_{a}$ is finite étale. The spaces ${\mathcal{Y}}_{m^{\prime }}^{\ast }$ are affinoid for $m^{\prime }$ sufficiently large, so write ${\mathcal{Y}}_{m^{\prime }}^{\ast }=\operatorname{Spa}(S_{m^{\prime }},S_{m^{\prime }}^{+})$. Assume further that $S_{\infty }:=H^{0}({\mathcal{Y}}_{\infty }^{\text{gd}},{\mathcal{O}}_{{\mathcal{Y}}_{\infty }^{\text{gd}}})$ is a perfectoid $\mathbb{Q}_{p}^{\text{cycl}}$-algebra and define ${\mathcal{Y}}_{\infty }^{\ast }:=\operatorname{Spa}(S_{\infty },S_{\infty }^{\circ })$. Then ${\mathcal{Y}}_{\infty }^{\ast }\sim \mathop{\varprojlim }\nolimits_{m^{\prime }}{\mathcal{Y}}_{m^{\prime }}^{\ast }$.

To simplify notation, we will not write out the subscripts ‘$\mathbb{Q}_{p}$’ for most of the discussion below until after the proof of Corollary 3.2.20, and we will put $\widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }:=\widetilde{P}_{\unicode[STIX]{x1D707},0}\cap \widetilde{G}^{\text{der}}(\mathbb{Z}_{p})$. Returning to the situation of the lemma above and keeping in mind these conventions, we recall that $\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{1}(p^{m})\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}$ is proved to be perfectoid via Lemma 3.2.18. In particular,

$$\begin{eqnarray}H^{0}(\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{1}(p^{m})\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a},{\mathcal{O}}_{\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{1}(p^{m})\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}})=H^{0}(\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{1}(p^{m})\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\text{gd}}(\unicode[STIX]{x1D716})_{a},{\mathcal{O}}_{\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{1}(p^{m})\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\text{gd}}(\unicode[STIX]{x1D716})_{a}})\end{eqnarray}$$

is a perfectoid $\mathbb{Q}_{p}^{\text{cycl}}$-algebra. As $\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}(p^{m})}^{\ast }(\unicode[STIX]{x1D716})_{a}\rightarrow \widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{1}(p^{m})}^{\ast }(\unicode[STIX]{x1D716})_{a}$ is finite étale by [Reference ScholzeSch15, Lemma 3.2.35]), one sees that

$$\begin{eqnarray}\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}(p^{m})\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}=\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}(p^{m})}^{\ast }(\unicode[STIX]{x1D716})_{a}\times _{\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{1}(p^{m})}^{\ast }(\unicode[STIX]{x1D716})_{a}}\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{1}(p^{m})\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}\end{eqnarray}$$

is affinoid perfectoid and a direct calculation using that $\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}(p^{m})}^{\ast }(\unicode[STIX]{x1D716})_{a}\rightarrow \widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{1}(p^{m})}^{\ast }(\unicode[STIX]{x1D716})_{a}$ is finite étale shows that

$$\begin{eqnarray}H^{0}(\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}(p^{m})\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a},{\mathcal{O}}_{\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}(p^{m})\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}})=H^{0}(\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}(p^{m})\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\text{gd}}(\unicode[STIX]{x1D716})_{a},{\mathcal{O}}_{\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}(p^{m})\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\text{gd}}(\unicode[STIX]{x1D716})_{a}}),\end{eqnarray}$$

so the right-hand side is perfectoid.

Proposition 3.2.19. We work over $\mathbb{Q}_{p}$. Let $\widetilde{K}\subset \widetilde{G}(\mathbb{Z}_{p})$ be a compact open subgroup. There exists an affinoid perfectoid space

$$\begin{eqnarray}\widetilde{{\mathcal{X}}}_{\widetilde{K}\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}\sim \mathop{\varprojlim }\nolimits_{m}\widetilde{{\mathcal{X}}}_{\widetilde{K}\cap \widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}.\end{eqnarray}$$

Moreover, the boundary $\widetilde{{\mathcal{Z}}}_{\widetilde{K}\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}(\unicode[STIX]{x1D716})_{a}\subset \widetilde{{\mathcal{X}}}_{\widetilde{K}\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}$ is strongly Zariski closed.

