Perfectoid Shimura varieties and the Calegari–Emerton conjectures

This paper is motivated by the notion of reciprocity in the Langlands program. Let $G/\mathbb{Q}$ be a connected reductive group. Roughly speaking, reciprocity is the expectation that there should be some precise relationship between

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  algebraic automorphic representations $\pi$ of $G(\mathbb{A_{Q}})$, and

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  $p$-adic Galois representations $\rho:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\,^{L}G(\overline{\mathbb{Q}}_{p})$ which are geometric in the sense of Fontaine-Mazur.

For a more precise conjectural formulation of this relationship, we refer the reader to [Clo90, BG14]. While there are many partial results, the general problem of reciprocity seems very difficult to attack, for (at least) two reasons:

1. (1)

   Algebraic automorphic representations are inherently of an archimedean/real-analytic nature, while $p$-adic Galois representations are (of course) inherently $p$-adic.

2. (2)

   Algebraic automorphic representations are rigid, while $p$-adic Galois representations naturally deform into positive-dimensional families.

These observations suggest that one should try to bridge the gap, by seeking a genuinely $p$-adic variant of the notion of automorphic representation, which is flexible enough to accommodate all $p$-adic Galois representations. At present, the most satisfactory theory of “$p$-adic automorphic representations” is the notion of completed (co)homology, introduced by Emerton [Eme06].

Let us recall the key definitions; we refer the reader to the body of the paper for any unexplained notation. Fix a connected reductive group $G/\mathbb{Q}$. Let $A\subseteq G$ be the maximal $\mathbb{Q}$-split central torus, and let $K_{\infty}\subseteq G(\mathbb{R})$ be a maximal compact subgroup. Let $X^{G}=G(\mathbb{R})/A(\mathbb{R})K_{\infty}$ be the (connected) symmetric space for $G$; we write $X$ for $X^{G}$ if $G$ is clear. For any open compact subgroup $K\subseteq G(\mathbb{A}_{f})$, we have the associated locally symmetric space $X_{K}=G(\mathbb{Q})^{+}\backslash(X\times G(\mathbb{A}_{f}))/K$.

By construction, these spaces admit commuting actions of $G(\mathbb{Q}_{p})$ and a “big” Hecke algebra $\mathbb{T}(K^{p})$, and the $G(\mathbb{Q}_{p})$-actions are continuous for the natural topologies. Moreover, these spaces are not “too big”. In particular, they are all $p$-adically separated and complete with bounded $p^{\infty}$-torsion. Additionally, $\widetilde{H}_{\ast}$ and $\widetilde{H}_{\ast}^{BM}$ are finitely generated as modules over the completed group ring $\mathbb{Z}_{p}\llbracket K_{p}\rrbracket$ for any open compact subgroup $K_{p}\subseteq G(\mathbb{Q}_{p})$, while $\widetilde{H}^{\ast}(K^{p})[\tfrac{1}{p}]$ and $\widetilde{H}^{\ast}_{c}(K^{p})[\tfrac{1}{p}]$ are naturally admissible unitary $\mathbb{Q}_{p}$-Banach space representations of $G(\mathbb{Q}_{p})$.

The main motivations for considering completed (co)homology are summarized in the following conjecture, which we don’t attempt to formulate precisely. For a more careful discussion, we refer the reader to [CE12] and [Eme14].

Here ${}^{C}G$ denotes the $C$-group of $G$ as defined in [BG14], which is an extension of ${}^{L}G$. When $G=\mathrm{GL}_{2}/\mathbb{Q}$, this is (an imprecise version of) a theorem of Emerton [Eme11]. However, in general, very little is known. As mentioned, the precise formulation of this conjecture should not be taken too seriously. The reader wondering about the appearance of the $C$-group and what “suitable” means might want to consider the case $G=\mathrm{PGL}_{2}/\mathbb{Q}$.

In this paper, we study the qualitative properties of completed (co)homology, which are encapsulated in a beautiful conjecture of Calegari–Emerton. To state this conjecture, we need a small amount of additional notation. If $G/\mathbb{Q}$ is a connected reductive group, we define nonnegative integers $l_{0}=\mathrm{rank}\,G(\mathbb{R})-\mathrm{rank}\,A(\mathbb{R})K_{\infty}$ and $q_{0}=\frac{\dim X^{G}-l_{0}}{2}$. Roughly speaking, for semisimple groups $l_{0}$ measures the failure of $G(\mathbb{R})$ to admit discrete series representations, while $q_{0}$ is the lowest degree in which the locally symmetric spaces $X_{K}$ should have “interesting” cohomology.

