# Torsion Vanishing for some Shimura Varieties

Linus Hamann and Si Ying Lee, with an Appendix by David Hansen

###### Abstract.

We generalize the torsion vanishing results of [CS17, CS19, Kos21, San23]. Our
results apply to the cohomology of general Shimura varieties $(\mathbf{G},X)$
of PEL type $A$ or $C$, localized at a suitable maximal ideal $\mathfrak{m}$
in the spherical Hecke algebra at primes $p$ such that
$\mathbf{G}_{\mathbb{Q}_{p}}$ is a group for which we know the Fargues-Scholze
local Langlands correspondence is the semi-simplification of a suitably nice
local Langlands correspondence, as shown in [FS21, Ham21, HKW22, BHN22]. This
is accomplished by combining Koshikawa’s technique [Kos21], the theory of
geometric Eisenstein series over the Fargues-Fontaine curve [Ham22], the work
of Santos [San23] describing the structure of the fibers of the minimally and
toroidally compactified Hodge-Tate period morphism for general PEL type
Shimura varieties of type $A$ or $C$, and ideas developed by Zhang [Zha23] on
comparing Hecke correspondences on the moduli stack of $G$-bundles with the
cohomology of Shimura varieties. In the process, we also establish a
description of the generic part of the cohomology that bears resemblance to
the work of Xiao-Zhu [XZ17]. Moreover, we also construct a filtration on the
compactly supported cohomology that differs from Manotovan’s filtration in the
case that the Shimura variety is non-compact, allowing us to circumvent some
of the circumlocutions taken in [CS19, Kos21]. Our method showcases a very
general strategy for proving such torsion vanishing results, and should bear
even more fruit once the inputs are generalized. Motivated by this, we
formulate an even more general torsion vanishing conjecture (Conjecture 6.6).

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on Shimura Varieties
    1. 2.1 Shimura Varieties
      1. 2.1.1 PEL type A and C
      2. 2.1.2 Compactifications
    2. 2.2 Igusa Varieties
      1. 2.2.1 Compactifications
      2. 2.2.2 Igusa Cusp Labels
      3. 2.2.3 Boundary components
  3. 3 Mantovan’s Formula and the Hodge-Tate Period Morphism
  4. 4 The Local Results
    1. 4.1 The Spectral Action
      1. 4.1.1 Assumption 4.4 under Central Isogenies
    2. 4.2 Perverse $t$-exactness
    3. 4.3 Verification of additional assumptions
  5. 5 The Proof of Theorems 1.15 and 1.17
    1. 5.1 Proof of Theorems 1.14 and 1.15
    2. 5.2 Proof of Corollary 1.19
  6. 6 Conjectures and Concluding Remarks
    1. 6.1 Relationship to Xiao-Zhu
    2. 6.2 A General Torsion Vanishing Conjecture
  7. A Spectral Decomposition of Sheaves on $\mathrm{Bun}_{G}$, by David Hansen

##  1\. Introduction

Let $\mathbf{G}$ be a connected reductive group over $\mathbb{Q}$ admitting a
Shimura datum $(\mathbf{G},X)$, and let $\mathbb{A}$ (resp. $\mathbb{A}_{f}$)
denote the adeles (resp. finite adeles) of $\mathbb{Q}$. Fix a prime number
$p>0$ and let $G:=\mathbf{G}_{\mathbb{Q}_{p}}$ be the base-change to
$\mathbb{Q}_{p}$. We will assume that $G$ is unramified so that there exists a
hyperspecial subgroup $K^{\mathrm{hs}}_{p}\subset G(\mathbb{Q}_{p})$ and a
Borel $B$ surjecting onto a maximal torus $T$ which we now fix. We consider
the open compact subgroup
$K:=K^{p}K_{p}^{\mathrm{hs}}\subset\mathbf{G}(\mathbb{A}_{f})$, where
$K^{p}\subset\mathbf{G}(\mathbb{A}_{f}^{p})$ denotes a sufficiently small
level away from $p$. Let $\mathrm{Sh}(\mathbf{G},X)_{K}$ denote the
corresponding Shimura variety defined over the reflex field $E$. Given a prime
$p\neq\ell$, we will be interested in understanding the $\ell$-torsion
cohomology groups

$R\Gamma_{c}(\mathrm{Sh}(\mathbf{G},X)_{K,\overline{E}},\overline{\mathbb{F}}_{\ell})$

and

$R\Gamma(\mathrm{Sh}(\mathbf{G},X)_{K,\overline{E}},\overline{\mathbb{F}}_{\ell}).$

In particular, since the level at $p$ is hyperspecial, the unramified Hecke
algebra

$H_{K_{p}^{\mathrm{hs}}}:=\overline{\mathbb{F}}_{\ell}[K_{p}^{\mathrm{hs}}\backslash
G(\mathbb{Q}_{p})/K_{p}^{\mathrm{hs}}]$

will act on these complexes via the right action. Given a maximal ideal
$\mathfrak{m}\subset H_{K_{p}^{\mathrm{hs}}}$, we can localize both of these
cohomology groups at $\mathfrak{m}$. We will be interested in describing this
localization. To do this, we recall that, given such a maximal ideal
$\mathfrak{m}\subset H_{K_{p}^{\mathrm{hs}}}$, this defines an unramified
$L$-parameter

$\phi_{\mathfrak{m}}:W_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}G(\overline{\mathbb{F}}_{\ell})$

specified by a semisimple element
$\phi_{\mathfrak{m}}(\mathrm{Frob}_{\mathbb{Q}_{p}})$. In particular, if $T$
denotes the maximal torus of $G$ then $\phi_{\mathfrak{m}}$ is induced from a
parameter
$\phi_{\mathfrak{m}}^{T}:W_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}T(\overline{\mathbb{F}}_{\ell})\subset\phantom{}^{L}G(\overline{\mathbb{F}}_{\ell})$
factoring through the $L$-group of the maximal torus. Now, recall that the
irreducible representations of $\phantom{}{}^{L}T$ correspond to the
$\Gamma$-orbits $\mathbb{X}_{*}(T_{\overline{\mathbb{Q}}_{p}})/\Gamma$ of
geometric dominant cocharacters of $G$. We have the following definition.

######  Definition 1.1.

[Ham22, Definition 1.4] Given a toral $L$-parameter
$\phi_{T}:W_{\mathbb{Q}_{p}}\rightarrow\phantom{}^{L}T(\overline{\mathbb{F}}_{\ell})$,
we say that $\phi_{T}$ is generic if, for all
$\alpha\in\mathbb{X}_{*}(T_{\overline{\mathbb{Q}}_{p}})/\Gamma$ corresponding
to a $\Gamma$-orbit of coroots, we have that the complex
$R\Gamma(W_{\mathbb{Q}_{p}},\alpha\circ\phi_{T})$ is trivial. Similarly, we
say that $\mathfrak{m}$ is generic if $\phi_{\mathfrak{m}}^{T}$ is a generic
toral parameter.

If $G=\mathrm{GL}_{n}$ then this coincides with the notion of decomposed
generic considered in [CS17, Definition I.9]. We set
$d=\mathrm{dim}(\mathrm{Sh}(\mathbf{G},X)_{K})$. Motivated by [CS17, Theorem
1.1] and [CS19, Theorem 1.1], we make the following conjecture.

######  Conjecture 1.2.

Let $(\mathbf{G},X)$ be a Shimura datum such that
$G=\mathbf{G}_{\mathbb{Q}_{p}}$ is unramified and $K=K_{p}K^{p}$ is a
sufficiently small level with $K_{p}=K_{p}^{\mathrm{hs}}$ hyperspecial. If
$\mathfrak{m}\subset H_{K_{p}^{\mathrm{hs}}}$ is a generic maximal ideal then
the cohomology of
$R\Gamma(\mathrm{Sh}(\mathbf{G},X)_{K,\overline{E}},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m}}$
(resp.
$R\Gamma_{c}(\mathrm{Sh}(\mathbf{G},X)_{K,\overline{E}},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m}}$)
is concentrated in degrees $d\leq i\leq 2d$ (resp. $0\leq i\leq d$).

We first recall the motivating situation of Caraiani-Scholze [CS17, CS19]. Let
$F/\mathbb{Q}$ be a CM field, and let $(B,*,V,\langle\cdot,\cdot\rangle)$ be a
PEL datum with $B$ a central simple $F$-algebra and $V$ a non-zero finite type
left $B$-module. Let $(\mathbf{G},X)$ denote the Shimura datum attached to it,
where $\mathbf{G}$ is a connected reductive group over $\mathbb{Q}$ defined by
the $B$-linear automorphisms of $V$ preserving a choice of pairing
$\langle\cdot,\cdot\rangle$. We have the following result.

######  Theorem 1.3.

[CS17, CS19, Kos21, San23] Assume that $(\mathbf{G},X)$ is a PEL type Shimura
datum of type $A$. If the prime $p$ splits completely in $F$ then Conjecture
1.2 is true.

######  Remark 1.4.

Koshikawa proved this under the assumption that $B=F$ and $V=F^{2n}$, and the
global unitary group $\mathbf{G}$ is quasi-split, as well as in the case when
$p$ is split in $F$ and the Shimura variety is compact. These additional
assumptions were removed in the PhD thesis of Santos [San23].

######  Remark 1.5.

Caraiani-Scholze actually proved a slightly different result. More precisely,
let $S$ be a set of finite places not containing $p$ such that $\mathbf{G}$ is
unramified and $K^{p}$ is hyperspecial away from $S$. Consider a maximal ideal
$\mathfrak{m}\subset\mathbb{T}^{S}$ in the spherical Hecke algebra such that
$\mathfrak{m}$ is generic at $p$. Caraiani-Scholze show that the localization
at $\mathfrak{m}^{p}\subset\mathbb{T}^{S\cup\\{p\\}}$ is concentrated in the
relevant degrees.

######  Remark 1.6.

In the case of Harris-Taylor Shimura varieties, there is also work of Boyer
[Boy19], which describes the localization at non-generic maximal ideals.

######  Remark 1.7.

We believe that Conjecture 1.2 is true under the weaker hypothesis that
$H^{2}(W_{\mathbb{Q}_{p}},\alpha\circ\phi_{T})$ is trivial for all
$\Gamma$-orbits of coroots $\alpha$, as is shown in [CS19, San23, Kos21] in
their particular case. However, the theory of geometric Eisenstein series
which we will invoke in this paper becomes more complicated in this case (See
the discussion around [Ham22, Conjecture 1.29]), and so a proof of this
Theorem using our methods would require more deeply understanding geometric
Eisenstein series when this assumption is dropped (cf. Remark 6.8).

Caraiani-Scholze [CS17, CS19] proved their results under some small
restrictions, which Koshikawa [Kos21] was able to remove by using
compatibility of the Fargues-Scholze local Langlands correspondence with the
semi-simplification of the Harris-Taylor correspondence for $\mathrm{GL}_{n}$.
In the process, Koshikawa exhibited a much more flexible method for proving
Theorem 1.3. The goal of the current paper is to expand the scope of
Koshikawa’s technique, motivated by work of the first author on geometric
Eisenstein series in the Fargues-Fontaine setting [Ham22]. We then carry the
strategy out in some particular cases using work on local-global compatibility
of the Fargues-Scholze local Langlands correspondence beyond the case of
$\mathrm{GL}_{n}$.

One of the basic ingredients is the perspective on Mantovan’s product formula
provided by the Hodge-Tate period morphism. To explain this, we let
$\mu\in\mathbb{X}_{*}(T_{\overline{\mathbb{Q}}_{p}})^{+}$ denote the minuscule
geometric dominant cocharacter of $G$ determined by the Hodge cocharacter of
$X$ and an isomorphism $j:\mathbb{C}\simeq\overline{\mathbb{Q}}_{p}$ which we
fix from now on. We consider the Kottwitz set $B(G)$ and with it the subset
$B(G,\mu)\subset B(G)$ of $\mu$-admissible elements. Let $\mathfrak{p}|p$ be
the prime dividing $p$ in the reflex field $E$, induced by the embedding
$\overline{\mathbb{Q}}\rightarrow\overline{\mathbb{Q}}_{p}$ given by the
isomorphism $j$. We let $E_{\mathfrak{p}}$ be the completion at
$\mathfrak{p}$, $C:=\hat{\overline{E}}_{\mathfrak{p}}$ be the completion of
the algebraic closure, and $\breve{E}_{\mathfrak{p}}$ be the compositum of
$E_{\mathfrak{p}}$ with the completion of the maximal unramified extension of
$\mathbb{Q}_{p}$. We recall that, attached to each element $b\in B(G,\mu)$,
there exists a diamond

$\mathrm{Sht}(G,b,\mu)_{\infty}\rightarrow\mathop{\rm
Spd}(\breve{E}_{\mathfrak{p}})$

parametrizing modifications

$\mathcal{E}_{b}\dashrightarrow\mathcal{E}_{0}$

of meromorphy $\mu$ between the $G$-bundle $\mathcal{E}_{b}$ corresponding to
$b$ and the trivial $G$-bundle. This space has an action by
$G(\mathbb{Q}_{p})=\mathrm{Aut}(\mathcal{E}_{0})$ and
$J_{b}(\mathbb{Q}_{p})\subset\mathrm{Aut}(\mathcal{E}_{b})$, where $J_{b}$ is
the $\sigma$-centralizer of $b$. This allows us to consider the quotients

$\mathrm{Sht}(G,b,\mu)_{\infty}/\underline{K_{p}}\rightarrow\mathop{\rm
Spd}(\breve{E}_{\mathfrak{p}})$

for varying compact open subgroups $K_{p}\subset G(\mathbb{Q}_{p})$. In
certain cases, these quotients are representable by rigid analytic varieties
called local Shimura varieties, but they are always representable as diamonds.
We can consider the compactly supported cohomology

$R\Gamma_{c}(\mathrm{Sht}(G,b,\mu)_{\infty,C}/\underline{K_{p}^{\mathrm{hs}}},\overline{\mathbb{F}}_{\ell})$

at hyperspecial level with torsion coefficients. This has an action of
$W_{E_{\mathfrak{p}}}\times J_{b}(\mathbb{Q}_{p})\times
H_{K_{p}^{\mathrm{hs}}}$. Now, the Mantovan product formula tells us that if
we look at
$R\Gamma(\mathrm{Sh}(\mathbf{G},X)_{K,\overline{E}},\overline{\mathbb{F}}_{\ell})$
then this should always admit a filtration in the derived category whose
graded pieces are

$(R\Gamma_{c}(\mathrm{Sht}(G,b,\mu)_{\infty,C}/\underline{K_{p}^{\mathrm{hs}}},\overline{\mathbb{F}}_{\ell}(d_{b}))[2d_{b}]\otimes_{\mathcal{H}(J_{b})}R\Gamma(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell})$

for varying $b\in B(G,\mu)$, where the objects are as follows.

  1. (1)

$\mathrm{Ig}^{b}$ is the perfect Igusa variety attached to an element $b\in
B(G,\mu)$ in the $\mu$-admissible locus inside $B(G)$ and
$d_{b}:=\mathrm{dim}(\mathrm{Ig}^{b})=\langle 2\rho_{G},\nu_{b}\rangle$, where
$\rho_{G}$ is the half sum of all positive roots and $\nu_{b}$ is the slope
cocharacter of $b$.

  2. (2)

$\mathcal{H}(J_{b}):=C^{\infty}_{c}(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})$
is the usual smooth Hecke algebra.

  3. (3)

$\overline{\mathbb{F}}_{\ell}(d_{b})$ is the sheaf on
$\mathrm{Sht}(G,b,\mu)_{\infty,C}/\underline{K_{p}^{\mathrm{hs}}}$ with
trivial Weil group action and $J_{b}(\mathbb{Q}_{p})$ action as defined in
[Kos21, Lemma 7.4].

Such a filtration should always exist, but is not currently proven in general.
In the case that the Shimura datum $(\mathbf{G},X)$ is PEL of type $A$ or $C$,
a modern proof of this result can be found in [Kos21, Theorem 7.1].

This filtration on the complex
$R\Gamma(\mathrm{Sh}(\mathbf{G},X)_{K,\overline{E}},\overline{\mathbb{F}}_{\ell})$
allows us to roughly split the verification of Conjecture 1.2 into two parts.

  1. (1)

Controlling the cohomology of the shtuka spaces
$R\Gamma_{c}(\mathrm{Sht}(G,b,\mu)_{\infty,C}/\underline{K_{p}^{\mathrm{hs}}},\overline{\mathbb{F}}_{\ell}(d_{b}))_{\mathfrak{m}}$.

  2. (2)

Controlling the cohomology of the Igusa varieties
$R\Gamma(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell})$.

We first discuss point (1). One of the key observations underlying Koshikawa’s
method was that the cohomology of the space $\mathrm{Sht}(G,b,\mu)_{\infty}$
computes the action of a Hecke operator $T_{\mu}$ corresponding to $\mu$ on
$\mathrm{Bun}_{G}$ the moduli stack of $G$-bundles of the Fargues-Fontaine
curve. The Hecke operators commute with the action of the excursion algebra on
$\mathrm{Bun}_{G}$, and the action of the excursion algebra on a smooth
irreducible representation $\rho$, viewed as a sheaf on $\mathrm{Bun}_{G}$,
determines the Fargues-Scholze parameter of $\rho$. It follows that
$R\Gamma_{c}(\mathrm{Sht}(G,b,\mu)_{\infty,C}/\underline{K_{p}^{\mathrm{hs}}},\overline{\mathbb{F}}_{\ell}(d_{b}))_{\mathfrak{m}}$
as a complex of smooth $J_{b}(\mathbb{Q}_{p})$-modules will have irreducible
constituents $\rho$ with Fargues-Scholze parameter $\phi_{\rho}^{\mathrm{FS}}$
equal to $\phi_{\mathfrak{m}}$ as conjugacy classes of parameters. When
$\mathbf{G}_{\mathbb{Q}_{p}}=G$ is a product of $\mathrm{GL}_{n}$s as in
Theorem 1.3 (by the assumption that $p$ splits in $F$), it follows from the
work of Hansen-Kaletha-Weinstein [HKW22, Theorem 1.0.3] that the Fargues-
Scholze correspondence for $J_{b}(\mathbb{Q}_{p})$ with rational coefficients
agrees with the semi-simplification of the Harris-Taylor correspondence, where
we recall that $J_{b}$ is a product of inner forms of $\mathrm{GL}_{n}$s in
this case. In particular, using that $\mathfrak{m}$ is generic, it follows
that $\phi_{\rho}^{\mathrm{FS}}=\phi_{\mathfrak{m}}$ must lift to a
$\overline{\mathbb{Z}}_{\ell}$ parameter which is also generic in the
analogous sense, and the condition of generic implies that the lift cannot
come from the semi-simplification of a parameter with non-trivial monodromy.
Using this, one can deduce that such a $\rho$ only exists if the group $J_{b}$
is quasi-split. In this particular case ($G$ is a product of
$\mathrm{GL}_{n}$s), this can only happen if $b\in B(G,\mu)$ corresponds to
the ordinary element.

