# Monodromy and irreducibility of Igusa varieties

Pol van Hoften and Luciena Xiao Xiao

###### Abstract.

We determine the irreducible components of Igusa varieties for Shimura
varieties of Hodge type and use that to determine the irreducible components
of central leaves. In particular, we show that the discrete Hecke-orbit
conjecture is false in general. Our method combines recent work of D’Addezio
on monodromy of compatible local systems with a generalisation of a method of
Hida, using the Honda-Tate theory for Shimura varieties of Hodge type
developed by Kisin–Madapusi Pera–Shin. We also determine the irreducible
components of Newton strata in Shimura varieties of Hodge type by combining
our results with recent work of Zhou-Zhu.

Pol van Hoften was supported by the Engineering and Physical Sciences Research
Council [EP/L015234/1], The EPSRC Centre for Doctoral Training in Geometry and
Number Theory (The London School of Geometry and Number Theory), University
College London and King’s College London. L.X. Xiao was supported by ERC
Advanced grant 742608 “GeoLocLang”, UMR 7586 IMJ-PRG and CNRS

##  1\. Introduction

Let $N\geq 4$, let $p$ be a prime number coprime to $N$, let $Y_{1}(N)$ be the
modular curve of level $\Gamma_{1}(N)$ over $\mathbb{F}_{p}$ and let
$Y_{1}(N)^{\text{ord}}$ be the ordinary locus. There is a tower of finite
étale covers (see [Igusa]) $\operatorname{Ig}_{m}\to Y_{1}(N)^{\text{ord}}$
with Galois groups $(\mathbb{Z}/p^{m}\mathbb{Z})^{\times}$, and we let
$\operatorname{Ig}_{\text{ord}}\to Y_{1}(N)^{\text{ord}}$ be their limit; it
is a classical result due to Igusa that $\operatorname{Ig}_{\text{ord}}$ is
irreducible.

Igusa varieties exist more generally as profinite étale covers of central
leaves in the special fibers of Shimura varieties of Hodge type (c.f.
[HamacherKim]*Section 5; see [MantovanPEL] for the PEL case). Understanding
the irreducible components of these Igusa varieties has important consequences
for the theory of $p$-adic automorphic forms. For example, in the work of
Eischen-Mantovan [eischen2017p] on $p$-adic automorphic forms for unitary
Shimura varieties, the irreducibility of Igusa varieties is assumed
throughout.

The irreducibility of Igusa varieties was proven for Siegel modular varieties
by Chai-Oort [ChaiOort], and their proof works more generally for Shimura
varieties of PEL type when hypersymmetric points exist (c.f.
[hida2011irreducibility, eischen2017p]). We would like to point out that even
in the $\mu$-ordinary locus, hypersymmetric points often do not exist (see
[LXXiao]*Corollary 7.5.). Hida [Hida] proves the irreducibility of the
ordinary Igusa tower over Shimura varieties of PEL type A and C without using
hypersymmetric points. Our results are the first to treat Hodge type Shimura
varieties and Igusa varieties over general central leaves (but see Remark
1.1.2); they are even new for the $\mu$-ordinary locus in many PEL type cases.

###  1.1. Main results

Let $(G,X)$ be a Shimura datum of Hodge type, let $p>2$ be a prime number, let
$U^{p}\subset G(\mathbb{A}_{f}^{p})$ be a sufficiently small compact open
subgroup and let $U_{p}\subset G(\mathbb{Q}_{p})$ be a hyperspecial subgroup.
Moreover, let $\operatorname{Sh}_{G}$ be the basechange to
$\overline{\mathbb{F}}_{p}$ of the canonical integral model of the Shimura
variety of level $U=U^{p}U_{p}$ over $\mathcal{O}_{E,v}$ for some prime $v\mid
p$ of the reflex field $E$ of $(G,X)$. Let $\operatorname{Sh}_{G,b}$ be a non-
basic Newton stratum and let $\operatorname{Ig}_{b}$ be the Igusa variety
associated to $b$.