The individual portions of this conjecture are far from independent, and in fact there are natural implications $(1)\Rightarrow(2)\Rightarrow(3)$. Amusingly, these implications are “asymmetric” in the sense that (1) for $\tilde{H}^{\ast}$ implies (2) for $\tilde{H}_{\ast}$ implies (3) for $\tilde{H}_{\ast}^{BM}$, and similarly (1) for $\tilde{H}^{\ast}_{c}$ implies (2) for $\tilde{H}_{\ast}^{BM}$ implies (3) for $\tilde{H}_{\ast}$.

Let us discuss what was previously known about this conjecture.

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  For some groups of small rank (e.g. $\operatorname{GL}_{2}$, or $\mathrm{Res}_{K/\mathbb{Q}}\operatorname{GL}_{2}$ for $K/\mathbb{Q}$ quadratic, or $\mathrm{GSp}_{4}$), one can prove Conjecture 1.3 by hand using various tricks involving the congruence subgroup property, the cohomological dimension bounds of [BS], Poincaré duality, etc. However, these methods quickly run out of steam.

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  When $G$ is semisimple and $l_{0}=0$, part (4) of the conjecture was proved by Calegari–Emerton [CE09], as a consequence of Matsushima’s formula and limit multiplicity results for discrete series representations.

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  When $G$ admits a Shimura datum of Hodge type, Scholze proved part (1) of Conjecture 1.3, but for $\widetilde{H}^{\ast}_{c}$ only, by perfectoid methods [Sch15]. Shen [She17] later treated the case when $G$ admits a compact Shimura variety of abelian type and satisfies $l_{0}(G)=0$.

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  For the unitary Shimura varieties treated in [CS19], Conjecture 1.3(1) for $\widetilde{H}^{\ast}$ follows from [CS19, Theorem 2.6.2, Lemma 4.6.2]. We make some further comments in Remark 5.23.

The main result of this paper is the following theorem (cf. Theorems 4.4, 4.5, and 4.9).

The assumptions on $G$ here guarantee that $l_{0}(G^{der})=0$, which allows us to prove part (4) of Conjecture 1.3 by a fairly straightforward analysis combining the results of [CE09] with the Leopoldt conjecture for $Z(G)$. By our previous remarks, the whole conjecture now follows if we can prove part (1). Note that when $l_{0}=0$ and $X$ is a Hermitian symmetric domain, part (1) of the conjecture simply asserts that $\widetilde{H}^{i}_{c}=\widetilde{H}^{i}=0$ for all $i>d=\dim_{\mathbb{C}}X$. It is this vanishing conjecture which we focus on.

Our proof of the vanishing conjecture builds on Scholze’s methods and combines them with some new ideas. Roughly speaking, we first reduce to the case where $(G,X)$ is a connected Shimura datum of pre-abelian type, and then proceed in two steps:

Step One. We prove the vanishing of $\widetilde{H}^{i}_{c}$ for $i>d$ by pushing Scholze’s methods to their limit.

Step Two. We prove the vanishing of $\widetilde{H}^{i}$ for $i>d$ by a careful analysis of boundary cohomology, using Step One for $G$ and for various auxiliary almost direct factors of Levi subgroups related to the boundary strata of the minimal compactification.

Let us now describe these steps in more detail.

As described above, the proof of Theorem 1.4 proceeds in two essentially distinct steps. In the first step, we prove the vanishing of $\widetilde{H}^{i}_{c}(K^{p})$ for $i$ above the middle degree, using the $p$-adic geometry of Shimura varieties. For Shimura data of Hodge type, this is one of the main results of [Sch15], where it is deduced from the existence of perfectoid Shimura varieties of Hodge type (with infinite level at $p$). We thus need to generalize the geometric results of [Sch15] to a wider class of Shimura data. To this end, we prove the following theorem.