This argument of Koshikawa was formalized and generalized further in work of
the first author [Ham22]. In particular, it was noted that, for a general
quasi-split $G$ and $\mathfrak{m}$ generic, the cohomology
$R\Gamma_{c}(\mathrm{Sht}(G,b,\mu)_{\infty,C}/\underline{K_{p}^{\mathrm{hs}}},\overline{\mathbb{F}}_{\ell}(d_{b}))_{\mathfrak{m}}$
will only be non-trivial if $b\in
B(G,\mu)_{\mathrm{un}}:=B(G)_{\mathrm{un}}\cap B(G,\mu)$, where
$B(G)_{\mathrm{un}}$ is the set of elements lying in the image of the map
$B(T)\rightarrow B(G)$, assuming that the Fargues-Scholze local Langlands
correspondence has certain expected properties (Assumption 4.4). These
unramified elements will be precisely the elements for which $J_{b}$ is quasi-
split. The set $B(G,\mu)_{\mathrm{un}}$ corresponds to Weyl group orbits of
weights in the representation $V_{\mu}$ of $\hat{G}$ restricted to
$\hat{G}^{\Gamma}$. In particular, if $G$ is split then, since $\mu$ is
minuscule, $B(G,\mu)_{\mathrm{un}}$ consists of only one element,
corresponding to the unique Weyl group orbit of the highest weight. This is
the situation occurring in the previous paragraph. Moreover, the contribution
of the cohomology of this shtuka space is easily understood, and the problem
completely reduces to controlling the cohomology of $\mathrm{Ig}^{b}$ when
$b\in B(G,\mu)_{\mathrm{un}}$ is the $\mu$-ordinary element. However, if $G$
is not split then the restriction of $V_{\mu}$ to $\hat{G}^{\Gamma}$ may have
multiple Weyl group orbits of weights. In particular, one needs to control the
cohomology groups

$R\Gamma_{c}(\mathrm{Sht}(G,b,\mu)_{\infty,C}/\underline{K^{\mathrm{hs}}_{p}},\overline{\mathbb{F}}_{\ell}(d_{b}))_{\mathfrak{m}}$

for all possible $b\in B(G,\mu)_{\mathrm{un}}$. This makes the situation much
more complicated; in fact, for non-split $G$, the basic element could be
unramified, and in this case the Igusa variety is just a profinite set, hence
the problem of torsion vanishing for the contribution of the basic locus is
completely reduced to controlling the generic part of the torsion cohomology
of the local shtuka space attached to the basic element.

Such control of the cohomology of shtuka spaces with torsion coefficients for
these more general situations was attained in [Ham22]. In order to understand
this, it is helpful to move away from the language of isotypic parts of shtuka
spaces and consider the action of Hecke operators on
$\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$, the derived
category of étale $\overline{\mathbb{F}}_{\ell}$-sheaves on
$\mathrm{Bun}_{G}$. Since we are interested in cohomology localized at a
generic maximal ideal $\mathfrak{m}$, we construct in appendix A a full-
subcategory
$\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}\subset\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$
together with an idempotent localization map
$(-)_{\phi_{\mathfrak{m}}}:\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})\rightarrow\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}$
such that, on smooth irreducible representations, the localization map is
either an isomorphism or $0$ depending on if the representation has Fargues-
Scholze parameter conjugate to $\phi_{\mathfrak{m}}$ or not (Lemma 4.2 (1)).
We let
$\mathrm{D}^{\mathrm{ULA}}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$
denote the full subcategory of ULA objects, where we recall by [FS21, Theorem
V.7.1], that this is equivalent to insisting that the restrictions to all the
HN-strata indexed by $b\in B(G)$ are valued in the full subcategories
$\mathrm{D}^{\mathrm{adm}}(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})$
of admissible complexes (i.e the invariants under all open compacts $K\subset
J_{b}(\mathbb{Q}_{p})$ is a perfect complex). Using the results of [Ham22], we
show, under various technical hypothesis including the genericity of
$\mathfrak{m}$, that one has a direct sum decomposition:

$\mathrm{D}^{\mathrm{ULA}}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}\simeq\bigoplus_{b\in
B(G)_{\mathrm{un}}}\mathrm{D}^{\mathrm{adm}}(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}.$

More precisely, we show that the $!$ and $*$ push-forwards with respect to the
inclusion of HN-strata agree on this sub-category, and so the excision semi-
orthogonal decomposition splits on
$\mathrm{D}^{\mathrm{ULA}}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}$.
This decomposition is a refinement of the fact mentioned above that only the
shtuka spaces corresponding to the unramified elements $b\in
B(G,\mu)_{\mathrm{un}}$ can contribute to the generic localization of the
cohomology of the Shimura variety. The desired control of the shtuka spaces is
now in turn encoded in understanding how Hecke operators interact with a
perverse $t$-structure on $\mathrm{Bun}_{G}$ after restricting to the
localized category
$\mathrm{D}^{\mathrm{ULA}}\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}$.

We recall $\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$ has an
action by Hecke operators. In particular, for each geometric dominant
cocharacter $\mu$, we have a correspondence

${\mathrm{Hck}_{G,\leq\mu}}$${\mathrm{Bun}_{G}}$${\mathrm{Bun}_{G}\times\mathop{\rm
Spd}(C)}$$\scriptstyle{h_{\mu}^{\leftarrow}}$$\scriptstyle{h_{\mu}^{\rightarrow}}$

where $\mathrm{Hck}_{G,\leq\mu}$ is the stack parametrizing modifications
$\mathcal{E}_{1}\rightarrow\mathcal{E}_{2}$ of a pair of $G$-bundles with
meromorphy bounded by $\mu$ at the closed Cartier divisor defined by the fixed
untilt given by $C$, and $h_{\mu}^{\rightarrow}$ (resp.
$h_{\mu}^{\leftarrow}$) remembers $\mathcal{E}_{1}$ (resp. $\mathcal{E}_{2}$).
We define

$T_{\mu}:\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})\rightarrow\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})^{BW_{E_{\mu}}}$
$A\mapsto
h_{\mu*}^{\rightarrow}(h_{\mu}^{\leftarrow*}(A)\otimes^{\mathbb{L}}\mathcal{S}_{\mu})$

where $E_{\mu}$ denotes the reflex field of $\mu$ and $\mathcal{S}_{\mu}$ is
the sheaf on $\mathrm{Hck}_{G,\leq\mu}$ attached to the highest weight tilting
module
$\mathcal{T}_{\mu}\in\mathrm{Rep}_{\overline{\mathbb{F}}_{\ell}}(\hat{G})$ of
highest weight $\mu$ by geometric Satake. The action of Hecke operators
commutes with the action of excursion operators and therefore the action of
the spectral Bernstein center. Moreover, it preserves the subcategory of ULA
objects. It follows that we have an induced map

$T_{\mu}:\mathrm{D}^{\mathrm{ULA}}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}\rightarrow\mathrm{D}^{\mathrm{ULA}}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})^{BW_{E_{\mu}}}_{\phi_{\mathfrak{m}}}$

on the localized category (See Lemma 4.2 (2)).

We are almost ready to state the result on Hecke operators we will need. To do
this, we recall that
$\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$ has a natural
perverse $t$-structure, which can be defined as follows. The $v$-stack
$\mathrm{Bun}_{G}$ is cohomologically smooth of $\ell$-dimension $0$.
Moreover, each one of the HN-strata $\mathrm{Bun}_{G}^{b}$ is isomorphic to
$[\ast/\mathcal{J}_{b}]$, which is cohomologically smooth of $\ell$-dimension
$-d_{b}=-\mathrm{dim}(\mathrm{Ig}^{b})$. Therefore, we can define a perverse
$t$-structure $\phantom{}{}^{\mathrm{p}}\mathrm{D}^{\geq
0}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$ (resp.
$\phantom{}{}^{\mathrm{p}}\mathrm{D}^{\leq
0}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$) on
$\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$ given by insisting
that the $!$ (resp. $*$) restrictions to $\mathrm{Bun}_{G}^{b}$ are
concentrated in degrees $\geq\langle 2\rho_{G},\nu_{b}\rangle$ (resp.
$\leq\langle 2\rho_{G},\nu_{b}\rangle$). The key result that follows from the
work of [Ham22] and various compatibility results for the Fargues-Scholze
correspondence is as follows.

######  Theorem 1.8.

(Corollary 4.24) Let $\mu$ be a minuscule geometric dominant cocharacter and
$G$ a product of groups satisfying the conditions of Table (1) with $p$ and
$\ell$ satisfying the corresponding conditions. Then if $\mathfrak{m}$ is
generic the restriction of the Hecke operator

$j_{1}^{*}T_{\mu}:\mathrm{D}^{\mathrm{ULA}}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}\rightarrow\mathrm{D}^{\mathrm{adm}}(G(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})^{BW_{E_{\mu}}}_{\phi_{\mathfrak{m}}}$

is perverse $t$-exact. In particular, it induces maps

$j_{1}^{*}T_{\mu}:\phantom{}^{\mathrm{p}}\mathrm{D}^{\mathrm{ULA},\geq
0}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}\rightarrow\mathrm{D}^{\mathrm{adm},\geq
0}(G(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})^{BW_{E_{\mu}}}_{\phi_{\mathfrak{m}}}$

and

$j_{1}^{*}T_{\mu}:\phantom{}^{\mathrm{p}}\mathrm{D}^{\mathrm{ULA},\leq
0}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}\rightarrow\mathrm{D}^{\mathrm{adm},\leq
0}(G(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})^{BW_{E_{\mu}}}_{\phi_{\mathfrak{m}}}$

on the halves of the perverse $t$-structure, where we note that the perverse
$t$-structure on
$\mathrm{D}(\mathrm{Bun}_{G}^{1},\overline{\mathbb{F}}_{\ell})\simeq\mathrm{D}(G(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})$
coincides with the usual $t$-structure.

Here is the table summarizing our local constraints:

(1) $G$ | Constraint on $G$ | $\ell$ | $p$
---|---|---|---
$\mathrm{Res}_{L/\mathbb{Q}_{p}}(\mathrm{GL}_{n})$ |  $L/\mathbb{Q}_{p}$ unramified | $(\ell,[L:\mathbb{Q}_{p}])=1$ |
$\mathrm{Res}_{L/\mathbb{Q}_{p}}(\mathrm{GSp}_{4})$ | $L=\mathbb{Q}_{p}$ | $(\ell,2(p^{4}-1))=1$ |
|  $L/\mathbb{Q}_{p}$ unramified | $(\ell,2[L:\mathbb{Q}_{p}](p^{4[L:\mathbb{Q}_{p}]}-1))=1$ | $p\neq 2$
$\mathrm{Res}_{L/\mathbb{Q}_{p}}(\mathrm{GU}_{2})$ |  $L/\mathbb{Q}_{p}$ unramified | $(\ell,[L:\mathbb{Q}_{p}])=1$ |
$G=\mathrm{U}_{n}(L/\mathbb{Q}_{p})$ |  $n$ odd $L$ unramified | $\ell\neq 2$ |
$G=\mathrm{GU}_{n}(L/\mathbb{Q}_{p})$ |  $n$ odd $L$ unramified | $\ell\neq 2$ |
$G(\mathrm{SL}_{2,L})$ |  $L/\mathbb{Q}_{p}$ unramified | $(\ell,[L:\mathbb{Q}_{p}])=1$ |
$G(\mathrm{Sp}_{4,L})$ |  $L/\mathbb{Q}_{p}$ unramified, $L\neq\mathbb{Q}_{p}$ | $(\ell,2[L:\mathbb{Q}_{p}](p^{4[L:\mathbb{Q}_{p}]}-1))=1$ | $p\neq 2$

The groups $G(\mathrm{SL}_{2,L})$ and $G(\mathrm{Sp}_{4,L})$ are the
similitude subgroup of $\mathrm{Res}_{L/\mathbb{Q}_{p}}(\mathrm{GL}_{2})$
(resp. $\mathrm{Res}_{L/\mathbb{Q}_{p}}(\mathrm{GSp}_{4})$), i.e. the subgroup
of elements such that the similitude factor lies in $\mathbb{Q}_{p}$. We will
recall the definition of these groups in §4.3.

######  Remark 1.9.

Assuming the Fargues-Scholze correspondence for $G$ behaves as expected with
rational coefficients, the analysis in [Ham22] allows one to verify this for
any $\mu$ after imposing some additional conditions on the toral parameter
$\phi_{T}^{\mathfrak{m}}$ attached to the maximal ideal $\mathfrak{m}$
([Ham22, Condition/Definition 3.6]). However, for the groups considered and
minuscule cocharacters, we will see that these additional conditions are
superfluous and all one needs is generic, except for the case where
$G=\mathrm{Res}_{L/\mathbb{Q}_{p}}(\mathrm{GSp}_{4})$ or
$G=G(\mathrm{Sp}_{4,L})$ with $L/\mathbb{Q}_{p}$ non-trivial, where we need an
extra banality assumption on the prime $\ell$. It is conjectured ([Ham22,
Conjecture 1.27]) that the results used to establish this theorem should
always be true just under the condition that $\mathfrak{m}$ is generic.

######  Remark 1.10.

We should warn the reader that some of the results of [Ham22] and in
particular this consequence, are currently contingent on the proof of some ULA
properties of sheaves on the moduli stack of $B$-structures [Ham22, Assumption
8.1]. However, this will appear in forthcoming work on geometric Eisenstein
series [HHS].

These local torsion vanishing results would allow us to prove Conjecture 1.2
in several new cases if one could get control over the Igusa varieties
$\mathrm{Ig}^{b}$. In Koshikawa’s argument, this is done by using a semi-
perversity result proven by Caraiani-Scholze [CS19, Theorem 4.6.1], which was
further generalized in work of Santos [San23]. Roughly speaking, we want to
show that $R\Gamma(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell})$ is
concentrated in degrees $\geq d_{b}$, so that the complex of
$J_{b}(\mathbb{Q}_{p})$-representations
$R\Gamma(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell})$ defines the stalk of a
semi-perverse sheaf on $\mathrm{Bun}_{G}$ at $b\in B(G)$, to which we can
apply the previous result. In the case that the Shimura vareities
$\mathrm{Sh}(\mathbf{G},X)_{K}$ are compact, there is a simpler way of seeing
this. In particular, $\mathrm{Ig}^{b}$ is known to be a perfect affine scheme
in this case, and so the desired semi-perversity just follows by applying
Artin vanishing and then using Poincaré duality on the global Shimura variety.
It turns out that this style of argument can be made to work even in the non-
compact case. In [CS17, CS19, Kos21, San23], the non-compactly supported
cohomology
$R\Gamma(\mathrm{Sh}(\mathbf{G},X)_{K},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m}}$
is studied together with its filtration involving
$R\Gamma(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell})$ coming from Mantovan’s
formula, and shown to be concentrated in degrees $\geq d$. However, one could
also study the compactly supported cohomology
$R\Gamma_{c}(\mathrm{Sh}(\mathbf{G},X)_{K},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m}}$
and show that it is concentrated in degrees $\leq d$, à la Poincaré duality.
To do this, we recall [CS19, Section 3.3] that, in the non-compact case, the
perfect scheme $\mathrm{Ig}^{b}$ is not affine, but it admits a partial
minimal compactification
$g_{b}:\mathrm{Ig}^{b}\hookrightarrow\mathrm{Ig}^{b,*}$ which is affine, as
proven in this more general setting of PEL type $A$ or $C$ by Santos [San23].
We define

$V_{b}:=R\Gamma_{c-\partial}(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell}):=R\Gamma(\mathrm{Ig}^{b,*},g_{b!}(\overline{\mathbb{F}}_{\ell}))$

the partially compactly supported cohomology, which is supported in degrees
$\leq d_{b}$ by Artin-vanishing (Proposition 3.7). Now, for
$K\subset\mathbf{G}(\mathbb{A}_{f})$ a sufficiently small open compact, we
define
$\mathcal{S}(\mathbf{G},X)_{K}:=(\mathrm{Sh}(\mathbf{G},X)_{K}\otimes_{E}E_{\mathfrak{p}})^{\mathrm{ad}}$
to be the adic space over $\mathop{\rm Spa}(E_{\mathfrak{p}})$ attached to the
Shimura variety. We can define the infinite level perfectoid Shimura varieties
$\mathcal{S}(\mathbf{G},X)_{K^{p}}$ by taking the inverse limit of
$\mathcal{S}(\mathbf{G},X)_{K^{p}K_{p}}$ as $K_{p}\rightarrow\\{1\\}$. The
base-change $\mathcal{S}(\mathbf{G},X)_{K^{p},C}$ is representable by a
perfectoid space if $(\mathbf{G},X)$ is of pre-abelian type, and in general it
is diamond. By the results of [Sch15, Han16], we have a Hodge-Tate period map

$\pi_{\mathrm{HT}}:[\mathcal{S}(\mathbf{G},X)_{K^{p},C}/\underline{G(\mathbb{Q}_{p})}]\rightarrow[\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}]$

recording the Hodge-Tate filtration on the abelian varieties with additional
structure that $\mathcal{S}(\mathbf{G},X)_{K^{p},C}$ parametrizes. Here
$\mathcal{F}\ell_{G,\mu^{-1}}:=(G_{C}/P_{\mu^{-1}})^{\mathrm{ad}}$ is the adic
flag variety attached to the parabolic in $G_{C}$ given by a dominant inverse
of $\mu$ and the dynamical method. We recall that the flag variety
$[\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}]$ admits a
locally closed stratification
$i_{b}:[\mathcal{F}\ell_{G,\mu^{-1}}^{b}/\underline{G(\mathbb{Q}_{p})}]\hookrightarrow[\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}]$
indexed by $b\in B(G,\mu)$, given by pulling the HN-stratification along the
natural map
$h^{\leftarrow}:[\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}]\rightarrow\mathrm{Bun}_{G}$.
We will now impose the following very mild assumption in what follows.

######  Assumption 1.11.

Write $\partial\mathrm{Ig}^{b,*}\subset\mathrm{Ig}^{b,*}$ for the closed
complement of $\mathrm{Ig}^{b}$ in $\mathrm{Ig}^{b,*}$. We assume that
$(\mathbf{G},X)$ is a PEL datum of type $A$ or $C$ such that, for all $b\in
B(G,\mu)$, the perfect scheme $\partial\mathrm{Ig}^{b,*}$ is empty or has
codimension in $\mathrm{Ig}^{b,*}$ greater than $2$.

######  Remark 1.12.

If $\mathbf{G}$ is simple then it is easy to show that this assumption will be
satisfied if $\mathrm{dim}(\mathcal{S}(\mathbf{G},X)_{K^{p},C})\geq 2$, by
using that the boundary of the partially minimally compactified Igusa
varieties is expressible as the Igusa varieties of Shimura varieties attached
to Levis of $\mathbb{Q}$-rational parabolics of $\mathbf{G}$, as we will
explain in §2.2.1. Moreover, if $\mathcal{S}(\mathbf{G},X)_{K^{p},C}$ is
compact then it is automatic that $\partial\mathrm{Ig}^{b,*}$ is empty.
Therefore, if $\mathbf{G}$ is simple, this is excluding the cases where
$\mathrm{dim}(\mathcal{S}(\mathbf{G},X)_{K^{p},C})=1$ and
$\mathcal{S}(\mathbf{G},X)_{K^{p},C}$ is non-compact. There are two
possibilities; either $(\mathbf{G},X)$ is the Shimura datum attached to the
modular curve, or it is the Shimura datum attached to the unitary Shimura
curve (See [Zha23, Proposition 1.9]). In the latter case, we have that the
connected components are given by modular curves. In these cases, the results
of [Kos21] are sufficient to prove Conjecture 1.2.