Proof. We check that the conditions of Lemma 3.2.18 apply to ${\mathcal{Y}}_{m}^{\ast }:=\widetilde{{\mathcal{X}}}_{\widetilde{K}\cap \widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}$ for any large enough $m$. The projection map

$$\begin{eqnarray}\widetilde{{\mathcal{X}}}_{\widetilde{K}\cap \widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})^{\prime }}^{\ast }\rightarrow \widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})^{\prime }}^{\ast }\end{eqnarray}$$

is finite étale away from the boundary, $\widetilde{{\mathcal{X}}}_{\widetilde{K}\cap \widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}$ is normal and no irreducible component of $\widetilde{{\mathcal{X}}}_{\widetilde{K}\cap \widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}$ maps into the boundary. Note that the space ${\mathcal{Y}}_{\infty }^{\text{gd}}:=\widetilde{{\mathcal{X}}}_{\widetilde{K}\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\text{gd}}(\unicode[STIX]{x1D716})_{a}$ exists and is perfectoid, since it is the pullback to the perfectoid space $\widetilde{{\mathcal{X}}}_{\widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\text{gd}}(\unicode[STIX]{x1D716})_{a}$ of $\widetilde{{\mathcal{X}}}_{\widetilde{K}\cap \widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})^{\prime }}(\unicode[STIX]{x1D716})_{a}\rightarrow \widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p^{m})^{\prime }}(\unicode[STIX]{x1D716})_{a}$, which is finite étale. The last thing we need to show is that

$$\begin{eqnarray}S_{\infty }^{\prime }:=H^{0}(\widetilde{{\mathcal{X}}}_{\widetilde{K}\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\text{gd}}(\unicode[STIX]{x1D716})_{a},{\mathcal{O}}_{\widetilde{{\mathcal{X}}}_{\widetilde{K}\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\text{gd}}(\unicode[STIX]{x1D716})_{a}})\end{eqnarray}$$

is a perfectoid $\mathbb{Q}_{p}^{\text{cycl}}$-algebra.

To see this, we argue as follows. There exists an integer $t\geqslant 1$ such that $\widetilde{\unicode[STIX]{x1D6E4}}(p^{t})\subset \widetilde{K}$; then the quotient $H:=\widetilde{K}/\widetilde{\unicode[STIX]{x1D6E4}}(p^{t})$ is a finite group. From the discussion preceding this proposition, we know that $\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}(p^{t})\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}$ is affinoid perfectoid and that

$$\begin{eqnarray}S_{\infty }:=H^{0}(\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}(p^{t})\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a},{\mathcal{O}}_{\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}(p^{t})\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}})=H^{0}(\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}(p^{t})\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\text{gd}}(\unicode[STIX]{x1D716})_{a},{\mathcal{O}}_{\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}(p^{t})\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\text{gd}}(\unicode[STIX]{x1D716})_{a}}).\end{eqnarray}$$

We now conclude using [Reference Kedlaya and LiuKL15, Proposition 3.6.22], since

$$\begin{eqnarray}S_{\infty }^{\prime }=(S_{\infty })^{H},\end{eqnarray}$$

which follows from the fact that $\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}(p^{t})\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\text{gd}}(\unicode[STIX]{x1D716})_{a}$ is an $H$-torsor over $\widetilde{{\mathcal{X}}}_{\widetilde{K}\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\text{gd}}(\unicode[STIX]{x1D716})_{a}$. This concludes the proof of the first part of the proposition, using Lemma 3.2.18.