Recall that a Shimura datum $(G,X)$ is of pre-abelian type if there exists a Shimura datum $(G^{\prime},X^{\prime})$ of Hodge type admitting an isomorphism of connected Shimura data $(G^{ad},X^{+})\simeq(G^{\prime ad},X^{\prime+})$. This is slightly more general than the (somewhat more well-known) notion of a Shimura datum of abelian type. While it is probably true that every tower of minimally compactified Shimura varieties with infinite level at $p$ is perfectoid, we expect that Theorem 1.5 is the most general result which can be proved via current technology. We also state and prove a similar result for connected Shimura varieties, cf. Theorem 5.21.

While the idea behind the proof of Theorem 1.5 is clear, the argument is unfortunately somewhat technical.¹¹1A glance at the proof of the key Proposition 5.20 should convince the reader of this. Roughly speaking, there are two key ingredients:

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  “Perfectoidization results” à la Bhatt–Scholze, building in particular on [BS19, Theorem 1.16(1)]. Roughly speaking, these techniques let us prove that if $(X_{i})_{i\in I}\overset{(f_{i})_{i\in I}}{\to}(Y_{i})_{i\in I}$ is a (pro-)finite morphism between two reasonable inverse systems of rigid analytic spaces, and $\varprojlim_{i\in I}Y_{i}$ is perfectoid, then $\varprojlim_{i\in I}X_{i}$ is also perfectoid. For a precise statement, see Lemma 5.10.

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  A new general and user-friendly existence result for quotients of perfectoid spaces by finite groups, cf. Theorem 5.8.

For open Shimura varieties of abelian type, the problem of proving perfectoidness at infinite level was previously considered by Shen [She17]. We remark that our method is more direct and uses very little from the theory of Shimura varieties and their connected components.

The second step is totally disjoint from the first, and doesn’t use any $p$-adic geometry. We content ourselves with a somewhat impressionistic sketch here. In what follows, assume $G$ is a semisimple group such that $(G,X)$ is a connected Shimura datum of pre-abelian type, and set $d=\dim_{\mathbb{C}}X$ as before.

First, we prove an isomorphism of the form $\widetilde{H}^{i}(K^{p})\cong H^{i}(X_{K^{p}K_{p}},\operatorname{Map}_{cts}(K_{p},\mathbb{Z}_{p}))$ for any choice of open compact subgroup $K_{p}\subseteq G(\mathbb{Q}_{p})$. Here $\operatorname{Map}_{cts}(K_{p},\mathbb{Z}_{p})$ denotes the $K_{p}$-module of continuous $\mathbb{Z}_{p}$-valued functions on $K_{p}$. This is essentially a version of Shapiro’s lemma, and goes back to a paper of Hill [Hil10]. Next, by standard properties of manifolds with boundary, this isomorphism induces an isomorphism $\widetilde{H}^{i}(K^{p})\cong H^{i}(\overline{X}_{K^{p}K_{p}},\operatorname{Map}_{cts}(K_{p},\mathbb{Z}_{p}))$, where $\overline{X}_{K^{p}K_{p}}$ denotes the Borel–Serre compactification of $X_{K^{p}K_{p}}$.

By repeated use of excision for compactly supported cohomology, it now suffices to prove that for some stratification $\overline{X}_{K^{p}K_{p}}=\cup_{Z\in\mathcal{Z}}Z$, we have $H^{i}_{c}(Z,\operatorname{Map}_{cts}(K_{p},\mathbb{Z}_{p})|_{Z})=0$ for all $i>d$ and all $Z\in\mathcal{Z}$. The key idea can now be phrased as follows:

($\dagger$) If we take $\mathcal{Z}$ to be the stratification of $\overline{X}_{K^{p}K_{p}}$ obtained by pulling back the usual stratification of $X^{\ast}_{K^{p}K_{p}}$ along the map $\pi:\overline{X}_{K^{p}K_{p}}\to X^{\ast}_{K^{p}K_{p}}$ constructed by Zucker [Zuc83], then $\mathcal{Z}$ is a stratification with the above property.