Now, assuming this, one can show that the stalk of
$R\pi_{\mathrm{HT}!}(\overline{\mathbb{F}}_{\ell})$ at a geometric point
$x:\mathop{\rm Spa}(C,C^{+})\rightarrow\mathcal{F}\ell_{G,\mu^{-1}}$ which
lies in the adic Newton strata $\mathcal{F}\ell_{G,\mu^{-1}}^{b}$ is given by
$V_{b}$. Moreover, if we write
$h_{b}^{\leftarrow}:[\mathcal{F}\ell_{G,\mu^{-1}}^{b}/\underline{G(\mathbb{Q}_{p})}]\rightarrow[\mathop{\rm
Spd}(C)/\mathcal{J}_{b}]\simeq\mathrm{Bun}_{G}^{b}$ for the pullback of
$h^{\leftarrow}$ to $\mathrm{Bun}_{G}^{b}$ then one can deduce that the
complex $i_{b!}i_{b}^{*}R\pi_{\mathrm{HT}!}(\overline{\mathbb{F}}_{\ell})$ is
isomorphic to $h^{\leftarrow*}j_{b!}(V_{b})$. Therefore, by excision, we
deduce that the complex of $G(\mathbb{Q}_{p})\times
W_{E_{\mathfrak{p}}}$-representations

$h_{*}^{\rightarrow}R\pi_{\mathrm{HT}!}(\overline{\mathbb{F}}_{\ell})\simeq
R\Gamma_{c}(\mathcal{S}_{K^{p},C},\overline{\mathbb{F}}_{\ell})\simeq\mathop{\rm
colim}\nolimits_{K_{p}\rightarrow\\{1\\}}R\Gamma_{c}(\mathcal{S}_{K^{p}K_{p},C},\overline{\mathbb{F}}_{\ell})\simeq\mathop{\rm
colim}\nolimits_{K_{p}\rightarrow\\{1\\}}R\Gamma_{c}(\mathrm{Sh}(\mathbf{G},X)_{K^{p}K_{p},C})$

has a filtration with graded pieces isomorphic to
$h^{\rightarrow}_{*}h^{\leftarrow*}(j_{b!}(V_{b}))$ for varying $b\in
B(G,\mu)$, where
$h^{\rightarrow}:[\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}]\rightarrow[\mathop{\rm
Spd}(C)/\underline{G(\mathbb{Q}_{p})}]$ is the structure map quotiented by
$G(\mathbb{Q}_{p})$. Here the second isomorphism follows since taking
compactly supported cohomology respects taking limits of spaces, and the third
isomorphism is a standard comparison result due to Huber [Hub96, Theorem
3.5.13].

Now, via the Bialynicki-Birula isomorphism, the flag variety
$[\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}]$ identifies with
an open substack of $\mathrm{Hck}_{G,\leq\mu}$ for the fixed minuscule $\mu$.
In particular, under this relationship the maps $h^{\rightarrow}_{\mu}$ and
$h^{\leftarrow}_{\mu}$ identify with $h^{\rightarrow}$ and $h^{\leftarrow}$,
and therefore we can relate the graded pieces of the excision filtration to
Hecke operators. We write

$R\Gamma_{c}(G,b,\mu):=\mathop{\rm
colim}\nolimits_{K_{p}\rightarrow\\{1\\}}R\Gamma_{c}(\mathrm{Sht}(G,b,\mu)/\underline{K_{p}},\overline{\mathbb{F}}_{\ell}(d_{b}))$

for the complex of $G(\mathbb{Q}_{p})\times J_{b}(\mathbb{Q}_{p})\times
W_{E_{\mathfrak{p}}}$-modules defined by the compactly supported cohomology of
this tower. Here $\overline{\mathbb{F}}_{\ell}(d_{b})$ is the sheaf with
$J_{b}(\mathbb{Q}_{p})$-action defined as in [Kos21, Lemma 7.4].

We deduce the following variant of the Mantovan product formula for the
compactly supported cohomology.

######  Theorem 1.13.

The complex $R\Gamma_{c}(\mathcal{S}_{K^{p},C},\overline{\mathbb{F}}_{\ell})$
has a filtration as a complex of $G(\mathbb{Q}_{p})\times
W_{E_{\mathfrak{p}}}$-representations with graded pieces isomorphic to
$j_{1}^{*}(T_{\mu}j_{b!}(V_{b}))[-d](-\frac{d}{2})$. More specifically, the
graded pieces are isomorphic to

$(R\Gamma_{c}(G,b,\mu)\otimes^{\mathbb{L}}_{\mathcal{H}(J_{b})}V_{b})[2d_{b}].$

as $G(\mathbb{Q}_{p})\times W_{E_{\mathfrak{p}}}$-modules.

######  Remark 1.14.

When the Shimura variety is compact, we have that
$R\Gamma_{c-\partial}(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell})\simeq
R\Gamma(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell})$, and this recovers
precisely [Kos21, Theorem 7.1].

We now apply our localization functor
$(-)_{\phi_{\mathfrak{m}}}:\mathrm{D}(\mathrm{Bun}_{G})\rightarrow\mathrm{D}(\mathrm{Bun}_{G})_{\phi_{\mathfrak{m}}}$
for a generic maximal ideal $\mathfrak{m}$ to get a complex
$R\Gamma_{c}(\mathcal{S}_{K^{p},C},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}\in\mathrm{D}^{\mathrm{ULA}}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}$,
which we view as a sheaf on $\mathrm{Bun}_{G}$ by $!$ extending along the
neutral strata. After applying $R\Gamma(K_{p}^{\mathrm{hs}},-)$, this agrees
with
$R\Gamma_{c}(\mathcal{S}_{K^{p}K_{p}^{\mathrm{hs}},C},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m}}$,
the usual localization under the unramified Hecke algebra, which is the object
we want to study. This in turn admits a filtration by
$R\Gamma(K_{p}^{\mathrm{hs}},(j_{1}^{*}T_{\mu}j_{b!}(V_{b})))_{\phi_{\mathfrak{m}}}[-d](\frac{-d}{2})$.
However, now we know, by the direct sum decomposition of
$\mathrm{D}^{\mathrm{ULA}}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}$
described above, that the natural map $j_{b!}(V_{b})\rightarrow j_{b*}(V_{b})$
is an isomorphism after applying $(-)_{\phi_{\mathfrak{m}}}$. Moreover, one
only has interesting contributions coming from the unramified elements
$B(G,\mu)_{\mathrm{un}}$. In particular, we can deduce the following
Corollary.

######  Theorem 1.15.

Suppose $(\mathbf{G},X)$ is a PEL datum of type $A$ or $C$ such that
$\mathbf{G}_{\mathbb{Q}_{p}}$ is a product of simple groups as in Table (1)
with $p$ and $\ell$ satisfying the corresponding conditions, the complex
$R\Gamma_{c}(\mathcal{S}_{K,C},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m}}\simeq
R\Gamma_{c}(\mathrm{Sh}(\mathbf{G},X)_{K,C},\overline{\mathbb{F}}_{\ell})_{\mathfrak{m}}$
breaks up as a direct sum

$\bigoplus_{b\in
B(G,\mu)_{\mathrm{un}}}(R\Gamma_{c}(\mathrm{Sht}(G,b,\mu)_{\infty,C}/\underline{K_{p}^{\mathrm{hs}}},\overline{\mathbb{F}}_{\ell}(d_{b}))_{\mathfrak{m}}\otimes^{\mathbb{L}}_{\mathcal{H}(J_{b})}R\Gamma_{c-\partial}(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell}))[2d_{b}]$

of $H_{K_{p}^{\mathrm{hs}}}\times W_{E_{\mathfrak{p}}}$-modules.

######  Remark 1.16.

As we will explain more in §6.1, in the case that the unique basic element
$b\in B(G,\mu)_{\mathrm{un}}$ is unramified, the contribution of the
corresponding summand to middle degree cohomology serves as a generic fiber
analogue of the description of the middle degree cohomology on the special
fiber of the integral model at hyperspecial level, as provided in [XZ17,
Theorem 1.1.4].

As a consequence, we deduce our main Theorem, by combining Theorem 1.7 with
the fact that
$R\Gamma_{c-\partial}(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell})\in\mathrm{D}^{\leq
d_{b}}(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})$, by Artin
vanishing.

######  Theorem 1.17.

Suppose $(\mathbf{G},X)$ is a PEL datum of type $A$ or $C$ such that
$\mathbf{G}_{\mathbb{Q}_{p}}$ is a product of simple groups as in Table (1)
with $p$ and $\ell$ satisfying the corresponding conditions then Conjecture
1.2 is true.

######  Remark 1.18.

This notably allows one to relax the assumption in [CS17, CS19, Kos21, San23]
that the prime $p$ splits in $F$, answering a question of Caraiani.

We can also easily deduce the result for some abelian type Shimura varieties,
such as Hilbert modular varieties, from the above result, which recovers work
of Caraiani-Tamiozzo [CT21] (See Corollary 5.3).

######  Corollary 1.19.

Suppose $(\mathbf{G},X)$ is an abelian-type Shimura datum which has an
associated PEL-type datum $(\mathbf{G}_{1},X_{1})$ of type $A$ or $C$ such
that $\mathbf{G}_{1,\mathbb{Q}_{p}}$ is a product of simple groups as in Table
(1) with $p$ and $\ell$ satisfying the corresponding conditions. Then
Conjecture 1.2 is true.

## Acknowledgements

We would like to thank Ana Caraiani, Jean-François Dat, David Hansen, Naoki
Imai, Teruhisa Koshikawa, and Chris Skinner, for helpful discussions
pertaining to this work. Special thanks go to Mafalda Santos for sharing with
us the results of her thesis, Peter Scholze for encouraging us to avoid
working with the good reduction locus by directly describing the fibers of the
Hodge-Tate period morphism, Matteo Tamiozzo for comments and corrections on an
earlier draft, as well as suggestions for the arguments in §5.2, and Mingjia
Zhang for very helpful discussions and filling in several gaps in the
arguments used in §3, as well as several comments and corrections. This
project was carried out while the second author was at the Max Planck
Institute for Mathematics in Bonn and she thanks them for their hospitality
and financial support.

## Notation

  * •

Fix distinct primes $\ell\neq p$.

  * •

We write $\mathbb{Q}_{p}$ for the $p$-adic numbers, and
$\breve{\mathbb{Q}}_{p}$ for the completion of the maximal unramified
extension with Frobenius $\sigma$.

  * •

We let $\overline{\mathbb{F}}_{\ell}$ denote the algebraic closure of the
finite field $\mathbb{F}_{\ell}$. We fix a choice of square root of $p$ in
$\overline{\mathbb{F}}_{\ell}$ and define all half Tate twists and square
roots of the norm character with respect to this choice.

  * •

For $L/\mathbb{Q}_{p}$ a finite extension, we write
$\breve{L}:=L\breve{\mathbb{Q}}_{p}$ for the compositum of $L$ with the
maximal unramified extension and $W_{L}$ for the Weil group of $L$. We let
$\mathrm{WD}_{L}:=W_{L}\times\mathrm{SL}(2,\overline{\mathbb{Q}}_{\ell})$
denote the Weil-Deligne group of $L$.

  * •

We let $\mathbb{A}$ (resp. $\mathbb{A}_{f}$) denote the adeles (resp. finite
adeles) over $\mathbb{Q}$.

  * •

A pair $(\mathbf{G},X)$ will denote a Shimura datum. We will use $E$ to denote
the reflex field. For $K\subset\mathbf{G}(\mathbb{A}_{f})$ a sufficiently
small open compact, we write
$\mathrm{Sh}(\mathbf{G},X)_{K}\rightarrow\mathop{\rm Spec}{E}$ for the
attached Shimura variety of level $K$.

  * •

We fix an isomorphism
$j:\overline{\mathbb{Q}}_{p}\xrightarrow{\simeq}\mathbb{C}$. Consider the
induced embedding $\overline{\mathbb{Q}}\rightarrow\overline{\mathbb{Q}}_{p}$
this gives a finite place $\mathfrak{p}$ of $E$. We write $E_{\mathfrak{p}}$
for the completion at $\mathfrak{p}$.

  * •

We let $C:=\hat{\overline{E}}_{\mathfrak{p}}$ be the completed algebraic
closure of $E_{\mathfrak{p}}$.

  * •

We use the symbol $G$ to always denote a connected reductive group over
$\mathbb{Q}_{p}$, usually taken to be $\mathbf{G}_{\mathbb{Q}_{p}}$. We will
always assume that $G$ is quasi-split with a fixed choice $T\subset B\subset
G$ of maximal torus and Borel, respectively.

  * •

If $G$ is unramified then we let $K_{p}^{\mathrm{hs}}\subset
G(\mathbb{Q}_{p})$ be a choice of hyperspecial subgroup. We set
$H_{K_{p}^{\mathrm{hs}}}:=\overline{\mathbb{F}}_{\ell}[K_{p}^{\mathrm{hs}}\backslash
G(\mathbb{Q}_{p})/K_{p}^{\mathrm{hs}}]$ to be the unramified Hecke algebra
with $\overline{\mathbb{F}}_{\ell}$-coefficients.

  * •

We let $\mathbb{X}_{*}(T_{\overline{\mathbb{Q}}_{p}})^{+}$ denote the set of
geometric dominant cocharacters of $G$ and let
$\mathbb{X}_{*}(T_{\overline{\mathbb{Q}}_{p}})^{+}/\Gamma$ denote the set of
Galois orbits, where
$\Gamma:=\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$.

  * •

Let $B(G):=G(\breve{\mathbb{Q}}_{p})/(g\sim hg\sigma(h)^{-1})$ denote the
Kottwitz set of $G$.

  * •

For $b\in B(G)$, we write $J_{b}$ for the $\sigma$-centralizer of $b$.

  * •

For $\mu\in\mathbb{X}_{*}(T_{\overline{\mathbb{Q}}_{p}})^{+}$, we let
$B(G,\mu)$ be the $\mu$-admissible locus (as defined in [RV14, Definition
2.3]).

  * •

Let $\mathrm{Perf}$ denote the category of affinoid perfectoid spaces in
characteristic $p$ over $\ast:=\mathop{\rm Spd}(\overline{\mathbb{F}}_{p})$
endowed with the $v$-topology. For a perfectoid space $S$, let
$\mathrm{Perf}_{S}$ denote the category of affinoid perfectoid spaces over the
tilt $S^{\flat}$.

  * •

For $S\in\mathrm{Perf}$, let $X_{S}$ denote the relative schematic Fargues-
Fontaine curve over $S$.

  * •

For $\mathop{\rm Spa}{(F,\mathcal{O}_{F})}\in\mathrm{Perf}$ a geometric point,
we will often drop the subscript on $X_{F}$ and just write $X$ for the
associated Fargues-Fontaine curve.

  * •

For $b\in B(G)$, we write $\mathcal{E}_{b}$ for the associated $G$-bundle on
$X$.

  * •

For $S\in\mathrm{Perf}$, we let $\mathcal{E}_{0}$ denote the trivial
$G$-bundle on $X_{S}$.

  * •

To a diamond or $v$-stack $X$ over $\ast$, we write
$\mathrm{D}(X,\overline{\mathbb{F}}_{\ell})$ for the category of étale
$\overline{\mathbb{F}}_{\ell}$-sheaves, as defined in [Sch18]. We let
$\mathrm{D}^{\mathrm{ULA}}(X,\overline{\mathbb{F}}_{\ell})$ denote the full
subcategory of ULA sheaves over $\ast$.

  * •

For an Artin $v$-stack $X$ and
$\Lambda\in\\{\overline{\mathbb{F}}_{\ell},\overline{\mathbb{Z}}_{\ell},\overline{\mathbb{Q}}_{\ell}\\}$,
we write $\mathrm{D}_{\blacksquare}(X,\Lambda)$ for the condensed
$\infty$-category of solid $\overline{\mathbb{F}}_{\ell}$-sheaves on $X$, and
write
$\mathrm{D}_{\mathrm{lis}}(X,\Lambda)\subset\mathrm{D}_{\blacksquare}(X,\Lambda)$
for the full sub-category of $\Lambda$-lisse-étale sheaves, as defined in
[FS21, Chapter VII].

  * •

If $X$ is an Artin $v$-stack ([FS21, Definition IV.V.1]) admitting a separated
cohomologically smooth surjection $U\rightarrow X$ from a locally spatial
diamond $U$ such that the etale site has a basis with bounded
$\ell$-cohomological dimension (which will always be the case for our
applications) then we will regard it as a condensed $\infty$-category via the
identification
$\mathrm{D}_{\mathrm{lis}}(X,\overline{\mathbb{F}}_{\ell})\simeq\mathrm{D}(X,\overline{\mathbb{F}}_{\ell})$
when viewed as objects in
$\mathrm{D}_{\blacksquare}(X,\overline{\mathbb{F}}_{\ell})$ [FS21, Proposition
VII.6.6].

  * •

We let $\hat{G}$ denote the Langlands dual group of $G$ with fixed splitting
$(\hat{T},\hat{B},\\{X_{\alpha}\\})$.

  * •

If $E$ denotes the splitting field of $G$ then the action of
$W_{\mathbb{Q}_{p}}$on $\hat{G}$ factors through
$Q:=W_{\mathbb{Q}_{p}}/W_{E}$. We let $\phantom{}{}^{L}G:=\hat{G}\rtimes Q$
denote the $L$-group.

  * •

For $I$ a finite index set, we let
$\mathrm{Rep}_{\overline{\mathbb{F}}_{\ell}}(\phantom{}^{L}G^{I})$ (resp.
$\mathrm{Rep}_{\overline{\mathbb{F}}_{\ell}}(\hat{G}^{I})$) denote the
category of finite-dimensional algebraic representations of
$\phantom{}{}^{L}G^{I}$ (resp. $\hat{G}^{I}$) over
$\overline{\mathbb{F}}_{\ell}$.

  * •

For $\mu\in\mathbb{X}_{*}(T_{\overline{\mathbb{Q}}_{p}})^{+}$, we write
$V_{\mu}\in\mathrm{Rep}_{\overline{\mathbb{F}}_{\ell}}(\hat{G})$ (resp.
$\mathcal{T}_{\mu}\in\mathrm{Rep}_{\overline{\mathbb{F}}_{\ell}}(\hat{G})$)
for the usual highest weight representation (resp. highest weight tilting
module, as in [Don93]) of highest weight $\mu$.

  * •

To any condensed $\infty$-category $\mathcal{C}$, we write
$\mathcal{C}^{BW^{I}_{\mathbb{Q}_{p}}}$ for the category of objects with
continuous $W_{\mathbb{Q}_{p}}^{I}$-action, as defined in [FS21, Section
IX.1].

  * •

For any separated $v$-stack, $X\rightarrow\mathop{\rm Spa}(K,\mathcal{O}_{K})$
where $\mathop{\rm Spa}(K,\mathcal{O}_{K})$ is a non-archimedean field, we
write $\overline{X}$ for the canonical compactification of $X$ with respect to
the structure map ([Sch18, Proposition 18.6], [Hub96, Theorem 5.15]).

  * •

For a reductive group $H/\mathbb{Q}_{p}$, we write
$\mathrm{D}(H(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})$ for the unbounded
derived category of smooth $\overline{\mathbb{F}}_{\ell}$-representations.

  * •

For an analytic adic space $X$, we will often abuse notation and use $X$ to
also denote the diamond $X^{\diamond}$ attached to it (as defined in [SW20,
Lecture X]).

##  2\. Preliminaries on Shimura Varieties

In this section we will recall some facts about Shimura varieties which we
will need later in this paper.

###  2.1. Shimura Varieties

We will mainly work with the following two types of Shimura varieties.

####  2.1.1. PEL type A and C

Let $(\mathcal{O}_{B},\ast,L,\langle\cdot,\cdot\rangle)$ be an integral PEL
datum, where $B$ is a finite-dimensional semisimple $\mathbb{Q}$-algebra,
$\ast$ is a $\mathbb{Q}$-linear involution of $B$, with fixed field $F$,
$\mathcal{O}_{B}$ is a $\ast$-stable $\mathbb{Z}$-order of $B$, $L$ is a
lattice with $\mathcal{O}_{B}$-actions, and $\langle\cdot,\cdot\rangle:L\times
L\rightarrow\mathbb{Z}(1)$ is a non-degenerate alternating form such that
$\langle bv,v^{\prime}\rangle=\langle v,b^{*}v^{\prime}\rangle$, for all
$b\in\mathcal{O}_{B}$ and $v,v^{\prime}\in L$.