To see that $\widetilde{{\mathcal{Z}}}_{\widetilde{K}\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}(\unicode[STIX]{x1D716})_{a}\subset \widetilde{{\mathcal{X}}}_{\widetilde{K}\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}$ is strongly Zariski closed, we note that $\widetilde{{\mathcal{Z}}}_{\widetilde{K}\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}(\unicode[STIX]{x1D716})_{a}$ is the pullback of $\widetilde{{\mathcal{Z}}}_{\widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}(\unicode[STIX]{x1D716})_{a}\subset \widetilde{{\mathcal{X}}}_{\widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}$ over the map $\widetilde{{\mathcal{X}}}_{\widetilde{K}\cap \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}\rightarrow \widetilde{{\mathcal{X}}}_{\widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}$, and use [Reference ScholzeSch15, Lemma 2.2.9].

We immediately deduce the following generalization.

We now go back to working over $C$, and deduce the following result.

Proof. First, assume that $\widetilde{H}\subseteq \widetilde{P}_{\unicode[STIX]{x1D707},0}^{\prime }$. Then

$$\begin{eqnarray}\widetilde{{\mathcal{X}}}_{\widetilde{H}}^{\ast }(\unicode[STIX]{x1D716})_{a}=(\widetilde{{\mathcal{X}}}_{\widetilde{H},\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a}\times _{\operatorname{Spa}(\mathbb{Q}_{p})^{\diamondsuit }}\operatorname{Spa}(\mathbb{Q}_{p}^{\text{cycl}})^{\diamondsuit })\times _{\operatorname{Spa}(\mathbb{Q}_{p}^{\text{cycl}})^{\diamondsuit }}\operatorname{Spa}(C)^{\diamondsuit },\end{eqnarray}$$

where the omitted ring of integral elements is the power-bounded elements. The first fiber product is affinoid perfectoid by Corollary 3.2.20, since it is equal to

$$\begin{eqnarray}\mathop{\varprojlim }\nolimits_{m}(\widetilde{{\mathcal{X}}}_{\widetilde{H},\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a}\times _{\operatorname{Spa}(\mathbb{Q}_{p})^{\diamondsuit }}\operatorname{Spa}(\mathbb{Q}_{p}(\unicode[STIX]{x1D701}_{p^{m}}))^{\diamondsuit });\end{eqnarray}$$

the second fiber product is then a fiber product of affinoid perfectoid spaces and hence affinoid perfectoid. The statement about the boundary follows by pullback, using the map $\widetilde{{\mathcal{X}}}_{\widetilde{H}}^{\ast }(\unicode[STIX]{x1D716})_{a}\rightarrow \widetilde{{\mathcal{X}}}_{\widetilde{H},\mathbb{Q}_{p}}^{\ast }(\unicode[STIX]{x1D716})_{a}$.

For general $\widetilde{H}\subseteq \widetilde{P}_{\unicode[STIX]{x1D707},0}$, set $\widetilde{H}^{\prime }=\widetilde{H}\cap G^{\text{der}}(\mathbb{Z}_{p})$. Then $\widetilde{{\mathcal{X}}}_{\widetilde{H}}^{\ast }(\unicode[STIX]{x1D716})_{a}$ is a closed union of connected components of $\widetilde{{\mathcal{X}}}_{\widetilde{H}^{\prime }}^{\ast }(\unicode[STIX]{x1D716})_{a}$, and the result for $\widetilde{H}$ follows from that for $\widetilde{H}^{\prime }$.

We now return to the Shimura varieties for $G$, and continue to work over $C$ for the rest of this paper. For any closed subgroup $H\subseteq \unicode[STIX]{x1D6E4}_{0}(p)$, set

$$\begin{eqnarray}\overline{{\mathcal{X}}}_{H}(\unicode[STIX]{x1D716})_{a}:=\overline{{\mathcal{X}}}_{H}\times _{\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p)}^{\ast }}\widetilde{{\mathcal{X}}}_{\widetilde{\unicode[STIX]{x1D6E4}}_{0}(p)}^{\ast }(\unicode[STIX]{x1D716})_{a};\end{eqnarray}$$

and ${\mathcal{X}}_{H}(\unicode[STIX]{x1D716})_{a}=\overline{{\mathcal{X}}}_{H}(\unicode[STIX]{x1D716})_{a}\cap {\mathcal{X}}_{H}$. These are non-empty, since the ordinary locus of the Shimura varieties of $G$ is non-empty. The following is the main result of this subsection.