The idea that ($\dagger$) is both true and provable is perhaps the most novel contribution of this paper; we make some additional remarks on the use of this stratification in Remark 3.14. Let us give a sketch of the key ideas. Let $S\subseteq X^{\ast}_{K^{p}K_{p}}$ be a boundary stratum, with preimage $Z=\pi^{-1}(S)\subseteq\overline{X}_{K^{p}K_{p}}$. By the structure theory of the minimal compactification, the strata $S$ are indexed by (equivalence classes of) pairs $(Q,\alpha)$ where $Q\subseteq G$ is a $\mathbb{Q}$-rational parabolic subgroup whose projection to each simple factor $G_{i}$ of $G^{ad}$ is either a maximal parabolic or equal to $G_{i}$, and $\alpha$ is some auxiliary data depending on the level structure. (We will suppress all dependences on level structures in the following discussion.) Moreover, the parabolic $Q$ comes equipped with a canonically defined almost direct product decomposition $Q=U\cdot L\cdot H$. Here $U$ is the unipotent radical of $Q$, $L$ is a reductive group (the linear part), $H$ is a semisimple group whose associated symmetric space is Hermitian (the Hermitian part); $L\cdot H$ is the full Levi subgroup of $Q$.

In parallel with this decomposition of $Q$, the stratum $Z$ almost admits a direct product decomposition $Z\approx Z_{U}\times Z_{L}\times Z_{H}$, where $Z_{U}$ is a compact nilmanifold, $Z_{L}$ is the Borel–Serre compactification of a locally symmetric space for the group $L$, and $Z_{H}\cong S$ is a locally symmetric space for the group $H$. The key idea now is that $H^{i}_{c}(Z,\operatorname{Map}_{cts}(K_{p},\mathbb{Z}_{p})|_{Z})$ can also be decomposed accordingly, by a Künneth-like formula, into contributions coming from each of these three factors, which can each be controlled:

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  The contribution of $Z_{U}$ is trivial, which follows from a well-known vanishing principle for completed cohomology of unipotent groups.

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  The contribution of $Z_{H}$ can be expressed in terms of compactly supported completed cohomology for $H$, which can be controlled by Step One.

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  The contribution of $Z_{L}$ can be expressed in terms of completed cohomology for $L$, which can be controlled using the bounds in [BS73], or even using the trivial bound.

The critical observation here is that Step One gives such good control over the contribution of $Z_{H}$ that we need very little control over the contribution of $Z_{L}$.

In reality, the above sketch is somewhat oversimplified, because $Z$ does not really admit a direct product decomposition; rather, it has the structure of an iterated fibration whose fibers are as described above. This makes the proof somewhat more complicated. Nevertheless, the essential idea follows the outline given above.

Let us briefly outline the contents of this paper. Section 2 collects some preliminaries on topology and arithmetic groups that are needed for the computations in later section. Section 3 discusses completed (co)homology and the Calegari–Emerton conjectures, carrying out the core of “Step Two” above. Section 4 introduces Shimura varieties and proves our main results on the Calegari–Emerton conjectures, including Theorem 1.4. Section 5 carries out “Step One”, proving Theorem 1.5 and deducing the vanishing theorem for compactly supported completed cohomology. We note that section 5 is completely independent of the previous sections. Conversely, the vanishing result Corollary 5.22 is the only part of section 5 that gets used in previous sections.

The authors would like to thank Bhargav Bhatt, Frank Calegari, Ana Caraiani, Matt Emerton, Michael Harris, Ben Heuer, Kai-Wen Lan, Vincent Pilloni, Peter Scholze, and Jack Thorne for conversations related to the material in this paper. They also wish to thank Mark Goresky, whose survey [Gor05] was very helpful in the initial stages of the project. Moreover, they wish to thank the Herchel Smith Foundation and the Max Planck Institute for Mathematics in Bonn for supporting visits to Cambridge and Bonn, respectively, during which work on this project was carried out. C.J. was supported by the Herchel Smith Foundation during part of this project.

Let $X$ be a topological space and let $\Gamma$ be a group acting from the left on $X$. In this paper most of our actions will be free²²2The most common terminology for this notion seems to be a free and properly discontinuous action, but we find this terminology rather cumbersome., by which we mean that every point $x\in X$ has an open neighborhood $U$ such that $U\cap\gamma U\neq\emptyset$ only if $\gamma=1$. The quotient map $\pi:X\to X_{\Gamma}:=\Gamma\backslash X$ is then a covering map, and we recall that any left $\Gamma$-module³³3By which we always mean a (left) $\mathbb{Z}[\Gamma]$-module, unless otherwise stated. $M$ defines a local system $\widetilde{M}$ on $X_{\Gamma}$ given by

where the right hand side denotes the locally constant functions $f:\pi^{-1}(U)\to M$ satisfying $f(\gamma x)=\gamma.f(x)$ for all $\gamma\in\Gamma$ and all $x\in\pi^{-1}(U)$. When $X$ is a manifold with boundary, this may be written as

where $\operatorname{Map}$ simply denotes set-theoretic functions (as $\pi_{0}(\pi^{-1}(U))$ is discrete). The following theorem is well known, and follows directly from the fact that the singular chain complex $C_{\bullet}(X)$ is a resolution of $\mathbb{Z}$ by free $\Gamma$-modules.