To our integral PEL datum, we have the following group scheme $\mathbf{G}$
over $\mathbb{Z}$ whose $R$-points, for each $\mathbb{Z}$-algebra $R$, are
given by

$\mathbf{G}(R):=\\{(g,r)\in\mathrm{End}_{\mathcal{O}_{B}\otimes_{\mathbb{Z}}R}(L\otimes_{\mathbb{Z}}R)\times
R|\langle gv,gw\rangle=r\langle v,w\rangle\text{ for all }v,w\in
L\otimes_{\mathbb{Z}}R\\}.$

The PEL data is of type A if $(B\otimes_{F}\overline{F},\ast)$ is isomorphic
to $\mathrm{End}(W)\times\mathrm{End}(W)^{\mathrm{op}}$ with
$(a,b)^{\ast}=(b,a)$ for some vector space $W$. The PEL data is of type C if
$(B\otimes_{F}\overline{F},\ast)$ is isomorphic to $\mathrm{End}(W)$ with
$\ast$ being the adjoint map with respect to a symmetric bilinear form on $W$.
We will assume from now on that we are in one of these cases.

We now further assume the data is unramified at $p$; namely, that each term in
the decomposition
$B_{\mathbb{Q}_{p}}=\prod_{\mathfrak{p}|p}B\otimes_{F}F_{\mathfrak{p}}$ is a
matrix algebra over an unramified extension of $\mathbb{Q}_{p}$. We will thus
moreover assume that $\mathcal{O}_{B}\otimes\mathbb{Z}_{p}$ is a maximal order
in $B_{\mathbb{Q}_{p}}$, and $L$ is self-dual after localization at $p$. This
can be arranged following [Lan13, Remark 1.3.4.8]. Note that these conditions
equivalently ensure that the group $G$ is unramified.

We will now briefly discuss what conditions we may need to impose on the prime
$p$ so that the form of the local group $G$ satisfies the conditions in Table
(1). Firstly, suppose we are in type A. Then the center $Z(B)=F_{c}$ is a
quadratic imaginary extension of $F$. Let $n$ be the $\mathcal{O}_{B}$ rank of
$L$. Observe that we will need to assume that the prime $p$ satisfies for all
primes $\mathfrak{p}$ of $F$ above $p$,

  1. (1)

$\mathfrak{p}$ is split in $F_{c}$; or

  2. (2)

$F_{\mathfrak{p}}=\mathbb{Q}_{p}$, and $n$ is odd.

These conditions imply that $G$ will be a similitude subgroup of
$\prod_{\mathfrak{p}}G_{\mathfrak{p}}$ where $G_{\mathfrak{p}}$ is either
$\mathrm{Res}_{F_{\mathfrak{p}}/\mathbb{Q}_{p}}(\mathbb{G}_{m}\times\mathrm{GL}_{n})$
or $\mathrm{GU}_{n}$ for an odd unitary group over $\mathbb{Q}_{p^{2}}$.

Now suppose we are in type C. Since the PEL data is unramified at $p$, we see
that $B\otimes_{F}F_{\mathfrak{p}}$ is indefinite for all primes
$\mathfrak{p}$ of $F$ above $p$, and thus $G$ will be a similitude subgroup of

$\prod_{\mathfrak{p}}\mathrm{Res}_{F_{\mathfrak{p}}/\mathbb{Q}_{p}}(\mathrm{GSp}_{2n}).$

Here, we will need to assume that the rank $n$ of $L$ as an $\mathcal{O}_{B}$
lattice is either $1$ or $2$ to satisfy the conditions in Table (1).

Both types of Shimura varieties will be moduli spaces of abelian varieties
with extra structures, which we will briefly describe. Let $K^{p}\subset
G(\mathbb{A}_{f}^{p})$ be an open compact subgroup. To any PEL data, the
Shimura variety $\mathfrak{S}(\mathbf{G},X)_{K}$ over
$\mathcal{O}_{E_{\mathfrak{p}}}$ is the scheme which represents the functor
that associates to each locally Noetherian scheme $S$ over
$\mathcal{O}_{E_{\mathfrak{p}}}$ the set of isomorphism classes of tuples
$(A,\lambda,\iota,\eta^{p})$ consisting of

  1. (1)

An abelian scheme $A/S$ of dimension $n[F:\mathbb{Q}]$ up to prime to
$p$-isogeny,

  2. (2)

A prime-to-$p$ polarization $\lambda:A\rightarrow A^{\vee}$,

  3. (3)

An embedding
$\iota:\mathcal{O}_{B}\otimes\mathbb{Z}_{(p)}\hookrightarrow\mathrm{End}(A)\otimes_{\mathbb{Z}}\mathbb{Z}_{(p)}$
of $\mathbb{Z}_{(p)}$-algebras such that

$\lambda\circ\iota(b^{*})=\iota(b)^{\vee}\circ\lambda,$

  4. (4)

A section
$\eta^{p}\in\Gamma(S,\mathrm{Isom}_{B}(L\otimes_{\mathbb{Z}}\mathbb{A}_{f}^{p},\hat{V}(A)^{p})/K^{p})$,
where
$\mathrm{Isom}_{B}(L\otimes_{\mathbb{Z}}\mathbb{A}_{f}^{p},\hat{V}(A)^{p})$ is
the étale torsor of isomorphisms that maps $\langle,\rangle$ to a
$\mathbb{A}_{f}^{p\times}$ multiple of the pairing on $\hat{V}(A)^{p}$ defined
by the Weil pairing,

satisfying the Kottwitz determinant condition that
$\mathrm{det}(b|\mathrm{Lie}(A))=\mathrm{det}(b|V^{-1,0})$ as polynomial
functions on $\mathcal{O}_{B}$, where $V=L\otimes\mathbb{Q}$ and
$V_{\mathbb{C}}=V^{-1,0}\oplus V^{0,-1}$ is the Hodge decomposition.

####  2.1.2. Compactifications

We will now recall some constructions from the theory of toroidal
compactifications of PEL type Shimura varieties from [Lan13]. To match the
setting in [Lan13], we will moreover assume from now on that the level
structure $K$ is a principal congruence subgroup for some $N\geq 3$, namely

$K=K(N)=\\{g\in\mathbf{G}(\hat{\mathbb{Z}})|g\equiv 1\pmod{N}\\}.$

We first recall the definition of a split, symplectic and admissible
filtration from [Lan13, §5.2.1]. Let $R$ be a commutative ring.

######  Definition 2.1.

A split, symplectic and admissible filtration on $L\otimes_{\mathbb{Z}}R$ is a
two-step filtration on $L\otimes_{\mathbb{Z}}R$ by
$(\mathcal{O}_{B}\otimes_{\mathbb{Z}}R)$-submodules, i.e. we have

$0=Z_{-3}\subset Z_{-2}\subset Z_{-1}\subset L\otimes_{\mathbb{Z}}R,$

such that if we put $\mathrm{Gr}^{Z}_{-i}=Z_{-i}/Z_{-i-1}$ for $0\leq i\leq
2$, and $\mathrm{Gr}^{Z}=\oplus_{0\leq i\leq 2}\mathrm{Gr}^{Z}_{-i}$, we have

  1. (1)

$\mathrm{Gr}^{Z}_{-i}$ is isomorphic to $M\otimes_{\mathbb{Z}}R$ for some
$\mathcal{O}_{B}$-lattice $M$

  2. (2)

There is some isomorphism of $\mathcal{O}_{B}$-lattices

$L\otimes_{\mathbb{Z}}R\simeq\mathrm{Gr}^{Z}$

  3. (3)

$Z_{-2}$ and $Z_{-1}$ are annihilators of each other under the pairing
$\langle,\rangle$ induced from $L$.

######  Definition 2.2.

Let $M$ be a finite $B$-module. Since $B\simeq\prod_{i}B_{i}$ where each
$B_{i}$ is simple, we have a decomposition $M\simeq M_{i}^{m_{i}}$ where
$M_{i}$ is the unique simple left $B_{i}$-module. We call the tuple $(m_{i})$
the $B$-multi-rank of $M$.

Let $R=\hat{\mathbb{Z}}$ and suppose that we have a split symplectic
admissible filtration $Z=Z_{\bullet}$ as above.

######  Definition 2.3.

A torus argument $\Phi$ for $Z$ is a tuple
$\Phi=(X,Y,\phi,\varphi_{-2},\varphi_{0})$, where

  1. (1)

$X$ and $Y$ are $\mathcal{O}_{B}$-lattices of the same $B$-multi-rank, and
$\phi:Y\hookrightarrow X$ is an $\mathcal{O}_{B}$-linear embedding

  2. (2)

We have isomorphisms
$\varphi_{-2}:\mathrm{Gr}^{Z}_{-2}\simeq\mathrm{Hom}_{R}(X\otimes_{\mathbb{Z}}R,R(1))$
and $\varphi_{0}:\mathrm{Gr}^{Z}_{0}\simeq Y\otimes_{\mathbb{Z}}R$ such that
the pairing
$\langle,\rangle_{20}:\mathrm{Gr}^{Z}_{-2}\times\mathrm{Gr}^{Z}_{0}\rightarrow
R(1)$ is the pullback under these isomorphisms of the pairing

$\langle\cdot,\cdot\rangle^{\phi}:\mathrm{Hom}_{R}(X\otimes
R,R(1))\times(Y\otimes
R)\xrightarrow{\mathrm{id}\times\phi}\mathrm{Hom}_{R}(X\otimes
R,R(1))\times(X\otimes R)\rightarrow R(1),$

where the last arrow is the tautological pairing.

We thus define a cusp label as a pair $(Z,\Phi)$, where $Z$ is a split
symplectic admissible filtration on $L\otimes_{\mathbb{Z}}\hat{\mathbb{Z}}$,
and $\Phi$ is a torus argument for $Z$. Note that this is the generalization
of the cusp labels $(Z,X)$ considered in [CS19, §2.5.2], as for the PEL type A
Shimura data they considered, the assumption of principal polarization means
we can set $X=Y$, and the torus argument $\Phi$ is determined by the
$\mathcal{O}_{F}$-isomorphism.

There is an action of $\mathbf{G}(\mathbb{A}_{f})$ on pairs $(Z,\Phi)$, as
defined in [Lan13, §5.4.3], and we define a cusp label at level $K$ to be a
$K$-orbit of pairs $(Z,\Phi)$.

To each cusp label $(Z,\Phi)$, we can associate a split torus $E_{\Phi}$ over
$\mathbb{Z}$, as constructed by Lan in [Lan13, §6.4]. Let
$S_{\Phi}=\mathbb{X}^{*}(E_{\Phi})$. Let
$S_{\Phi}^{\vee}:=\mathrm{Hom}_{\mathbb{Z}}(S_{\Phi},\mathbb{Z})$ be the
$\mathbb{Z}$-dual of $S_{\Phi}$, and let
$(S_{\Phi})^{\vee}_{\mathbb{R}}:=S^{\vee}_{\Phi}\otimes_{\mathbb{Z}}\mathbb{R}$.
The $\mathbb{R}$-vector space $(S_{\Phi})^{\vee}_{\mathbb{R}}$ is isomorphic
to the space of Hermitian pairings
$|\cdot,\cdot|:(Y\otimes\mathbb{R})\times(Y\otimes\mathbb{R})\rightarrow\mathcal{O}_{B}\otimes\mathbb{R}$
by sending a Hermitian pairing $|\cdot,\cdot|$ to the function
$y\otimes\phi(y^{\prime})\mapsto\mathrm{Tr}_{B/\mathbb{Q}}(|y,y^{\prime}|)$ in
$\mathrm{Hom}_{\mathbb{Z}}(S_{\Phi},\mathbb{R})$ (c.f. [Lan13, §6.2.5]).

Thus, we have an $\mathbb{R}$-vector space $(S_{\Phi})_{\mathbb{R}}^{\vee}$ of
Hermitian pairings, and we define $P_{\Phi}$ to be the subset of
$(S_{\Phi})_{\mathbb{R}}^{\vee}$ corresponding to positive semi-definite
Hermitian pairings with admissible radicals (see [Lan13, Definition 6.2.5.4]
and subsequent discussion for the precise definition of admissible radical).
$P_{\Phi}$ will a rational polyhedral cone in
$(S_{\Phi})_{\mathbb{R}}^{\vee}$. Moreover, to every torus argument $\Phi$ we
can also associate a stabilizer group $\Gamma_{\Phi}$. We thus let
$\Sigma_{\Phi}$ be a $\Gamma_{\Phi}$-admissible rational polyhedral cone
decomposition of $P_{\Phi}$, as in [Lan13, Definition 6.1.1.14].

From now on, we will assume that we have fixed a compatible choice of
admissible smooth rational polyhedral cone decomposition data (rpcd) $\Sigma$
for $K$; namely, we have

  1. (1)

A complete set of representatives $(Z,\Phi)$ of cusp labels at level $K$,

  2. (2)

A $\Gamma_{\Phi}$ -admissible smooth rational polyhedral cone decomposition
$\Sigma_{\Phi}$ for each cusp $(Z,\Phi)$ so that the cone decompositions are
pairwise compatible.

The precise definition and proof of existence of such smooth admissible rpcd
is [Lan13, §6.3.3.2,§6.6.3.3]. Associated to this admissible smooth rpcd, we
have a toroidal compactification of $\mathfrak{S}(G,X)_{K}$, as in the
following theorem of Lan [Lan13, Theorem 6.4.1.1].

######  Theorem 2.4.

To each compatible choice $\Sigma=\\{\Sigma_{\Phi}\\}$ of admissible smooth
rational polyhedral cone decomposition data, there is an associated proper
smooth algebraic scheme $\mathfrak{S}(G,X)_{K}^{\mathrm{tor}}$ over
$\mathcal{O}_{E,\mathfrak{p}}$ containing $\mathfrak{S}(G,X)_{K}$ as an open
dense subscheme, together with a semiabelian family $\mathcal{A}$ over
$\mathfrak{S}(G,X)_{K}^{\mathrm{tor}}$. Moreover, we have a stratification

$\mathfrak{S}(G,X)_{K}^{\mathrm{tor}}=\coprod_{(\Phi,\sigma)}Z_{(\Phi,\sigma)},$

where $\sigma$ is a face of $P_{\Phi}$.

Finally, observe that there is a cusp label corresponding to taking the
filtration $Z_{-2}=0,Z_{-1}=L\otimes R$, and the torus argument $X=Y=0$. This
trivial cusp label will correspond to the original Shimura variety
$\mathfrak{S}(G,X)_{K}$ in the stratification above.

###  2.2. Igusa Varieties

We fix now some $b\in B(G,\mu)$, and consider a geometric point
$x\in\mathfrak{S}(\mathbf{G},X)_{K}(\overline{\mathbb{F}}_{p})$ lying in the
Newton strata for $b$. This corresponds to some abelian variety
$\mathcal{A}_{x}$, which has $p$-divisible group with $G$-structure
$\mathbb{X}:=\mathcal{A}_{x}[p^{\infty}]$ given by $b$. Up to replacing $x$ by
another element in its isogeny class, we can assume $\mathbb{X}$ is completely
slope divisible. Thus, we can write
$\mathbb{X}=\oplus_{i=1}^{r}\mathbb{X}_{i}$, where the $\mathbb{X}_{i}$ are
isoclinic $p$-divisible groups of strictly decreasing slopes.

We consider the following subset

$\mathscr{C}_{\mathbb{X}}:=\\{x\in\mathfrak{S}(\mathbf{G},X)_{K,\overline{\mathbb{F}}_{p}}:\exists\text{
isomorphism }\rho:\mathcal{A}_{x}[p^{\infty}]\times
k(\overline{x})\simeq\mathbb{X}\times k(\overline{x})\text{ preserving
$G$-structure}\\},$

where we denote by $\mathfrak{S}(\mathbf{G},X)_{K,\overline{\mathbb{F}}_{p}}$
the (geometric) special fiber of $\mathfrak{S}(\mathbf{G},X)_{K}$. This turns
out to be a closed subset of
$\mathfrak{S}(\mathbf{G},X)_{K,\overline{\mathbb{F}}_{p}}$, and thus we can
give this subset the induced reduced scheme structure. We will continue to
denote the associated scheme by $\mathscr{C}_{\mathbb{X}}$, and it turns out
this scheme is smooth.

Let $\mathscr{G}$ be the $p$-divisible group of the restriction to
$\mathscr{C}_{\mathbb{X}}$ of the universal abelian variety over
$\mathfrak{S}(\mathbf{G},X)_{K}$. We further define $\mathrm{Ig}^{b}$ as the
scheme over $\mathscr{C}_{\mathbb{X}}$ parametrizing, for any perfect
$\mathscr{C}_{\mathbb{X}}$-scheme $\mathscr{T}$, isomorphisms
$\mathscr{G}\times_{\mathscr{C}_{\mathbb{X}}}\mathscr{T}\simeq\mathbb{X}\times_{\overline{\mathbb{F}}_{p}}\mathscr{T}$
which preserve $G$-structure. Equivalently, we can define $\mathrm{Ig}^{b}$ as
the functor sending an $\overline{\mathbb{F}}_{p}$-algebra $R$ to the set

(2)
${}\mathrm{Ig}^{b}(R)=\\{(\rho,x):x\in\mathfrak{S}(\mathbf{G},X)_{K}(R),\rho:\mathcal{A}_{x}[p^{\infty}]\xrightarrow{\sim}\mathbb{X}_{R}\text{
preserving $G$-structure}\\}.$

By [CS17, Corollary 4.3.5], we know that this scheme is perfect, and hence it
lifts uniquely to a flat $p$-adic formal scheme, which we denote by
$\mathfrak{Ig}^{b}$ over $\mathrm{Spf}(W(\overline{\mathbb{F}}_{p}))$.

We write $\mathfrak{Ig}_{C}^{b}$ for the perfectoid space attached to the adic
generic fiber of $\mathfrak{Ig}^{b}$ over $C$. These spaces are supposed to
model the fibers of the Hodge-Tate period morphism, a connection we will
elaborate upon in §3.

####  2.2.1. Compactifications

In order to understand (partial) minimal and toroidal compactifications of
$\mathrm{Ig}^{b}$, we must first consider compactifications of the central
leaf $\mathscr{C}_{\mathbb{X}}$. As in the discussion in [CS19, §3.1], the
central leaf $\mathscr{C}_{\mathbb{X}}$ is a well-positioned subset of
$\mathfrak{S}(\mathbf{G},X)_{K,\overline{\mathbb{F}}_{p}}$, and thus admits
partial toroidal and minimal compactifications, which we will denote by
$\mathscr{C}_{\mathbb{X}}^{\mathrm{tor}}$ and $\mathscr{C}_{\mathbb{X}}^{*}$
respectively. Moreover, let $Z$ be a cusp label at level $K(N)$. This
determines a locally closed boundary stratum
$\mathscr{C}_{\mathbb{X},Z}\subset\mathscr{C}_{\mathbb{X}}^{\mathrm{tor}}$.

The Igusa variety $\mathrm{Ig}^{b}$ over $\mathscr{C}_{\mathbb{X}}$ extends to
a perfect scheme $\mathrm{Ig}^{b,\mathrm{tor}}$ over
$\mathscr{C}_{\mathbb{X}}^{\mathrm{tor}}$. More precisely, we can define
$\mathrm{Ig}^{b,\mathrm{tor}}$ as follows. Let $\mathcal{A}$ denote the
universal semi-abelian scheme over $\mathscr{C}_{\mathbb{X}}^{\mathrm{tor}}$.
This is the restriction to $\mathscr{C}_{\mathbb{X}}^{\mathrm{tor}}$ of the
universal semi-abelian scheme over
$\mathfrak{S}(\mathbf{G},X)_{K,\overline{\mathbb{F}}_{p}}$. Then, we know from
[CS19, Proposition 3.2.1] that the connected part
$\mathcal{A}[p^{\infty}]^{\circ}$ of $\mathcal{A}[p^{\infty}]$ is a
$p$-divisible group. Moreover, if we denote by $\mathcal{A}[p^{\infty}]^{\mu}$
the multiplicative part, then this is also a $p$-divisible group. We thus let
$\mathcal{A}[p^{\infty}]^{(0,1)}=\mathcal{A}[p^{\infty}]^{\circ}/\mathcal{A}[p^{\infty}]^{\mu}$
be the biconnected part. We can similarly define
$\mathbb{X}^{\circ},\mathbb{X}^{(0,1)}$ as the connnted and biconnected parts
of $\mathbb{X}$.