######  Theorem 1.

Assume that $G^{\text{der}}$ is simply connected, that $G^{\text{ad}}$ is
$\mathbb{Q}$-simple and that $p$ does not divide the order of the Weyl group
$W_{G}$ of $G$. Then there is a bijection (where $G^{\text{ab}}$ is the
maximal abelian quotient)

(1)
$\displaystyle\pi_{0}(\operatorname{Ig}_{b})\cong\pi_{0}(\operatorname{Sh}_{G})\times
G^{\text{ab}}(\mathbb{Z}_{p}).$

In the case of the modular curve, the ordinary Igusa variety
$\operatorname{Ig}_{b}$ is a $(\mathbb{Z}_{p}^{\times})^{2}$-torsor over the
ordinary locus, and our theorem says us that its connected components are in
bijection with $\mathbb{Z}_{p}^{\times}$; here $(\mathbb{Z}_{p}^{\times})^{2}$
acts on $\mathbb{Z}_{p}^{\times}$ via the product map. This recovers the
result of Igusa from the introduction, because the Igusa tower
$\operatorname{Ig}_{\text{ord}}$ introduced there is the inverse image of $1$
under $\operatorname{Ig}_{b}\to\mathbb{Z}_{p}^{\times}$.

We can use Theorem 1 to determine the irreducible (equivalently, connected)
components of central leaves. Let $C\subset\operatorname{Sh}_{G,b}$ be a
central leaf and let $\operatorname{Ig}_{b}\to C$ be the Igusa variety over
$C$, this is a pro-étale torsor for a compact open subgroup $H_{C}\subset
J_{b}(\mathbb{Q}_{p})$, where $J_{b}$ is the twisted centraliser attached to
$b$ (see Section 2).

######  Corollary 1.1.1.

There is a bijection

(2)
$\displaystyle\pi_{0}(C)\simeq\pi_{0}(\operatorname{Sh}_{G,b})\times\frac{G^{\text{ab}}(\mathbb{Z}_{p})}{H_{C}^{\text{ab}}},$

where $H_{C}^{\text{ab}}$ is the image of $H_{C}$ in
$G^{\text{ab}}(\mathbb{Z}_{p})$ via the natural map $J_{b}\to G^{\text{ab}}$.
The group $H_{C}^{\text{ab}}$ is of finite index in
$G^{\text{ab}}(\mathbb{Z}_{p})$ because it always contains the image of
$Z_{G}(\mathbb{Z}_{p})$.

The discrete part of the Hecke orbit conjecture (c.f. [ChaiOort2, LXXiao])
predicts that $\pi_{0}(C)\simeq\pi_{0}(\operatorname{Sh}_{G})$. In an earlier
version of [vanHoften], the first-named author claimed to have proven this
conjecture; however, the supplied proof contained an error. Using Corollary
6.2.2, the conjecture comes down to the equality
$H_{C}^{\text{ab}}=G^{\text{ab}}(\mathbb{Z}_{p})$ for all central leaves C. We
will show that if $H_{C}$ is a parahoric subgroup, then
$H_{C}^{\text{ab}}=G^{\text{ab}}(\mathbb{Z}_{p})$, this is also used in our
proof of Theorem 1. We will also give an example, due to Rong Zhou, that shows
that this equality does not always hold. In particular, the discrete Hecke
orbit conjecture is false in general.

Our second main result is about irreducible components of Newton strata.

######  Theorem 2.

Assume that $G^{\text{der}}$ is simply connected, that $G^{\text{ad}}$ is
$\mathbb{Q}$-simple and that $p$ does not divide the order of the Weyl group
$W_{G}$ of $G$. Then the natural map

(3)
$\displaystyle\pi_{0}(\operatorname{Sh}_{G,b})\to\pi_{0}(\operatorname{Sh}_{G})$

is a bijection, and the number of irreducible components in each connected
component of $\operatorname{Sh}_{G,b}$ is given by the representation
theoretic constant

(4) $\displaystyle\operatorname{Dim}V_{\mu}(\lambda_{b})_{\text{rel}},$

introduced in [ZhouZhu]. In particular, when $G_{\mathbb{Q}_{p}}$ is split,
the connected components of $\operatorname{Sh}_{G,b}$ are irreducible.