We now consider a relative version of Theorem 2.1. Let $p:E\to B$ be a fibre bundle with contractible fibre $F$ (all spaces are manifolds with boundary). Assume that we have a group $\Gamma$ acting (from the left) on both $E$ and $B$, making $p$ $\Gamma$-equivariant. We assume further that the action of $\Gamma$ is free on $E$, and that the action of $\Gamma$ on $B$ factors through a quotient $\Delta$ which acts freely on $B$. Set $N=\operatorname{Ker}(\Gamma\to\Delta)$; $N$ then acts freely on the fibres of $p$. Consider the induced map

on quotients.

We will also use a (less precise but more general) version for pushforwards with proper support.

Next, let $X$ be a manifold with boundary with a free left action of a group $\Gamma$, and assume that $\Gamma^{\prime}\subseteq\Gamma$ is a finite index subgroup. Consider the natural map $q:X_{\Gamma^{\prime}}\to X_{\Gamma}$. If $M$ is a $\Gamma^{\prime}$-module, we put

which is a left $\Gamma$-module under right translation $(\gamma.f)(x)=f(x\gamma)$. We then have the following.

We move on to results on the commutation of $M\mapsto\widetilde{M}$ with direct limits. First, let $\overline{X}$ be a manifold with boundary, with a free left action of a group $\Gamma$. Write $\overline{X}_{\Gamma}:=\Gamma\backslash\overline{X}$; we assume that $\overline{X}_{\Gamma}$ is compact, so it has a finite triangulation. Fix such a triangulation and pull it back to $\overline{X}$; this gives a triangulation whose corresponding complex of simplicial chains $C_{\bullet}^{\Delta}(\overline{X})$ is a bounded complex of finite free $\mathbb{Z}[\Gamma]$-modules. Let $(M_{i})_{i\in I}$ be a directed system of $\Gamma$-modules with direct limit $M=\varinjlim_{i}M_{i}$.

We can then prove the result in greater generality. With $\overline{X}$ and $\Gamma$ as above, let $U\subseteq\overline{X}$ be a $\Gamma$-invariant open subset containing the interior of $\overline{X}$. Set $U_{\Gamma}:=\Gamma\backslash U$, $Z:=\overline{X}\setminus U$ and $Z_{\Gamma}:=\Gamma\backslash Z$.

In this subsection we make some definitions and recall a theorem of Hill which we will use to handle completed cohomology later. To begin with, we make the following general definition. Let $R=\varprojlim_{i}R/I^{n}$ be an adic ring, with $I$ a finitely generated ideal of definition.

We now recall the computation of completed cohomology as the cohomology of a “big” local system at finite level in some circumstances, which first appeared in [Hil10]. Let $\overline{X}$ be a manifold with boundary, equipped with a left action of a group $G$. We assume that there is a subgroup $\Gamma\subseteq G$ which acts freely on $\overline{X}$, and suppose that $\Gamma=\Gamma_{0}\supseteq\Gamma_{1}\supseteq\Gamma_{2}\supseteq\dots$ is a sequence of finite index subgroups of $\Gamma$. Let $X\subseteq\overline{X}$ be a $\Gamma$-stable open subset containing the interior of $\overline{X}$. Set

and define

this is a profinite set with a right action of $\Gamma$. Assume that $\overline{X}_{\Gamma}$ is compact. Then we get the following formula for completed cohomology of $\widehat{X}$ (cf. [Hil10, Corollary 1]):

We will also encounter local systems slightly bigger than the one appearing in Proposition 2.9. We keep the notation and assumptions of Proposition 2.9, except that we forget the groups denoted by $K$ and $\Gamma_{i}$, $i\geq 1$. Let $G$ be a profinite group with closed subgroups $K\subseteq H\subseteq G$, and assume that $K$ is normal in $H$. For simplicity, we assume that there is a countable basis of neighborhoods of $1\in G$. Suppose that we have a group homomorphism $\Gamma\to H/K$; then $\operatorname{Map}_{cts}(H/K,R)$ and $\operatorname{Map}_{cts}(G/K,R)$ become left $\Gamma$-modules via right translation, and hence induce local systems on the space $X_{\Gamma}$. Then we have the following simple but useful lemma.