Thus, we can define $\mathrm{Ig}^{b,\mathrm{tor}}$ to be the scheme which, for
a perfect $\mathscr{C}_{\mathbb{X}}^{\mathrm{tor}}$-scheme $\mathscr{T}$,
parametrizes $\mathcal{O}_{B}$ -linear isomorphisms
$\rho:\mathcal{A}[p^{\infty}]^{\circ}\times_{\mathscr{C}_{\mathbb{X}}^{\mathrm{tor}}}\mathscr{T}\xrightarrow{\sim}\mathbb{X}^{\circ}_{\mathscr{T}}$
and a scalar in $\mathbb{Z}^{\times}_{p}(\mathscr{T})$ such that the induced
isomorphism
$\rho^{(0,1)}:\mathcal{A}[p^{\infty}]^{(0,1)}\times_{\mathscr{C}_{\mathbb{X}}^{\mathrm{tor}}}\mathscr{T}\xrightarrow{\sim}\mathbb{X}_{\mathscr{T}}^{(0,1)}$
obtained by quotienting by the multiplicative parts commutes with the
polarizations up to the given element of
$\mathbb{Z}_{p}^{\times}(\mathscr{T})$. Here,
$\mathbb{Z}_{p}^{\times}(\mathscr{T})$ is the set of $\mathscr{T}$-points of
the group scheme
$\mathbb{Z}_{p}^{\times}\times\mathscr{C}_{\mathbb{X}}^{\mathrm{tor}}$.

Finally, we define the partial minimal compactification $\mathrm{Ig}^{b,*}$ as
the normalization of $\mathscr{C}_{\mathbb{X}}^{*}$ in $\mathrm{Ig}^{b}$.
Since we have
$\mathrm{Ig}^{b}\subset\mathrm{Ig}^{b,*},\mathrm{Ig}^{b,\mathrm{tor}}$, we
will denote the boundaries by $\partial\mathrm{Ig}^{b,*}$ and
$\partial\mathrm{Ig}^{b,\mathrm{tor}}$, respectively. These schemes are all
perfect, and we can lift them to $p$-adic formal schemes over
$\mathrm{Spf}(W(\overline{\mathbb{F}}_{p}))$. We similarly denote by
$\mathfrak{Ig}_{C}^{b,*}$, $\mathfrak{Ig}_{C}^{b,\mathrm{tor}}$,
$\partial\mathfrak{Ig}_{C}^{b,*}$, and
$\partial\mathfrak{Ig}_{C}^{b,\mathrm{tor}}$ the associated perfectoid spaces
over $C$.

####  2.2.2. Igusa Cusp Labels

In order to understand the boundary components $\partial\mathrm{Ig}^{b,*}$ and
$\partial\mathrm{Ig}^{b,\mathrm{tor}}$, we will recall the notion of Igusa
cusp labels, as in [San23, Definition 3.2.19] (which we have slightly modified
to match the definition of cusp labels previously introduced). We let
$\mathbb{X}_{b}:=\mathbb{X}$ be the completely slope divisible $p$-divisible
group attached to $b$ defined above. We reintroduce $b$ in the notation to
emphasise that all constructions here depend on $b$. Finally, observe that,
since from the moduli problem the polarization on $\mathcal{A}_{x}$ is prime-
to-$p$, the $p$-divisible group $\mathbb{X}_{b}=\mathcal{A}_{x}[p^{\infty}]$
is principally polarized.

######  Definition 2.5.

We define an Igusa cusp label as a tuple
$(Z_{b},Z^{p},X,Y,\phi,\varphi_{0},\varphi_{-2},\tilde{\varphi}_{0},\delta_{b})$
where

  1. (1)

$Z_{b}$ is an $\mathcal{O}_{B}$ -stable filtration of $\mathbb{X}_{b}$ by
$p$-divisible subgroups of the form

$0=Z_{b,-3}\subset Z_{b,-2}\subset Z_{b,-1}\subset\mathbb{X}_{b},$

where $\mathrm{Gr}_{-2}^{Z_{b}}=Z_{b,-2}$ is multiplicative, and
$\mathrm{Gr}_{0}^{Z_{b}}=\mathbb{X}_{b}/Z_{b,-1}$ is étale, and $Z_{b,-1}$,
$Z_{b,-2}$ are Cartier dual to each other under the principal polarization on
$\mathbb{X}_{b}$.

  2. (2)

$\delta_{b}$ is an $\mathcal{O}_{B}$-linear isomorphism

$\delta_{b}:\mathrm{Gr}^{Z_{b}}\simeq\mathbb{X}_{b}$

  3. (3)

$Z^{p}$ is an $\mathcal{O}_{B}$ -stable split, symplectic and admissible
filtration

$0=Z^{p}_{-3}\subset Z^{p}_{-2}\subset Z^{p}_{-1}\subset
L\otimes_{\mathbb{Z}}\hat{\mathbb{Z}}^{p}$

  4. (4)

$X,Y$ are $\mathcal{O}_{B}$-lattices of the same $B$-multirank, together with
an $\mathcal{O}_{B}$-linear embedding $\phi:Y\hookrightarrow X$, and we have
isomorphisms

$\varphi_{0}:Y\otimes_{\mathbb{Z}}\hat{\mathbb{Z}}^{p}\simeq\mathrm{Gr}_{0}^{Z^{p}}$
$\varphi_{-2}:\mathop{\rm
Hom}\nolimits(X\otimes_{\mathbb{Z}}\hat{\mathbb{Z}}^{p},\hat{\mathbb{Z}}^{p}(1))\simeq\mathrm{Gr}_{-2}^{Z^{p}}$
$\tilde{\varphi}_{0}:Y\otimes(\mathbb{Q}_{p}/\mathbb{Z}_{p})\simeq\mathrm{Gr}_{0}^{Z_{b}}$

such that the pairing
$\langle,\rangle_{20}:\mathrm{Gr}^{Z^{p}}_{-2}\times\mathrm{Gr}^{Z^{p}}_{0}\rightarrow\hat{\mathbb{Z}}^{p}(1)$
induced from the one on $L$ is the pullback via $\varphi_{-2},\varphi_{0}$ of
the one defined on $X,Y$.

There is an action of
$J_{b}(\mathbb{Q}_{p})\times\mathbf{G}(\mathbb{A}^{p}_{f})$ on Igusa cusp
labels. If $K\subset
J_{b}(\mathbb{Q}_{p})\times\mathbf{G}(\mathbb{A}^{p}_{f})$ is a compact open
subgroup then an Igusa cusp label at level $K$ is a $K$-orbit of Igusa cusp
labels. For a general closed subgroup $H\subset
J_{b}(\mathbb{Q}_{p})\times\mathbf{G}(\mathbb{A}^{p}_{f})$, an Igusa cusp
label at level $H$ is a compatible family of Igusa cusp labels at level $K$
for all $K\supset H$.

####  2.2.3. Boundary components

We can decompose the boundary $\partial\mathrm{Ig}^{b,\mathrm{tor}}$ according
to Igusa cusp labels of prime-to-$p$ level $K^{p}(N)$, in the following way.
For every positive integer $m$, there is a level $p^{m}$-Igusa variety
$\mathrm{Ig}^{b}_{m}$, defined as in [CS17, Definition 4.3.6], and we let
$\Gamma_{m,b}$-denote the Galois group of the finite étale cover
$\mathrm{Ig}^{b}_{m}\rightarrow\mathscr{C}_{\mathbb{X}}$. We also have a
toroidal extension of $\mathrm{Ig}^{b}_{m}$ to a level $p^{m}$-Igusa variety
$\mathrm{Ig}^{b,\mathrm{tor}}_{m}$, as defined in [CS19, Definition 3.2.5]. We
let
$\Gamma_{b}(p^{m}):=\mathrm{ker}(\mathrm{Aut}(\mathbb{X})\rightarrow\Gamma_{m,b})$,
and note that if we let $K=\Gamma_{b}(p^{m})K^{p}(N)$ then such Igusa cusp
labels at level $K$ have the same data as triples $(Z_{m,b},Z,\Phi)$, where
$Z=(Z,\Phi)$ is a cusp label as level $K(N)$, and $Z_{m,b}$ is an
$\mathcal{O}_{B}$-filtration on $\mathbb{X}_{b}[p^{m}]$ together with an
isomorphism $X/p^{m}\simeq\mathrm{Gr}_{0}^{Z_{m,b}}$. In particular, we can
consider the locally closed boundary stratum $\mathscr{C}_{\mathbb{X},Z}$, and
the $p^{m}$-Igusa variety $\mathrm{Ig}^{b,\mathrm{tor}}_{m,Z}$ which is the
preimage of $\mathscr{C}_{\mathbb{X},Z}$. Moreover, from [San23, Theorem
3.2.22], we see that we have a decomposition

$\mathrm{Ig}^{b,\mathrm{tor}}_{m,Z}=\coprod_{\tilde{Z}_{m}}\mathrm{Ig}^{b,\mathrm{tor}}_{m,\tilde{Z}_{m}},$

where $\tilde{Z}_{m}$ denotes Igusa cusp labels of level
$\Gamma_{b}(p^{m})K^{p}(N)$ lying over $Z$.

Now, let $\tilde{Z}$ denote an Igusa cusp label of level $K^{p}(N)$. This is
by definition a compatible system $\\{\tilde{Z}_{m}\\}$ of Igusa cusp labels
for $\Gamma_{b}(p^{m})K^{p}(N)$, for all positive integers $m$. We can thus
define

$\mathrm{Ig}^{b,\mathrm{tor}}_{\tilde{Z}}=\mathop{\rm
lim}\nolimits\limits_{\begin{subarray}{c}\leftarrow\\\
m\end{subarray}}\mathrm{Ig}^{b,\mathrm{tor}}_{m,\tilde{Z}_{m}}.$

For later use, we will want to have a moduli description of points in
$\mathrm{Ig}^{b,\mathrm{tor}}_{\tilde{Z}}$. We first recall some facts about
degenerations of abelian schemes, from [Lan13, §3.4] and [CS19, §2.5.1]. Let
$C^{\prime}$ be a complete algebraically closed nonarchimedean field with ring
of integers $\mathcal{O}_{C^{\prime}}$. Consider a polarized abelian variety
$(A,\lambda)$ over $C^{\prime}$ with $\mathcal{O}_{B}$-structure, and a
degeneration $\mathcal{A}$ of $A$ over $\mathcal{O}_{C^{\prime}}$. Then this
uniquely determines a short exact sequence

$0\rightarrow T\rightarrow\mathcal{G}\rightarrow\mathcal{B}\rightarrow 0$

where $T$ is a torus, $\mathcal{B}$ is an abelian scheme over
$\mathcal{O}_{C^{\prime}}$, and $\mathcal{G}$ is the Raynaud extension. Let
$X=\mathbb{X}_{*}(T)$, which is a free abelian group over $\mathcal{O}_{C}$.
The lattice $X$ has an action of $\mathcal{O}_{B}$, and so does $\mathcal{B}$.
Then, $\mathcal{G}$ determines, and is uniquely determined by, an
$\mathcal{O}_{B}$-linear map $c:X\rightarrow\mathcal{B}^{\vee}$. Similarly, we
can consider a degeneration $\mathcal{A}^{\vee}$ over
$\mathcal{O}_{C^{\prime}}$ of the dual $A^{\vee}/C^{\prime}$, which gives us a
short exact sequence

$0\rightarrow
T^{\vee}\rightarrow\mathcal{G}^{\vee}\rightarrow\mathcal{B}^{\vee}\rightarrow
0,$

and if we similarly let $Y=\mathbb{X}_{*}(T^{\vee})=\mathbb{X}^{*}(T)$, the
extension $\mathcal{G}$ determines, and is uniquely determined by an
$\mathcal{O}_{B}$-linear map $c^{\vee}:Y\rightarrow\mathcal{B}$.

Moreover, the polarization $\mathcal{G}\rightarrow\mathcal{G}^{\vee}$
determines and is uniquely determined by the data of the polarization on
$\mathcal{B}$, and an injective $\mathcal{O}_{B}$-linear map
$\phi:Y\rightarrow X$.

Given $\tilde{Z}$, an Igusa cusp label of level $K^{p}(N)$, we want to
understand the $(C,\mathcal{O}_{C})$-valued points of
$\mathfrak{Ig}^{b,\mathrm{tor}}$ which specialize to points in
$\mathrm{Ig}^{b,\mathrm{tor}}_{\tilde{Z}}$. In particular, note that the data
of the Igusa cusp label means we have fixed an $\mathcal{O}_{B}$-stable
filtration

$0=Z_{b,-3}\subset Z_{b,-2}\subset Z_{b,-1}\subset\mathbb{X}_{b},$

as well as a $\mathcal{O}_{B}$ -linear splitting $\delta_{b}$ of this
filtration. From the proof of [San23, Theorem 4.3.10], we see that
$(C^{\prime},\mathcal{O}_{C^{\prime}})$-valued points are hence given by the
following data:

  1. (1)

A polarized abelian variety $\mathcal{B}$ over $\mathcal{O}_{C^{\prime}}$ with
$\mathcal{O}_{B}$-action,

  2. (2)

An $\mathcal{O}_{B}$ -linear extension

$0\rightarrow T\rightarrow\mathcal{G}\rightarrow\mathcal{B}\rightarrow 0$

where $\mathbb{X}_{*}(T)=X$; equivalently, an $\mathcal{O}_{B}$ -linear map
$c:X\rightarrow\mathcal{B}^{\vee}$.

  3. (3)

An $\mathcal{O}_{B}$-linear isomorphism

$\rho:\mathcal{G}[p^{\infty}]\simeq Z_{b,-1}$

that is compatible with the identification $T[p^{\infty}]=Z_{b,-2}$. By the
splitting $\delta_{b}$, this induces a splitting of

$0\rightarrow
T[p^{\infty}]\rightarrow\mathcal{G}[p^{\infty}]\rightarrow\mathcal{B}[p^{\infty}]\rightarrow
0,$

and in particular $c$ extends to a map
$c:X[1/p]\rightarrow\mathcal{B}^{\vee}$.

  4. (4)

An $\mathcal{O}_{B}$ -linear extension

$0\rightarrow
T^{\vee}\rightarrow\mathcal{G}^{\vee}\rightarrow\mathcal{B}^{\vee}\rightarrow
0$

where $\mathbb{X}^{*}(T)=Y$. Equivalently, an $\mathcal{O}_{B}$ -linear map
$c^{\vee}:Y\rightarrow\mathcal{B}$. By the splitting $\delta_{b}$, as well as
by duality (using that $\mathcal{G}[p^{\infty}]$ and $\mathcal{B}[p^{\infty}]$
will be principally polarized), we have a splitting of

$0\rightarrow
T^{\vee}[p^{\infty}]\rightarrow\mathcal{G}^{\vee}[p^{\infty}]\rightarrow\mathcal{B}^{\vee}[p^{\infty}]\rightarrow
0$

and in particular we extend $c^{\vee}$ to a map
$c^{\vee}:Y[1/p]\rightarrow\mathcal{B}$.

  5. (5)

An $\mathcal{O}_{C^{\prime}}$ -point of $\mathcal{P}^{\prime}_{\Sigma_{\Phi}}$
whose special fibre lies in the boundary. Here, we note that away from the
boundary we have a torsor $\mathcal{P}^{\prime}$ over
$\mathcal{O}_{C^{\prime}}$ for the torus $E_{\Phi}$ with character group
$S_{\Phi}$, parametrizing lifts of $c^{\vee}$ to
$\iota:Y[1/p]\rightarrow\mathcal{G}$.
$\mathcal{P}^{\prime}_{\Sigma_{\Phi}}\supset\mathcal{P}^{\prime}$ is the torus
embedding defined by the admissible rational polyhedral cone decomposition
$\Sigma_{\Phi}$ for the cusp $(Z,\Phi)$.

##  3\. Mantovan’s Formula and the Hodge-Tate Period Morphism

For $K\subset\mathbf{G}(\mathbb{A}_{f})$ a sufficiently small open compact, we
define
$\mathcal{S}_{K}:=(\mathrm{Sh}(\mathbf{G},X)_{K}\otimes_{E}E_{\mathfrak{p}})^{\mathrm{ad}}$
to be the adic space over $\mathop{\rm Spa}(E_{\mathfrak{p}})$ attached to the
Shimura variety. When $K=K_{p}^{\mathrm{hs}}K^{p}$ with $K_{p}^{\mathrm{hs}}$
a hyperspecial subgroup, the space $\mathcal{S}_{K}$ has a canonical integral
model $\mathfrak{S}_{K}$ over $\mathcal{O}_{E,\mathfrak{p}}$. Let
$\mathcal{S}_{K}^{\circ}\subset\mathcal{S}_{K}$ be the good reduction locus,
i.e. the open subspace of $\mathcal{S}_{K}$ obtained from the adic generic
fiber of the $p$-adic completion $\mathfrak{S}^{\wedge}_{K}$ of the scheme
$\mathfrak{S}_{K}$. We define
$\mathcal{S}_{K^{\prime}}^{\circ}\subset\mathcal{S}_{K^{\prime}}$ for
$K^{\prime}\subset K$ by taking the preimage under the natural map from
$\mathcal{S}_{K^{\prime}}$ to $\mathcal{S}_{K}$. We also consider the adic
spaces $\mathcal{S}^{*}_{K}$ and $\mathcal{S}^{\mathrm{tor}}_{K}$ attatched to
the minimal and toroidal compactification of the Shimura variety
$\mathcal{S}^{*}_{K}$.

Associated to the $G(\mathbb{R})$-conjugacy class $X$, we have a minuscule
cocharacter $\mu$ of $G_{\mathbb{Q}_{p}}$ which is defined over
$E_{\mathfrak{p}}$. Let $\mathcal{F}\ell_{G,\mu^{-1}}$ be the flag variety
over $\mathop{\rm Spa}(C)$ associated to $\mu^{-1}$ the dominant inverse of
$\mu$. Since $\mu$ is minuscule, via the Bialynicki-Birula isomorphism, when
viewed as a diamond the flag variety $\mathcal{F}\ell_{G,\mu^{-1}}$ represents
the following functor on $\mathrm{Perf}_{C}$. Given any
$S\in\mathrm{Perf}_{C}$, $\mathcal{F}\ell_{G,\mu^{-1}}(S)$ is the set of
modifications of vector bundles $\mathcal{E}\dashrightarrow\mathcal{E}_{0}$ of
meromorphy $\mu$ on $X_{S}$, the relative Fargues-Fontaine curve over $S$,
such that the modification occurs over the untilt of $S$ corresponding to the
map $S\rightarrow\mathop{\rm Spd}(C)$.