Our proof uses the Mantovan product-formula [MantovanPEL], due to Hamacher-Kim
[HamacherKim] in this generality, and results of Chen-Kisin-Viehmann
[ChenKisinViehmann] and Zhou-Zhu [ZhouZhu]. Theorems 1 and 2 were proven for
Siegel modular varieties by Chai and Oort in their seminal paper [ChaiOort].
Amusingly, they prove irreducibility of Newton strata first, irreducibility of
central leaves second and irreducibility of Igusa varieties last. Their proof
of the Hecke orbit conjecture depends on the existence of hypersymmetric
points.

Our proofs also work when $G_{\mathbb{Q}_{p}}$ is only assumed to be quasi-
split and split over a tamely ramified extension, under some additional
assumptions, see Theorems 6.2.1 and 6.2.4 for precise versions of our main
theorems in that generality. The assumption that $p$ does not divide the order
of $W_{G}$ is relaxed there. For example, this assumption is not necessary for
the Igusa tower over the $\mu$-ordinary locus. When $b$ is basic, the Igusa
variety $\operatorname{Ig}_{b}$ is zero-dimensional and highly reducible, in
which case it will have many more irreducible components than the Igusa
varieties for non-basic Newton strata. This means that the theorem is false
for products of Shimura varieties with $b$ basic in one factor and non-basic
in the other; this is where the assumption that $G^{\text{ad}}$ is
$\mathbb{Q}$-simple comes from.

######  Remark 1.1.2.

In recent work [KretShin], Kret and Shin also determine the irreducible
components of Igusa varieties and deduce results about the discrete Hecke
orbit conjecture.111Kret and Shin pointed out to us that there is a mistake in
their proof of Lemma 8.1.1 in the current version of their paper. This lemma
asserts that $H_{C}^{\text{ab}}=G^{\text{ab}}(\mathbb{Z}_{p})$ for all central
leaves $C$ and would imply the discrete Hecke orbit conjecture. Their proof
uses harmonic analysis and automorphic forms and is completely different from
ours. They compute the $0$-th étale cohomology of $\operatorname{Ig}_{b}$ as a
representation of $G(\mathbb{A}_{f}^{p})\times J_{b}(\mathbb{Q}_{p})$ using
the Langlands-Kottwitz method ([KretShin]*Theorem A), and then determine the
irreducible components of $\operatorname{Ig}_{b}$ using that computation. It
should be possible to recover their computation of the $0$-th étale cohomology
from Theorem 1; we plan to include this in future work.

###  1.2. Strategy

####  1.2.1. Setup

The Igusa variety is a profinite étale cover $\operatorname{Ig}_{b}\to C$ of a
central leaf $C$ inside the Newton stratum $\operatorname{Sh}_{G,b}$. To be
precise, it is a torsor for a compact open subgroup $H_{C}\subset
J_{b}(\mathbb{Q}_{p})$, where $J_{b}$ is the twisted centraliser attached to
$b$ (see Section 2).

There are many different central leaves $C$ inside the Newton stratum, and
they all have the same Igusa variety $\operatorname{Ig}_{b}$ but the groups
$H=H_{C}$ depend on $C$. For the purposes of our proof we will always choose
$C$ to be a distinguished central leaf, i.e., a central leaf that is also an
Ekedahl-Oort stratum; these exist by Theorem D of [ShenZhang]. Under the
assumptions of our main theorem, these distinguished central leaves are
irreducible. To be precise, [vanHoften]*Theorem 2 tells us that

(5) $\displaystyle\pi_{0}(C)\to\pi_{0}(\operatorname{Sh}_{G})$

is a bijection. Therefore the main theorem would follow if we could show that
the fibers of $\pi_{0}(\operatorname{Ig}_{b})\to\pi_{0}(C)$ are in bijection
with $G^{\text{ab}}(\mathbb{Z}_{p})$, equivariant for the action of $H$.