Here we quickly recall some material on arithmetic and congruence subgroups. Let $G$ be a connected linear algebraic group over $\mathbb{Q}$. Congruence subgroups of $G(\mathbb{Q})$ are subgroups of the form $G(\mathbb{Q})\cap K$, where $K\subseteq G(\mathbb{A}_{f})$ is a compact open subgroup and the intersection is taken inside $G(\mathbb{A}_{f})$. A subgroup in $G(\mathbb{Q})$ is usually said to be arithmetic if it is commensurable with one (equivalently any) congruence subgroup. In fact, one can require a slightly stronger condition.

Moving on, let $H$ be another connected linear algebraic group, and let $\Gamma\subseteq G(\mathbb{Q})$ be an arithmetic subgroup. If $H\subseteq G$ is a subgroup, then directly from the definitions we see that $\Gamma\cap H(\mathbb{Q})$ is an arithmetic subgroup in $H(\mathbb{Q})$, which is congruence if $\Gamma$ is. If we instead have a surjection $f:G\to H$, then $f(\Gamma)$ is an arithmetic subgroup (see [PR94, Theorem 4.1]); this will be important in this paper and we will use it freely. Before moving on, we recall that group cohomology for any torsion-free arithmetic subgroup $\Gamma$ commutes with direct limits.

We recall the notion of neatness from [Bor69, §17.1]. An element $\gamma\in G(\mathbb{Q})$ is called neat if there is a faithful representation $r:G\to\operatorname{GL}(V)$ such that the multiplicative group generated by the eigenvalues of $r(\gamma)$ (in one, or equivalently any, algebraically closed field containing $\mathbb{Q}$) is torsion-free. A neat element cannot have finite order. An arithmetic subgroup $\Gamma\subseteq G(\mathbb{Q})$ is called neat if all its elements are neat; such subgroups are in particular torsion-free. From the definitions, we see that if $H\subseteq G$ is a connected linear algebraic subgroup and $\Gamma\subseteq G(\mathbb{Q})$ is neat, then $\Gamma\cap H(\mathbb{Q})$ is neat. If an element $\gamma$ is neat, then for any representation $\rho:G\to GL(W)$, the subgroup generated by the eigenvalues of $\rho(\gamma)$ is torsion-free [Bor69, Corollaire 17.3]. An easy consequence of this is that if $f:G\to H$ is a surjection of linear algebraic groups and $\Gamma$ is neat, then $f(\Gamma)$ is neat.

For language reasons, let us also introduce notions of neatness for adelic and $p$-adic groups. The notion of neatness for an element $g=(g_{p})_{p}\in G(\mathbb{A}_{f})$ and a subgroup $K\subseteq G(\mathbb{A}_{f})$ is defined in [Pin90, §0.6]. For $p$-adic groups, we make the definition analogous to the case of arithmetic groups: An element $g\in G(\mathbb{Q}_{p})$ is called neat if there is a faithful representation $\rho:G_{\mathbb{Q}_{p}}\to\operatorname{GL}(W)$ over $\mathbb{Q}_{p}$ such that the multiplicative group generated by the eigenvalues of $\rho(g)$ (in one, or equivalently any, algebraically closed field containing $\mathbb{Q}_{p}$) is torsion-free. Again, this is independent of the choice of $\rho$. A subgroup $K_{p}\subseteq G(\mathbb{Q}_{p})$ is called neat if all of its elements are neat. We note the following implications among these concepts: If $K_{p}\subseteq G(\mathbb{Q}_{p})$ is a neat compact open subgroup, then $K^{p}K_{p}\subseteq G(\mathbb{A}_{f})$ is neat for any compact open $K^{p}\subseteq G(\mathbb{A}_{f}^{p})$. If a compact open $K\subseteq G(\mathbb{A}_{f})$ is neat, then $\Gamma=\Gamma(\mathbb{Q})\cap K$ is a neat congruence subgroup of $G$.

We record the following version of the standard result that “sufficiently small” congruence subgroups are neat; it will be important for us to be able to only impose congruence conditions at a fixed prime $p$.