Let

$\mathcal{S}_{K^{p}}^{\circ}:=\mathop{\rm
lim}\nolimits\limits_{\begin{subarray}{c}\leftarrow\\\
K_{p}\end{subarray}}\mathcal{S}_{K^{p}K_{p}}^{\circ}\subset\mathcal{S}_{K^{p}}:=\mathop{\rm
lim}\nolimits\limits_{\begin{subarray}{c}\leftarrow\\\
K_{p}\end{subarray}}\mathcal{S}_{K^{p}K_{p}}\subset\mathcal{S}_{K^{p}}^{\mathrm{tor}}:=\mathop{\rm
lim}\nolimits\limits_{\begin{subarray}{c}\leftarrow\\\
K_{p}\end{subarray}}\mathcal{S}^{\mathrm{tor}}_{K^{p}K_{p}}\rightarrow\mathcal{S}_{K^{p}}^{*}:=\mathop{\rm
lim}\nolimits\limits_{\begin{subarray}{c}\leftarrow\\\
K_{p}\end{subarray}}\mathcal{S}^{*}_{K^{p}K_{p}}$

be the associated perfectoid Shimura varieties. We also consider
$\overline{\mathcal{S}}^{\circ}_{K^{p},C}$, the canonical compactification of
the good reduction locus. This will be a subspace of $\mathcal{S}_{K^{p},C}$,
since $\mathcal{S}_{K^{p},C}$ is partially proper. Caraiani-Scholze [CS17,
§2.1] consider the Hodge-Tate period morphism on $\mathcal{S}_{K^{p}}$

$\pi_{\mathrm{HT}}:\mathcal{S}_{K^{p},C}\rightarrow\mathcal{F}\ell_{G,\mu^{-1}},$

which records the relative position of the Hodge-Tate filtration associated
with the $p$-divisible group. This extends [CS19, §4.1] to a Hodge-Tate period
morphism on the minimal compactification

$\pi_{\mathrm{HT}}^{*}:\mathcal{S}_{K^{p},C}^{*}\rightarrow\mathcal{F}\ell_{G,\mu^{-1}}$

and toroidal compactification

$\pi_{\mathrm{HT}}^{\mathrm{tor}}:\mathcal{S}_{K^{p},C}^{\mathrm{tor}}\rightarrow\mathcal{F}\ell_{G,\mu^{-1}}.$

We write $\pi_{\mathrm{HT}}^{\circ}$ for the restriction to the good reduction
locus, and $\overline{\pi}_{\mathrm{HT}}^{\circ}$ for the canonical
compactification of $\pi_{\mathrm{HT}}^{\circ}$, where we note that this again
maps to $\mathcal{F}\ell_{G,\mu^{-1}}$ as this is proper over $\mathop{\rm
Spa}(C)$.

These maps have the following properties:

  1. (1)

$\pi_{\mathrm{HT}}^{*}$ and $\pi_{\mathrm{HT}}^{\mathrm{tor}}$ are partially
proper and qcqs; hence, proper.

  2. (2)

$\pi_{\mathrm{HT}}$ and $\overline{\pi}_{\mathrm{HT}}^{\circ}$ are partially
proper, but not always qcqs.

  3. (3)

$\pi_{\mathrm{HT}}^{\circ}$ is qcqs, but not partially proper.

With these properties in mind, let us study the fibers of these maps. For our
purposes, we will focus on the compactly supported cohomology of
$\mathcal{S}_{K^{p},C}$ and in turn the sheaf
$R\pi_{\mathrm{HT}!}(\overline{\mathbb{F}}_{\ell})$ on
$\mathcal{F}\ell_{G,\mu^{-1}}$.

######  Remark 3.1.

We note that it is always true that the compactly supported cohomology at
infinite level is the colimit of the compactly supported cohomology at finite
levels, but, for usual cohomology one needs to assume the spaces are qcqs for
this to be true (e.g the tower defined by the good reduction locus or the
minimal/toroidal compactifications).

Our goal is to describe the stalks of
$R\pi_{\mathrm{HT}!}(\overline{\mathbb{F}}_{\ell})$ at a geometric point
$x:\mathop{\rm Spa}(C,C^{+})\rightarrow\mathcal{F}\ell_{G,\mu^{-1}}$. We
assume the geometric point $x$ factors through the adic Newton strata
$\mathcal{F}\ell_{G,\mu^{-1}}^{b}$ for $b\in B(G,\mu)$, and choose a
completely slope divisible $p$-divisible group $\mathbb{X}_{b}$ over
$\overline{\mathbb{F}}_{p}$ corresponding to $b\in B(G,\mu)$. Let
$\mathrm{Ig}^{b}$ be the associated perfect Igusa variety as defined in §2.2,
with toroidal compactification $\mathrm{Ig}^{b,\mathrm{tor}}$ and minimal
compactification $\mathrm{Ig}^{b,*}$. Recall that we have associated
perfectoid Igusa varieties,
$\mathfrak{Ig}^{b}_{C},\mathfrak{Ig}^{b,\mathrm{tor}}_{C},\mathfrak{Ig}^{b,*}_{C}$,
which should model the fibers of $\overline{\pi}_{\mathrm{HT}}^{\circ}$,
$\pi_{\mathrm{HT}}^{\mathrm{tor}}$, and $\pi^{*}_{\mathrm{HT}}$, respectively.
We let $\partial\mathfrak{Ig}_{C}^{b,*}$ and
$\partial\mathfrak{Ig}_{C}^{b,\mathrm{tor}}$ be the Zariski closed subspaces
attached to the boundaries $\partial\mathrm{Ig}^{b,*}$ and
$\partial\mathrm{Ig}^{b,\mathrm{tor}}$, respectively.

Let $g_{b}:\mathrm{Ig}^{b}\hookrightarrow\mathrm{Ig}^{b,*}$ be the natural
open immersion of $\overline{\mathbb{F}}_{p}$-schemes. We define the partially
compactly supported cohomology

$R\Gamma_{c-\partial}(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell}):=R\Gamma(\mathrm{Ig}^{b,*},g_{b!}(\overline{\mathbb{F}}_{\ell})).$

Our goal is to show that this computes the fibers of
$R\pi_{\mathrm{HT}!}(\overline{\mathbb{F}}_{\ell})$ for geometric points in
$\mathcal{F}\ell_{G,\mu^{-1}}^{b}$.

To get a clearer picture of how these spaces interact with each other, we have
the following theorem.

######  Theorem 3.2.

[CS19, San23] There exists a diagram of spaces of the form

${(\pi_{\mathrm{HT}}^{\circ})^{-1}(x)}$${\pi_{\mathrm{HT}}^{-1}(x)}$${(\pi_{\mathrm{HT}}^{\mathrm{tor}})^{-1}(x)}$${(\pi_{\mathrm{HT}}^{*})^{-1}(x)}$${\mathfrak{Ig}^{b}_{C}}$${\mathfrak{Ig}^{b,\mathrm{tor}}_{C},}$${\mathfrak{Ig}^{b,*}_{C}}$$\scriptstyle{i}$$\scriptstyle{\phantom{}{}^{\mathrm{tor}}i}$$\scriptstyle{\phantom{}{}^{*}i}$

where the maps $i$, $\phantom{}{}^{*}i$, and $\phantom{}{}^{\mathrm{tor}}i$
are open immersions whose image contains all rank $1$ points. Moreover, the
fibers $(\pi_{\mathrm{HT}}^{*})^{-1}(x)$ and
$(\pi_{\mathrm{HT}}^{\mathrm{tor}})^{-1}(x)$ are partially proper, so in
particular $\phantom{}{}^{*}i$ and $\phantom{}{}^{\mathrm{tor}}i$ are
canonical compactifications in the sense of [Sch18, Proposition 18.6].

###### Proof.

This theorem in the case where $(\mathbf{G},X)$ is of PEL type $A$ attached to
a globally quasi-split unitary group of even dimension is [CS19, Theorems
2.7.2,Theorem 4.5.1], and the general case of PEL type $A$ or $C$ is proven in
[San23, Theorems 4.3.10,4.3.12]. ∎

We will also combine this with the following result.

######  Theorem 3.3.

[CS19, San23] The partially minimally compactified Igusa variety
$\mathrm{Ig}^{b,*}$ is affine; in particular, the attatched adic space
$\mathfrak{Ig}^{b,*}_{C}$ is affinoid perfectoid. Moreover, there exists a
proper map

$\mathfrak{Ig}^{b,\mathrm{tor}}_{C}\rightarrow\mathfrak{Ig}^{b,*}_{C}$

which induces an isomorphism on global sections.

###### Proof.

For the affineness, the case of PEL type $A$ attached to a global quasi-split
unitary group of even dimension is covered by [CS19, Theorem 1.7] and [CS19,
Lemma 4.5.2]. The general case of PEL type $A$ or $C$ is covered in [San23,
Lemma 3.3.7]. To see the properness, we note that the map

$\mathfrak{Ig}_{C}^{b,\mathrm{tor}}\rightarrow\mathfrak{Ig}_{C}^{b,*}$

is the one appearing in the Stein factorization described in [CS19,
Proposition 3.3.4] and [San23, Proposition 3.3.5]. This also shows that one
has an isomorphism on global sections. ∎

We will also need the following Corollary.

######  Corollary 3.4.

The boundary $\partial\mathfrak{Ig}_{C}^{b,\mathrm{tor}}$ is quasi-compact and
$\partial\mathfrak{Ig}_{C}^{b,*}$ is affinoid perfectoid (In particular,
quasi-compact).

###### Proof.

The fact that $\mathfrak{Ig}_{C}^{b,*}$ is affinoid perfectoid follows from
the previous Theorem. Moreover, $\partial\mathfrak{Ig}_{C}^{b,*}$ is a Zariski
closed subspace, since it came from considering the adic generic fiber of a
formal model of the perfect closed subscheme
$\partial\mathrm{Ig}^{b,*}\subset\mathrm{Ig}^{b,*}$. The claim for the
toroidal compactification follows since the map
$\mathfrak{Ig}_{C}^{b,\mathrm{tor}}\rightarrow\mathfrak{Ig}_{C}^{b,*}$ is
proper, and maps the boundary $\partial\mathfrak{Ig}_{C}^{b,\mathrm{tor}}$ to
$\partial\mathfrak{Ig}_{C}^{b,*}$. ∎

It is natural to wonder how one could describe the fiber of
$\pi_{\mathrm{HT}}$ in terms of the spaces described above. In particular, we
now deduce the following Corollary.

######  Corollary 3.5.

For $x:\mathop{\rm Spa}(C,C^{+})\rightarrow\mathcal{F}\ell_{G,\mu^{-1}}^{b}$ a
geometric point, we have isomorphisms

$\pi_{\mathrm{HT}}^{-1}(x)\simeq\overline{\mathfrak{Ig}}_{C}^{b,*}\setminus\overline{\partial\mathfrak{Ig}}_{C}^{b,*}\simeq\overline{\mathfrak{Ig}}_{C}^{b,\mathrm{tor}}\setminus\overline{\partial\mathfrak{Ig}}_{C}^{b,\mathrm{tor}}$

induced by the natural open immersions
$\pi_{\mathrm{HT}}^{-1}(x)\hookrightarrow(\pi_{\mathrm{HT}}^{*})^{-1}(x)\simeq\overline{\mathfrak{Ig}}_{C}^{b,*}$
(resp.
$\pi_{\mathrm{HT}}^{-1}(x)\hookrightarrow(\pi_{\mathrm{HT}}^{\mathrm{tor}})^{-1}(x)\simeq\overline{\mathfrak{Ig}}_{C}^{b,\mathrm{tor}}$),
as given by Theorem 3.2. Here $\overline{\partial\mathfrak{Ig}}_{C}^{b,*}$
(resp. $\overline{\partial\mathfrak{Ig}}_{C}^{b,\mathrm{tor}}$) is the
Zarisiki closed subset of $\overline{\mathfrak{Ig}}_{C}^{b,*}$ (resp.
$\overline{\mathfrak{Ig}}_{C}^{b,\mathrm{tor}}$) defined by the canonical
compactification of the boundaries
$\partial\mathfrak{Ig}^{b}_{C}\subset\mathfrak{Ig}^{b}_{C}$ (resp.
$\partial\mathfrak{Ig}_{C}^{b,\mathrm{tor}}\subset\mathfrak{Ig}^{b,\mathrm{tor}}_{C}$).

###### Proof.

We first establish the claim for the toroidal compactification. We consider
the closed immersion

$\overline{\mathfrak{Ig}}_{C}^{b,\mathrm{tor}}\times_{\mathcal{S}^{\mathrm{tor}}_{K^{p},C}}\partial\mathcal{S}^{\mathrm{tor}}_{K^{p},C}\hookrightarrow\overline{\partial\mathfrak{Ig}}_{C}^{b,\mathrm{tor}}$

obtained by base-changing the closed immersion
$\partial\mathcal{S}^{\mathrm{tor}}_{K^{p},C}\hookrightarrow\mathcal{S}^{\mathrm{tor}}_{K^{p},C}$
to the fiber $(\pi_{\mathrm{HT}}^{\mathrm{tor}})^{-1}(x)$ and applying Theorem
3.2. To show the desired claim, it suffices to show this is an isomorphism.
Note that both the LHS and RHS are partially proper; therefore, to show this
is an isomorphism, it suffices to show it induces an isomorphism on rank $1$
points. In particular, given $C^{\prime}/C$ a complete algebraically closed
non-archimedean field, we claim that there exists a Cartesian diagram of the
form

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}\hbox{${\partial\mathfrak{Ig}^{b,\mathrm{tor}}_{C}(C^{\prime},\mathcal{O}_{C^{\prime}})}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
33.34006pt\hfil&\hfil\hskip 54.6855pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{${\mathfrak{Ig}^{b,\mathrm{tor}}_{C}(C^{\prime},\mathcal{O}_{C^{\prime}})}$}
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}\hbox{${\partial\mathcal{S}^{\mathrm{tor}}_{K^{p},C}(C^{\prime},\mathcal{O}_{C^{\prime}})}$}
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This can be checked using the moduli interpretation, as in the proof of [CS19,
Theorem 4.4.1]. In particular, given a $\mathop{\rm
Spa}(C^{\prime},\mathcal{O}_{C^{\prime}})$ point of
$\mathfrak{Ig}^{b,\mathrm{tor}}_{C}$ specializing to a boundary component
indexed by an Igusa cusp label $\tilde{Z}$ (See §2.2.2 for the definition of
Igusa cusp label), then from the discussion in §2.2.3 this corresponds to the
datum of $(\mathcal{B},\mathcal{G},\mathcal{G}^{\vee},\rho,y)$, where

  1. (1)

$\mathcal{B}$ is an abelian scheme over $\mathcal{O}_{C^{\prime}}$ with
polarization, and $\mathcal{O}_{B}$-structure,

  2. (2)

$\mathcal{G}$ is a Raynaud extension

$0\rightarrow T\rightarrow\mathcal{G}\rightarrow\mathcal{B}\rightarrow 0$

of $\mathcal{B}$ by a torus $T$ with cocharacter group $X$,

  3. (3)

$\mathcal{G}^{\vee}$ is a Raynaud extension

$0\rightarrow
T^{\vee}\rightarrow\mathcal{G}^{\vee}\rightarrow\mathcal{B}^{\vee}\rightarrow
0$

of $\mathcal{B}^{\vee}$ by a torus $T^{\vee}$ with cocharacter group $Y$,

  4. (4)

$\rho$ is an $\mathcal{O}_{B}$-linear isomorphism
$\rho:\mathcal{G}[p^{\infty}]\simeq Z_{b,-1}$, extending the isomorphism
$T[p^{\infty}]\simeq Z_{b,-2}$,

  5. (5)

$y\in\mathcal{P}_{\Sigma_{\Phi}}(\mathcal{O}_{C^{\prime}})$, where
$\mathcal{P}_{\Sigma_{\Phi}}$ is the toroidal compactification (determined by
an admissible rpcd $\Sigma_{\Phi}$, as in §2.1.2) of a torsor under a torus
$E_{\Phi}/\mathcal{O}_{C^{\prime}}$, whose character group is given by
$S_{\Phi}$, and $y$ is a point whose special fiber lies in the boundary.

The natural map
$\mathfrak{Ig}^{b,\mathrm{tor}}(C^{\prime},\mathcal{O}_{C^{\prime}})\hookrightarrow\mathcal{S}_{K^{p},C}^{\mathrm{tor}}(C^{\prime},\mathcal{O}_{C^{\prime}})$
is determined by forgetting the trivialization
$\rho:Z_{b,-1}\simeq\mathcal{G}[p^{\infty}]$. The image under this map of a
point lying in the boundary $\partial\mathfrak{Ig}^{b,\mathrm{tor}}_{C}$ is
equivalent to insisting that $y$ is a point whose special fiber lies in the
boundary of $\mathcal{P}_{\Sigma_{\Phi}}$ for some Igusa cusp label
$\tilde{Z}$, which is not the trivial one. Let $Z=(Z,\Phi)$ be the cusp label
which $\tilde{Z}$ lives over. This is then equivalent to the condition
guaranteeing that the image lies in the component of
$\mathcal{S}_{K^{p},C}^{\mathrm{tor}}$ indexed by $Z=(Z,\Phi)$, where $Z$ is
not the trivial cusp label, and thus lies in the boundary
$\partial\mathcal{S}^{\mathrm{tor}}_{K^{p},C}$. The claim follows.

It remains to see the analogous claim for the minimal compactification. This
follows easily using Theorem 3.2 and the fact that the proper surjective map
$\mathfrak{Ig}_{C}^{b,\mathrm{tor}}\rightarrow\mathfrak{Ig}_{C}^{b,*}$ sends
$\partial\mathfrak{Ig}_{C}^{b,\mathrm{tor}}$ to
$\partial\mathfrak{Ig}_{C}^{b,*}$ by construction. ∎

We have the following Corollary.

######  Corollary 3.6.

For a geometric point $x:\mathop{\rm
Spa}(C,C^{+})\rightarrow\mathcal{F}\ell_{G,\mu^{-1}}^{b}$, we have an
identification:

$R\Gamma_{c-\partial}(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell})\simeq
R\pi_{\mathrm{HT}!}(\overline{\mathbb{F}}_{\ell})_{x}.$

###### Proof.

We have an identification
$\pi_{\mathrm{HT}}^{-1}(x)\simeq\overline{\mathfrak{Ig}}^{b,*}_{C}\setminus\overline{\partial\mathfrak{Ig}}^{b,*}_{C}$
by the previous Corollary, so, by proper base-change, we are tasked with
computing the compactly supported cohomology of this space. We note, by
Theorem 3.3, the adic spaces $\mathfrak{Ig}^{b,*}_{C}$ are affinoid
perfectoid. It follows that the canonical compactification
$\overline{\mathfrak{Ig}}^{b,*}_{C}\simeq(\pi_{\mathrm{HT}}^{*})^{-1}(x)$ is
also affinoid perfectoid, by [Sch18, Proposition 18.7 (iv)]. In particular, it
is quasi-compact and partially proper, so in particular proper. It therefore
follows by excision111One easily checks that the excision sequence is exact on
points, and this is sufficient by [Sch18, Proposition 14.3]. that we have a
distinguished triangle

(3)
${}R\Gamma_{c}(\pi_{\mathrm{HT}}^{-1}(x),\overline{\mathbb{F}}_{\ell})\rightarrow
R\Gamma(\overline{\mathfrak{Ig}}^{b,*}_{C},\overline{\mathbb{F}}_{\ell})\rightarrow
R\Gamma(\overline{\partial\mathfrak{Ig}}^{b,*}_{C},\overline{\mathbb{F}}_{\ell})\xrightarrow{+1}.$

Applying Theorem 3.2 again, we know that
$k:\mathfrak{Ig}^{b,*}\hookrightarrow\overline{\mathfrak{Ig}}^{b,*}_{C}$ is a
qcqs open immersion of perfectoid spaces inducing an isomorphism on rank $1$
points, and it follows that the same is true for the induced map on the
Zariski closed subspaces
$\phantom{}{}^{\partial}k:\partial\mathfrak{Ig}^{b,*}\hookrightarrow\overline{\partial\mathfrak{Ig}}^{b,*}_{C}$.
Therefore, we can apply [CS17, Lemma 4.4.2], this tells us that the natural
maps

$\overline{\mathbb{F}}_{\ell}\rightarrow k_{*}(\overline{\mathbb{F}}_{\ell})$
$\overline{\mathbb{F}}_{\ell}\rightarrow\phantom{}^{\partial}k_{*}(\overline{\mathbb{F}}_{\ell})$

are isomorphisms, giving identifications
$R\Gamma(\overline{\mathfrak{Ig}}_{C}^{b,*},\overline{\mathbb{F}}_{\ell})\simeq
R\Gamma(\mathfrak{Ig}^{b,*}_{C},\overline{\mathbb{F}}_{\ell})$ and
$R\Gamma(\overline{\partial\mathfrak{Ig}}_{C}^{b,*},\overline{\mathbb{F}}_{\ell})\simeq
R\Gamma(\partial\mathfrak{Ig}^{b,*}_{C},\overline{\mathbb{F}}_{\ell})$. Now,
by [CS17, Lemma 4.4.3], we have further identifications of
$R\Gamma(\partial\mathfrak{Ig}^{b,*}_{C},\overline{\mathbb{F}}_{\ell})$ and
$R\Gamma(\mathfrak{Ig}^{b,*}_{C},\overline{\mathbb{F}}_{\ell})$ with the
cohomology of the perfect schemes $\partial\mathrm{Ig}^{b,*}$ and
$\mathrm{Ig}^{b,*}$, respectively. Substituting this into the triangle (3), we
get a distinguished triangle

$R\Gamma_{c}(\pi_{\mathrm{HT}}^{-1}(x),\overline{\mathbb{F}}_{\ell})\rightarrow
R\Gamma(\mathrm{Ig}^{b,*},\overline{\mathbb{F}}_{\ell})\rightarrow
R\Gamma(\partial\mathrm{Ig}^{b,*},\overline{\mathbb{F}}_{\ell})\xrightarrow{+1}.$

By applying quasi-compact base-change [Sch18, Proposition 17.6] and then using
that the inclusion
$\partial\mathfrak{Ig}_{C}^{b,*}\subset\mathfrak{Ig}_{C}^{b,*}$ is induced
from taking the rigid generic fiber over $C$ of Witt vectors applied to
$\partial\mathrm{Ig}^{b,*}\subset\mathrm{Ig}^{b,*}$, we identify the last map
with the natural restriction map on the cohomology. However, this identifies
the first term with precisely the partially compactly supported cohomology, as
desired. ∎

We will combine this with the following proposition, which already hints at
our expectation that $R\pi_{\mathrm{HT}!}(\overline{\mathbb{F}}_{\ell})$ is
connective in some suitable perverse $t$-structure.

######  Proposition 3.7.

If $d_{b}:=\langle
2\rho_{G},\nu_{b}\rangle=\mathrm{dim}(\mathrm{Ig}^{b,*})=\mathrm{dim}(\mathrm{Ig}^{b})$
then the cohomology of the complex

$R\Gamma_{c-\partial}(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell})\simeq
R\pi_{\mathrm{HT}!}(\overline{\mathbb{F}}_{\ell})_{x}$

is concentrated in degrees $\leq d_{b}$.

###### Proof.

We saw in Theorem 3.3 that $\mathrm{Ig}^{b,*}$ is an affine scheme. So we
would like to apply Artin vanishing; however, $\mathrm{Ig}^{b,*}$ is also a
perfect scheme so in particular not of finite type. To remedy this, consider
the pro-étale cover

$\mathrm{Ig}^{b}\rightarrow\mathcal{C}_{\mathbb{X}_{b}}^{\mathrm{perf}},$

with Galois group $\mathrm{Aut}(\mathbb{X})(\overline{\mathbb{F}}_{p})$ over
the perfection of the central leaf attached to $\mathbb{X}_{b}$. This is
obtained as the perfection of the limit of the finite étale covers

$\mathrm{Ig}^{b}_{m}\rightarrow\mathcal{C}_{\mathbb{X}_{b}},$

described in 2.2.3, as shown in [CS17, Proposition 4.3.8]. These spaces are of
finite type over $\overline{\mathbb{F}}_{p}$. We now define
$\mathrm{Ig}^{b,*}_{m}$ to be the normalization of
$\mathcal{C}_{\mathbb{X}}^{*}$ in $\mathrm{Ig}^{b}_{m}$ of the finite étale
cover $\mathrm{Ig}^{b}_{m}\rightarrow\mathcal{C}_{\mathbb{X}_{b}}$. By [San23,
Theorem 3.33], $\mathcal{C}_{\mathbb{X}}^{*}$ is affine; therefore, it follows
that $\mathrm{Ig}^{b,*}_{m}$ is a normal and affine scheme which will be of
finite type, since $\mathrm{Ig}^{b}_{m}$ is. It follows, by definition of
$\mathrm{Ig}^{b,*}$, that it is the perfection of $\mathop{\rm
lim}\nolimits_{m\geq 1}\mathrm{Ig}_{m}^{b,*}$. Therefore, since passing to
perfections doesn’t change the étale cohomology, we can conclude by combining
Artin vanishing with an application of [Sta23, Tag 09QY] to the system of
sheaves $g_{b,m!}(\overline{\mathbb{F}}_{\ell})$, where
$g_{b,m}:\mathrm{Ig}^{b}_{m}\rightarrow\mathrm{Ig}^{b,*}_{m}$ is the natural
open inclusion at finite level. ∎

Now we would like to link this analysis with the semi-perversity of certain
sheaves on $\mathrm{Bun}_{G}$. We consider the Hodge-Tate period morphism

$\pi_{\mathrm{HT}}:[\mathcal{S}_{K^{p}}/\underline{G(\mathbb{Q}_{p})}]\rightarrow[\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}]$

quotiented out by $G(\mathbb{Q}_{p})$. We let
$h^{\rightarrow}:[\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}]\rightarrow[\mathop{\rm
Spd}(C)/\underline{G(\mathbb{Q}_{p})}]\simeq\mathrm{Bun}_{G}^{1}$ be the
structure map quotiented out by $G(\mathbb{Q}_{p})$. Note this is a proper
map, since $\mathcal{F}\ell_{G,\mu^{-1}}$ is proper over $\mathop{\rm
Spd}(C)$.

Then we have an identification

(4) $R\Gamma_{c}(\mathcal{S}_{K^{p},C},\overline{\mathbb{F}}_{l})\simeq
h^{\rightarrow}_{*}R\pi_{\mathrm{HT}!}(\overline{\mathbb{F}}_{\ell})$

of $G(\mathbb{Q}_{p})$-representations, and this computes the compactly
supported torsion cohomology of the Shimura variety.

Similarly, we have a map

$h^{\leftarrow}:[\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}]\rightarrow\mathrm{Bun}_{G}$

remembering the isomorphism class of the bundle $\mathcal{E}_{1}$ in the
moduli interpretation of $\mathcal{F}\ell_{G,\mu^{-1}}$ as a diamond. This
defines a cohomologically smooth map by [FS21, Theorem IV.1.19], and the image
identifies with the open subset $B(G,\mu)\subset B(G)$ under the
identification $|\mathrm{Bun}_{G}|\simeq B(G)$ of topological spaces, where
$|\mathrm{Bun}_{G}|$ denotes the underlying topological space of
$\mathrm{Bun}_{G}$ and $B(G)$ has the topology given by its natural partial
ordering [Vie21]. For each $b$, we have a locally closed Harder-Narasimhan
stratum $j_{b}:\mathrm{Bun}_{G}^{b}\hookrightarrow\mathrm{Bun}_{G}$, and we
can define the locally closed subset
$[\mathcal{F}\ell_{G,\mu^{-1}}^{b}/\underline{G(\mathbb{Q}_{p})}]$, by pulling
back this HN-strata along $h^{\leftarrow}$. This defines a locally closed
stratification of
$[\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}]$. Let
$i_{b}:[\mathcal{F}\ell_{G,\mu^{-1}}^{b}/\underline{G(\mathbb{Q}_{p})}]\hookrightarrow[\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}]$
denote the associated locally closed immersion. We write
$\pi_{\mathrm{HT}}^{b}:[\mathcal{S}^{b}_{K^{p},C}/\underline{G(\mathbb{Q}_{p})}]\rightarrow[\mathcal{F}\ell_{G,\mu^{-1}}^{b}/\underline{G(\mathbb{Q}_{p})}]$
(resp.
$\pi_{\mathrm{HT}}^{b,*}:[\mathcal{S}^{b,*}_{K_{p},C}/\underline{G(\mathbb{Q}_{p})}]\rightarrow[\mathcal{F}\ell_{G,\mu^{-1}}^{b}/\underline{G(\mathbb{Q}_{p})}]$)
for the pullbacks of $\pi_{\mathrm{HT}}$ (resp. $\pi^{*}_{\mathrm{HT}}$) along
$i_{b}$. On the good reduction locus, we also have an additional
stratification coming from pulling back the Newton stratification on the
special fiber along the specialization map. There is a rather subtle point
that this does not agree with the pullback of the locally closed strata
$\mathcal{F}\ell_{G,\mu^{-1}}^{b}$ (namely, the closure relationships are
opposite with respect to the partial ordering on $B(G)$). We write
$\mathcal{S}^{b,\circ,\mathrm{rd}}_{K^{p},C}$ for these Newton strata coming
from the special fiber. There exists a natural map

$\mathcal{S}^{b,\circ,\mathrm{rd}}_{K^{p},C}\times_{\mathcal{F}\ell_{G,\mu^{-1}}}\mathcal{F}\ell_{G,\mu^{-1}}^{b}\hookrightarrow\mathcal{S}^{b,\circ}_{K^{p},C}$

which is a qcqs open immersion containing all rank $1$ points ([CS17, Page 68]
[Kos21, Page 8]). We write
$\pi_{\mathrm{HT}}^{b,\circ}:[(\mathcal{S}^{b,\circ,\mathrm{rd}}_{K^{p},C}\times_{\mathcal{F}\ell_{G,\mu^{-1}}}\mathcal{F}\ell_{G,\mu^{-1}}^{b})/\underline{G(\mathbb{Q}_{p})}]\rightarrow[\mathcal{F}\ell_{G,\mu^{-1}}^{b}/\underline{G(\mathbb{Q}_{p})}]$
for the Hodge-Tate period map on this locus, and similarly we write
$\overline{\pi}_{\mathrm{HT}}^{b,\circ}:[\overline{\mathcal{S}}^{b,\circ}_{K^{p},C}/\underline{G(\mathbb{Q}_{p})}]\rightarrow[\mathcal{F}\ell_{G,\mu^{-1}}^{b}/\underline{G(\mathbb{Q}_{p})}]$
for the induced map on the canonical compactification, where we note that this
agrees with the canonical compactification of
$[(\mathcal{S}^{b,\circ,\mathrm{rd}}_{K^{p},C}\times_{\mathcal{F}\ell_{G,\mu^{-1}}}\mathcal{F}\ell_{G,\mu^{-1}}^{b})/\underline{G(\mathbb{Q}_{p})}]$
by the previous remark on rank $1$ points and the fact that
$\mathcal{F}\ell_{G,\mu^{-1}}^{b}$ is partially proper222The partial
properness of these strata follows directly from the moduli interpretation,
since the category of vector bundles on the Fargues-Fontaine curve is
insensitive to the ring of definition..

Define the group diamond $\mathcal{J}_{b}:=\mathrm{Aut}(\mathcal{E}_{b})$, as
in [FS21, Proposition III.5.1]. We have an isomorphism
$j_{b}:\mathrm{Bun}_{G}^{b}\simeq[\mathop{\rm
Spd}(C)/\mathcal{J}_{b}]\hookrightarrow\mathrm{Bun}_{G}$ with the locally
closed HN-strata in $\mathrm{Bun}_{G}$ defined by $b$. There is a
$\mathcal{J}_{b}$-torsor over the adic Newton strata
$\mathcal{F}\ell_{G,\mu^{-1}}^{b}$ given by rigidifying the bundle
$\mathcal{E}$ to be isomorphic to $\mathcal{E}_{b}$. This gives a map

$h^{\leftarrow}_{b}:[\mathcal{F}\ell_{G,\mu^{-1}}^{b}/\underline{G(\mathbb{Q}_{p})}]\rightarrow[\mathop{\rm
Spd}(C)/\mathcal{J}_{b}]\simeq\mathrm{Bun}_{G,C}^{b}$

such that $h^{\leftarrow}\circ i_{b}=j_{b}\circ h^{\leftarrow}_{b}$.

The perfectoid Igusa variety $\mathfrak{Ig}^{b}_{C}$ comes equipped with an
action of $\mathcal{J}_{b}$. Namely, using [CS17, Proposition 4.2.11], we get
an action on the trivialization of $\mathbb{X}_{b}$, as in the moduli
description in equation (2). This action extends to the formal model, giving
rise to the action of $\mathcal{J}_{b}$ the generic fiber. This allows us to
form the map of $v$-stacks:

$\pi_{\mathfrak{Ig}}^{b}:[\mathfrak{Ig}_{C}^{b}/\mathcal{J}_{b}]\rightarrow[\mathop{\rm
Spd}(C)/\mathcal{J}_{b}].$

We would like to say $\pi_{\mathfrak{Ig}}^{b}$ pulls back to the map
$\pi^{b}_{\mathrm{HT}}$. However, as seen in Corollary 3.6, we need to account
for the additional points in the fiber of the Hodge-Tate period morphism that
are not seen by the perfectoid Igusa varieties $\mathfrak{Ig}^{b}_{C}$. To
capture this, we need to show that $\pi_{\mathfrak{Ig}}^{b}$ also extends to
the partial minimal compactification. We have the following.

######  Proposition 3.8.

[Zha23, Corollary 9.43] Assuming 1.11, the action of $\mathcal{J}_{b}$ on the
perfectoid Igusa variety $\mathfrak{Ig}^{b}_{C}$ extends uniquely to an action
on $\mathfrak{Ig}^{b,*}_{C}$. In particular, by functorality of the formation
of the canonical compactification ([Sch18, Proposition 18.6]), we have a map

$\pi_{\mathfrak{Ig}}^{b,*}:[\overline{\mathfrak{Ig}}_{C}^{b,*}/\mathcal{J}_{b}]\rightarrow[\mathop{\rm
Spd}(C)/\mathcal{J}_{b}]$

extending $\pi_{\mathfrak{Ig}}^{b}$. This action preserves the boundary
$\partial\mathfrak{Ig}^{b,*}$, so, in particular, we also get a map

$\pi_{\mathfrak{Ig}}^{b,\partial}:[(\overline{\mathfrak{Ig}}^{b,*}_{C}\setminus\overline{\partial\mathfrak{Ig}}^{b,*}_{C})/\mathcal{J}_{b}]\rightarrow[\mathop{\rm
Spd}(C)/\mathcal{J}_{b}]$

by restriction.

###### Proof.

We consider the open immersion

$g_{b}:\mathrm{Ig}^{b}\rightarrow\mathrm{Ig}^{b,*}$

of perfect schemes, which we claim induces an isomorphism on global sections.
To show this, we write $g_{b}$ as the perfection of the limit of the
corresponding maps at finite level

$g_{b,m}:\mathrm{Ig}^{b}_{m}\hookrightarrow\mathrm{Ig}^{b,*}_{m},$

as explained in the proof of Proposition 3.7. Under Assumption 1.11, we can
apply the algebraic form of Hartogs’ principle (See for example [Sch15,
Proposition III.2.9]) to the open inclusion $g_{b,m}$ to conclude an
isomorphism of global sections via restriction. This gives the corresponding
claim for the map $g_{b}$ of perfect schemes. In particular, we have an
isomorphism

(5) ${}\mathcal{O}(\mathrm{Ig}^{b})\simeq\mathcal{O}(\mathrm{Ig}^{b,*})$

on global sections. Taking generic fibers of the corresponding integral
models, we claim that we obtain an isomorphism

(6)
${}\mathcal{O}(\mathfrak{Ig}^{b}_{C})\simeq\mathcal{O}(\mathfrak{Ig}^{b,*}_{C}),$

of global sections. We need to be a bit careful here with analytic
sheafification. In particular, for an index set $I$, we let $\\{U_{i}\\}_{i\in
I}$ be an affine covering of $\mathrm{Ig}^{b}$, and compute global sections
via the Čech complex

(7) ${}0\rightarrow\mathcal{O}(\mathrm{Ig}^{b})\rightarrow\prod_{i\in
I}\mathcal{O}(U_{i})\rightarrow\prod_{i,j\in I}\mathcal{O}(U_{i}\cap
U_{j})\rightarrow\cdots.$

We let $\mathfrak{U}_{i}$ be the formal schemes obtained by taking Witt
vectors of this affine covering, with adic generic fibers $\mathfrak{U}_{i,C}$
over $C$. These form an affinoid perfectoid covering of the adic space
$\mathfrak{Ig}^{b}_{C}$, and, by the acyclicity of affinoid perfectoids
[Sch14, Theorem 1.8 (iv)], we have that

(8) ${}0\rightarrow\mathcal{O}(\mathfrak{Ig}_{C}^{b})\rightarrow\prod_{i\in
I}\mathcal{O}(\mathfrak{U}_{i,C})\rightarrow\prod_{i,j\in
I}\mathcal{O}(\mathfrak{U}_{i,C}\cap\mathfrak{U}_{j,C})\rightarrow\cdots.$

It follows that the Čech complex (8) is obtained from Čech complex (7) by
taking Witt vectors followed by taking the completed tensor product with $C$.
Using this, we deduce that the identification (6) follows from the
identification (5), as desired.

Now $\mathcal{J}_{b}$ acts on the LHS of (6), as discussed above. Using that
$\mathfrak{Ig}^{b,*}$ is affinoid, this will give the desired action on
$\mathfrak{Ig}^{b,*}$, and one can see that it preserves the boundary by using
the moduli interpretation of the toroidal compactifications, and the
description of the $J_{b}(\mathbb{Q}_{p})$ action on cusp labels, as discussed
in §2.2.2. ∎

Lastly, we will consider the map
$\overline{\pi}^{b}_{\mathfrak{Ig}}:[\overline{\mathfrak{Ig}}^{b}_{C}/\mathcal{J}_{b}]\rightarrow[\ast/\mathcal{J}_{b}]$,
given by taking the canonical compactification of $\pi^{b}_{\mathfrak{Ig}}$,
where we note, by [CS17, Proposition 4.2.22], the $v$-stack $[\mathop{\rm
Spd}(C)/\mathcal{J}_{b}]$ is partially proper over $\mathop{\rm Spd}(C)$. We
now have the following Proposition.

######  Proposition 3.9.

The maps constructed above fit into the following Cartesian squares333We
emphasize that these are really diagrams of $v$-stacks and that all fiber
products are formed in this category.

(9)
${\left[(\mathcal{S}^{b,\circ,\mathrm{rd}}_{K^{p},C}\times_{\mathcal{F}\ell_{G,\mu^{-1}}}\mathcal{F}\ell_{G,\mu^{-1}}^{b})/\underline{G(\mathbb{Q}_{p})}\right]}$${\left[\mathcal{F}\ell^{b}_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}\right]}$${\left[\mathfrak{Ig}_{C}^{b}/\mathcal{J}_{b}\right]}$${\left[\mathop{\rm
Spd}(C)/\mathcal{J}_{b}\right]}$$\scriptstyle{\pi_{\mathrm{HT}}^{b,\circ}}$$\scriptstyle{\tilde{h}_{b}^{\leftarrow}}$$\scriptstyle{h^{\leftarrow}_{b}}$$\scriptstyle{\pi_{\mathfrak{Ig}}^{b}}$

and

(10)
${\left[\overline{\mathcal{S}}^{b,\circ}_{K^{p},C}/\underline{G(\mathbb{Q}_{p})}\right]}$${\left[\mathcal{F}\ell^{b}_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}\right]}$${\left[\overline{\mathfrak{Ig}}_{C}^{b}/\mathcal{J}_{b}\right]}$${\left[\mathop{\rm
Spd}(C)/\mathcal{J}_{b}\right].}$$\scriptstyle{\overline{\pi}_{\mathrm{HT}}^{b,\circ}}$$\scriptstyle{\tilde{h}_{b}^{\leftarrow}}$$\scriptstyle{h^{\leftarrow}_{b}}$$\scriptstyle{\overline{\pi}_{\mathfrak{Ig}}^{b}}$

###### Proof.

Consider the moduli space of local shtukas $\mathrm{Sht}(G,b,\mu)_{\infty,C}$,
as defined in [SW20, §23]. This represents the functor sending
$S\in\mathrm{Perf}_{C}$ to the set of all pairs $(S^{\\#},\alpha)$ where
$S^{\\#}$ is the untilt of $S$ coming from the map $S\rightarrow\mathop{\rm
Spd}(C)$, and $\alpha$ is a modification from
$\mathcal{E}_{b}\dashrightarrow\mathcal{E}_{0}$ with meromorphy along
$S^{\\#}$ and bounded by $\mu$. We have a local Hodge-Tate period morphism

$\mathrm{Sht}(G,b,\mu)_{\infty,C}\rightarrow\mathcal{F}\ell_{G,\mu^{-1}}^{b},$

which fits into the following Cartesian diagram coming from the definition of
$\mathrm{Sht}(G,b,\mu)_{\infty,C}$.

(11) ${\mathrm{Sht}(G,b,\mu)_{\infty,C}}$${\mathop{\rm
Spd}(C)}$${\mathcal{F}\ell_{G,\mu^{-1}}^{b}}$${\left[\mathop{\rm
Spd}(C)/\mathcal{J}_{b}\right]}$

Let
$\mathrm{Sht}(G,b,\mu)_{\infty,C}\times^{\mathcal{J}_{b}}\mathfrak{Ig}_{C}^{b}$
denote the quotient of
$\mathrm{Sht}(G,b,\mu)_{\infty,C}\times_{C}\mathfrak{Ig}_{C}^{b}$ by
$\\{(jx,j^{-1}y):j\in\mathcal{J}_{b},x\in\mathrm{Sht}(G,b,\mu)_{\infty,C},y\in\mathfrak{Ig}_{C}^{b}\\}$.
To see that the diagram (9) above is Cartesian, observe that (11) implies we
have an isomorphism

$\mathcal{F}\ell^{b}_{G,\mu^{-1}}\times_{[\mathop{\rm
Spd}(C)/\mathcal{J}_{b}]}[\mathfrak{Ig}_{C}^{b}/\mathcal{J}_{b}]\simeq\mathrm{Sht}(G,b,\mu)_{\infty,C}\times^{\mathcal{J}_{b}}\mathfrak{Ig}_{C}^{b}.$

Moreover, we see, by [CS17, Corollary 4.3.19,Lemma 4.3.20], that we have an
isomorphism

(12)
$\mathrm{Sht}(G,b,\mu)_{\infty,C}\times_{C}\mathfrak{Ig}_{C}^{b}\simeq\mathcal{S}^{b,\circ,\mathrm{rd}}_{K^{p}}\times_{\mathcal{F}\ell_{G,\mu^{-1}}}\mathrm{Sht}(G,b,\mu)_{\infty,C},$

and again applying (11) implies that
$\mathcal{S}^{b,\circ,\mathrm{rd}}_{K^{p}}\times_{\mathcal{F}\ell_{G,\mu^{-1}}}\mathcal{F}\ell_{G,\mu^{-1}}^{b}$
is isomorphic to the quotient of
$\mathcal{S}^{b,\circ,\mathrm{rd}}_{K^{p}}\times_{\mathcal{F}\ell_{G,\mu^{-1}}}\mathrm{Sht}(G,b,\mu)_{\infty,C}$
by the action of $\mathcal{J}_{b}$ (here $\mathcal{J}_{b}$ acts via the action
on the second factor). Thus, we have an isomorphism:

$\mathrm{Sht}(G,b,\mu)_{\infty,C}\times^{\mathcal{J}_{b}}\mathfrak{Ig}_{C}^{b}\simeq\mathcal{S}^{b,\circ,\mathrm{rd}}_{K^{p}}\times_{\mathcal{F}\ell_{G,\mu^{-1}}}\mathcal{F}\ell_{G,\mu^{-1}}^{b}.$

This gives us the Cartesian diagram (10). Now, the natural map,

$\mathcal{S}_{K^{p},C}^{b,\circ,\mathrm{rd}}\times_{\mathcal{F}\ell_{G,\mu^{-1}}}\mathcal{F}\ell_{G,\mu^{-1}}^{b}\hookrightarrow\mathcal{S}^{b,\circ}_{K^{p},C}$

is a qcqs open immersion, which is an isomorphism on rank $1$ points. In turn,
it induces an isomorphism of canonical compactifications over the partially
proper strata $\mathcal{F}\ell_{G,\mu^{-1}}^{b}$. Therefore, by passing to
canonical compactifications over $\mathcal{F}\ell_{G,\mu^{-1}}^{b}$, we deduce
that diagram (10) is also Cartesian. ∎

We now invoke a result of Zhang [Zha23].

######  Theorem 3.10.

[Zha23, Theorem 1.3] Assuming 1.11, for all $b\in B(G,\mu)$ the Cartesian
diagram (10) extends to a Cartesian diagram

(13)
${\left[\mathcal{S}^{b,*}_{K^{p},C}/\underline{G(\mathbb{Q}_{p})}\right]}$${\left[\mathcal{F}\ell^{b}_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}\right]}$${\left[\overline{\mathfrak{Ig}}_{C}^{b,*}/\mathcal{J}_{b}\right]}$${\left[\mathop{\rm
Spd}(C)/\mathcal{J}_{b}\right]}$$\scriptstyle{\pi_{\mathrm{HT}}^{b,*}}$$\scriptstyle{\phantom{}{}^{*}\tilde{h}_{b}^{\leftarrow}}$$\scriptstyle{h^{\leftarrow}_{b}}$$\scriptstyle{\pi_{\mathfrak{Ig}}^{b,*}}$

of $v$-stacks.

######  Remark 3.11.

In fact Zhang shows a much stronger claim, that there exists a series of
larger Cartesian diagrams living over $\mathrm{Bun}_{G.C}$ such that the
diagrams (10) and (13), are the base-change along the inclusions
$j_{b}:\mathrm{Bun}_{G,C}^{b}\hookrightarrow\mathrm{Bun}_{G,C}$ of HN-strata
for $b\in B(G,\mu)$ varying.

######  Remark 3.12.

The rough idea behind proving this is to apply a relative $\mathop{\rm Spa}$
construction in the category of diamonds to the horizontal maps of the diagram
(13) by invoking Hartogs’ principle, as in Proposition 3.8.

We now state the key Corollary that we will need.

######  Corollary 3.13.

Assuming 1.11, for all $b\in B(G,\mu)$ we have a Cartesian diagram

(14)
${\left[\mathcal{S}^{b}_{K^{p},C}/\underline{G(\mathbb{Q}_{p})}\right]}$${\mathcal{F}\ell^{b}_{G,\mu^{-1}}}$${\left[(\overline{\mathfrak{Ig}}_{C}^{b,*}\setminus\overline{\partial\mathfrak{Ig}}_{C}^{b,*})/\mathcal{J}_{b}\right]}$${\left[\mathop{\rm
Spd}(C)/\mathcal{J}_{b}\right].}$$\scriptstyle{\pi_{\mathrm{HT}}^{b}}$$\scriptstyle{\phantom{}{}^{\partial}\tilde{h}_{b}^{\leftarrow}}$$\scriptstyle{h^{\leftarrow}_{b}}$$\scriptstyle{\pi_{\mathfrak{Ig}}^{b,\partial}}$

###### Proof.

This follows from the Cartesian diagram (13) and Corollary 3.5. ∎

By the Cartesian diagram (14), if we look at the sheaf

$i_{b}^{*}R\pi_{\mathrm{HT}!}(\overline{\mathbb{F}}_{\ell})$

we can see that this is canonically identified with

$h_{b}^{\leftarrow*}R\pi_{\mathfrak{Ig}!}^{b,\partial}(\overline{\mathbb{F}}_{\ell})$

via proper base change. Moreover, we can identify
$R\pi_{\mathfrak{Ig}!}^{b,\partial}(\overline{\mathbb{F}}_{\ell})$ simply with
the complex
$V_{b}:=R\Gamma_{c-\partial}(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell})$ of
$J_{b}(\mathbb{Q}_{p})$-modules under the identification
$\mathrm{D}(\mathrm{Bun}_{G}^{b},\overline{\mathbb{F}}_{\ell})\simeq\mathrm{D}(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})$,
as in Corollary 3.6. We can further refine this using the following lemma.

######  Lemma 3.14.

We have isomorphisms

$i_{b!}h_{b}^{\leftarrow*}(V_{b})\simeq h^{\leftarrow*}j_{b!}(V_{b})$

and

$i_{b*}h_{b}^{\leftarrow*}(V_{b})\simeq h^{\leftarrow*}j_{b*}(V_{b})$

of sheaves on $\mathcal{F}\ell_{G,\mu^{-1}}$.

###### Proof.

The first isomorphism follows from proper base-change. For the second
isomorphism, we note that

$h^{\leftarrow}:[\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}]\rightarrow\mathrm{Bun}_{G}$

is cohomologically smooth separated and representable in locally spatial
diamonds; therefore, the result follows by smooth base-change [Sch18,
Proposition 23.16 (2)]. ∎

######  Remark 3.15.

In particular, the graded pieces of the filtration

$R\Gamma([\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}],i_{b!}i_{b}^{*}(R\pi_{\mathrm{HT}!}(\overline{\mathbb{F}}_{\ell})))$

on the cohomology of the Shimura variety are identified with
$h^{\rightarrow}_{*}h^{\leftarrow*}j_{b!}(V_{b})\in\mathrm{D}(G(\mathbb{Q}_{p}),\overline{\mathbb{F}}_{\ell})$,
and similarly for
$R\Gamma([\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}],i_{b*}i_{b}^{*}(R\pi_{\mathrm{HT}!}(\overline{\mathbb{F}}_{\ell})))$
and $h^{\rightarrow}_{*}h^{\leftarrow*}j_{b*}(V_{b})$.

All in all, we get the following.

######  Proposition 3.16.

Assuming 1.11, we have a filtration on
$R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\overline{\mathbb{F}}_{\ell})$
by complexes of smooth representations of $G(\mathbb{Q}_{p})$, with graded
pieces isomorphic to $h^{\rightarrow}_{*}h^{\leftarrow*}j_{b!}(V_{b})$, where
$V_{b}\simeq
R\Gamma_{c-\partial}(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell})$.

The functor $h^{\rightarrow}_{*}h^{\leftarrow*}(-)$ appearing on the graded
pieces is manifestly related to the action on
$\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$ by Hecke
operators. In particular, for each geometric dominant cocharacter
$\mu\in\mathbb{X}_{*}(T_{\overline{\mathbb{Q}}_{p}})^{+}$, we have a
correspondence

${\mathrm{Hck}_{G,\leq\mu}}$${\mathrm{Bun}_{G}}$${\mathrm{Bun}_{G}\times\mathop{\rm
Spd}(C)}$$\scriptstyle{h_{\mu}^{\leftarrow}}$$\scriptstyle{h_{\mu}^{\rightarrow}}$

where $\mathrm{Hck}_{G,\leq\mu}$ is the stack parametrizing modifications
$\mathcal{E}_{1}\rightarrow\mathcal{E}_{2}$ of a pair of $G$-bundles with
meromorphy bounded by $\mu$ over the fixed untilt defined by $C$. We define
the Hecke operator [FS21, Section IX.2]

$T_{\mu}:\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})\rightarrow\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})^{BW_{E_{\mu}}}$
$A\mapsto
h_{\mu*}^{\rightarrow}(h_{\mu}^{\leftarrow*}(A)\otimes^{\mathbb{L}}\mathcal{S}_{\mu})$

where $E_{\mu}$ is the reflex field of $\mu$ and $\mathcal{S}_{\mu}$ is a
sheaf on $\mathrm{Hck}_{G,\leq\mu}$ attached to the highest weight tilting
module $\mathcal{T}_{\mu}$ by geometric Satake444We note that, using [FS21,
Proposition VII.5.2], we can replace the natural push-forward in the category
of solid sheaves with the * push-forward in the usual category of étale
$\overline{\mathbb{F}}_{\ell}$-sheaves when defining the Hecke operator.. Here
$E_{\mu}$ denotes the reflex field of $\mu$.

If we now let $\mu$ be the minuscule cocharacter appearing above then the
Bialynicki-Birula map gives an isomorphism of diamonds between the open locus
of $\mathrm{Hck}_{G,\leq\mu}$ where $\mathcal{E}_{1}$ is isomorphic to the
trivial bundle and
$[\mathcal{F}\ell_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}]$, which
identifies $h_{\mu}^{\rightarrow}$ (resp. $h_{\mu}^{\leftarrow})$) with
$h^{\rightarrow}$ (resp. $h^{\leftarrow}$). Moreover, this is a
cohomologically smooth space of dimension $d:=\langle 2\rho_{G},\mu\rangle$,
and we have an isomorphism
$\mathcal{S}_{\mu}\simeq\overline{\mathbb{F}}_{\ell}[d](\frac{d}{2})$555This
is true for the highest weight module $V_{\mu}$ and this agrees with the
highest weight tilting module $\mathcal{T}_{\mu}$, since $\mu$ is minuscule..
It follows, by proper base-change, that we have an isomorphism

$h^{\rightarrow}_{*}h^{\leftarrow*}j_{b!}(V_{b})\simeq
j_{1}^{*}T_{\mu}(j_{b!}(V_{b}))[-d](-\frac{d}{2})$

of $G(\mathbb{Q}_{p})\times W_{E_{\mathfrak{p}}}$-modules, where $1\in B(G)$
is the trivial element. The $W_{E_{\mathfrak{p}}}$-equivariance follows since
the above Cartesian diagrams all descend to $\breve{E}_{\mathfrak{p}}$ and are
also compatible with the Frobenius descent datum on
$\mathrm{Sht}(G,b,\mu)_{\infty,C}\rightarrow\mathop{\rm
Spd}(\breve{E}_{\mathfrak{p}})$ (This is true for (9) by the results of [CS17]
and all the other diagrams are constructed from this one). This gives us
Theorem 1.13.

######  Corollary 3.17.

Assuming 1.11, the complex
$R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p},C},\overline{\mathbb{F}}_{\ell})$
has a $G(\mathbb{Q}_{p})\times W_{E_{\mathfrak{p}}}$-equivariant filtration
with graded pieces given by
$j_{1}^{*}T_{\mu}(j_{b!}(V_{b}))[-d](-\frac{d}{2})$ for varying $b\in
B(G,\mu)$, where $V_{b}\simeq
R\Gamma_{c-\partial}(\mathrm{Ig}^{b},\overline{\mathbb{F}}_{\ell})$.

Moreover, we obtain that each graded piece is isomorphic to

$(R\Gamma_{c}(G,b,\mu)\otimes^{\mathbb{L}}_{\mathcal{H}(J_{b})}V_{b})[2d_{b}]$

as $G(\mathbb{Q}_{p})\times W_{E_{\mathfrak{p}}}$-modules. Here

$R\Gamma_{c}(G,b,\mu):=\mathop{\rm
colim}\nolimits_{K_{p}\rightarrow\\{1\\}}R\Gamma_{c}(\mathrm{Sht}(G,b,\mu)_{\infty,C}/\underline{K_{p}},\overline{\mathbb{F}}_{\ell}(d_{b}))$

is a complex of $G(\mathbb{Q}_{p})\times J_{b}(\mathbb{Q}_{p})\times
W_{E_{\mathfrak{p}}}$-modules, where $\mathrm{Sht}(G,b,\mu)_{\infty,C}$ is as
defined above, and $\overline{\mathbb{F}}_{\ell}(d_{b})$ is the sheaf with
trivial Weil group action and $J_{b}(\mathbb{Q}_{p})$-action given as in
[Kos21, Lemma 7.4].

###### Proof.

It remains to explain the description of
$j_{1}^{*}T_{\mu}(j_{b!}(V_{b}))[-d](-\frac{d}{2})$. By applying the second
part of [Ham22, Proposition 11.12] and noting that
$\mathcal{S}_{\mu}\simeq\overline{\mathbb{F}}_{\ell}[d](\frac{d}{2})$ since
$\mu$ is minuscule (where we recall that, since $\mu$ is minuscule, the
representation $\mathcal{T}_{\mu}$ agrees with the usual highest weight
representation), we obtain that the graded pieces are isomorphic to

$\mathop{\rm
colim}\nolimits_{K_{p}\rightarrow\\{1\\}}(R\Gamma_{c}(\mathrm{Sht}(G,b,\mu)_{\infty,C}/\underline{K_{p}},\overline{\mathbb{F}}_{\ell})\otimes^{\mathbb{L}}_{\mathcal{H}(J_{b})}V_{b}\otimes\kappa^{-1})[2d_{b}]$

as desired, where $\kappa$ is the character of $J_{b}(\mathbb{Q}_{p})$ defined
by the action of $J_{b}(\mathbb{Q}_{p})$ on the compactly supported cohomology
of the $\ell$-adically constractible group diamond $\mathcal{J}_{b}^{>0}$,
where $\mathcal{J}_{b}\simeq\mathcal{J}_{b}^{>0}\ltimes J_{b}(\mathbb{Q}_{p})$
is the semi-direct product structure given by allowing
$\mathrm{Aut}(\mathcal{E}_{b})$ to act on its canonical reduction. However, by
combining this with [Kos21, Lemma 7.6] and its proof, we can rewrite this as

$(\mathop{\rm
colim}\nolimits_{K_{p}\rightarrow\\{1\\}}R\Gamma_{c}(\mathrm{Sht}(G,b,\mu)_{\infty,C}/\underline{K_{p}},\overline{\mathbb{F}}_{\ell}(d_{b}))\otimes^{\mathbb{L}}_{\mathcal{H}(J_{b})}V_{b})[2d_{b}],$

as desired. ∎

##  4\. The Local Results

###  4.1. The Spectral Action

Let $G/\mathbb{Q}_{p}$ be a quasi-split connected reductive group with a
choice of Borel $B$ and maximal torus $T$ as before. We will work with
$\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$, the derived
category of étale $\overline{\mathbb{F}}_{\ell}$-sheaves on the moduli stack
of $G$-bundles. Our goal in this section will be to describe a localization
$\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})_{\phi_{\mathfrak{m}}}\subset\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$
for $\mathfrak{m}\subset H_{K_{p}^{\mathrm{hs}}}$ a generic maximal ideal with
associated semi-simple L-parameter $\phi_{\mathfrak{m}}$ in the case that $G$
is unramified. We will do this in slightly more generality using the spectral
action [FS21, Section X.2]. We assume that $\ell$ is very good as in [FS21,
Page 33], and consider the moduli stack $\mathfrak{X}_{\hat{G}}/\mathop{\rm
Spec}{\overline{\mathbb{F}}_{\ell}}$ of $\overline{\mathbb{F}}_{\ell}$-valued
Langlands parameters, as defined in [Dat+20, Zhu20]. We let
$\mathrm{Perf}(\mathfrak{X}_{\hat{G}})$ denote the derived category of perfect
complexes on this stack, and we write
$\mathrm{Perf}(\mathfrak{X}_{\hat{G}})^{BW_{\mathbb{Q}_{p}}^{I}}$ for the
derived category of objects with a continuous $W_{\mathbb{Q}_{p}}^{I}$ action
for a finite index set $I$, and
$\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})^{\omega}$ for the
triangulated sub-category of compact objects in
$\mathrm{D}(\mathrm{Bun}_{G},\overline{\mathbb{F}}_{\ell})$. By [FS21,