The algebraic group $J_{b}$ is an inner form of a Levi subgroup of $G$, and
therefore has a surjective map $J_{b}\to G^{\text{ab}}$. Let $J_{b}^{\prime}$
be the kernel of this map and let $H^{\prime}=H\cap
J_{b}^{\prime}(\mathbb{Q}_{p})$. In the last section, we will show that
$H/H^{\prime}$ is isomorphic to $G^{\text{ab}}(\mathbb{Z}_{p})$ and therefore
it suffices to show that the stabiliser of a component
$\pi_{0}(\operatorname{Ig}_{b})$ under the action of $H$ is given by
$H^{\prime}$.

From now on we work with a fixed connected component
$\operatorname{Sh}_{G}^{\circ}$ of the Shimura variety, and hence a fixed
connected component $\operatorname{C}^{\circ}$ of $C$. The $H$-torsor
$\operatorname{Ig}_{b}^{\circ}\to C^{\circ}$ corresponds to a morphism

(6)
$\displaystyle\rho_{\operatorname{Ig}}:\pi_{1}^{\text{\'{e}t}}(C^{\circ})\to
H$

and showing that $H^{\prime}$ acts trivially on
$\pi_{0}(\operatorname{Ig}_{b})$ is the same as showing that the image of
monodromy contains $H^{\prime}$. Let $N$ be this image and let $\mathcal{N}$
be its Zariski closure inside $J_{b}(\overline{\mathbb{Q}}_{p})$, which is an
algebraic group over $\overline{\mathbb{Q}}_{p}$.

It will follow from [vanHoften]*Theorem 2 and two results of D’Addezio
[d2020monodromy, AdezzioII] that $\mathcal{N}=J_{b}^{\prime}$. This implies
that $N$ is a compact open subgroup of $J_{b}^{\prime}(\mathbb{Q}_{p})$, and
therefore that $N\cap H^{\prime}$ is of finite index in both $N$ and
$H^{\prime}$. This proves the theorem up to a finite error, but it is a finite
error over which we have absolutely no control.

####  1.2.2. Outline of the proof

In order to show that $N=H^{\prime}$, we will make use of the fact that the
action of $H$ on $\operatorname{Ig}_{b}$ extends to an action of
$J_{b}(\mathbb{Q}_{p})$. If we could show that the action of
$J_{b}^{\prime}(\mathbb{Q}_{p})$ on $\pi_{0}(\operatorname{Ig}_{b})$ was
trivial, then we would know that $N=H\cap
J_{b}^{\prime}(\mathbb{Q}_{p})=H^{\prime}$.

In Section 3, we show via a group-theoretic argument that the isotropic
factors of $J_{b}^{\text{der}}(\mathbb{Q}_{p})$ act trivially on
$\pi_{0}(\operatorname{Ig}_{b})$. The main ingredients are the equality
$\mathcal{N}=J_{b}^{\prime}$ mentioned above and the fact that the
$\mathbb{Q}_{p}$-points of semisimple and simply connected isotropic groups
have no nontrivial finite quotients.

We show in Section 4 that $T^{\prime}(\mathbb{Q}_{p})$ acts trivially on
$\pi_{0}(\operatorname{Ig}_{b})$, where $T^{\prime}$ is a maximal torus of
$J_{b}^{\prime}$. In fact, we can show this for many maximal tori of
$J_{b}^{\prime}$. This is done by generalising an argument of Hida [Hida]
using the Honda-Tate theory for Shimura varieties of [KMPS].

In Section 5.1, we show that the anisotropic factors of
$J_{b}^{\text{der}}(\mathbb{Q}_{p})$ are generated by
$T_{1}^{\text{der}}(\mathbb{Q}_{p})$ and $T_{2}^{\text{der}}(\mathbb{Q}_{p})$
for some well chosen maximal tori $T_{1},T_{2}\subset J_{b}$. Our arguments
here are completely explicit, relying on the classification of semisimple and
simply connected anisotropic groups over local fields.

At this point we have shown that $J_{b}^{\text{der}}(\mathbb{Q}_{p})$ acts
trivially on $\pi_{0}(\operatorname{Ig}_{b})$, and that we can find many
maximal tori of $J_{b}^{\prime}(\mathbb{Q}_{p})$ that act trivially on
$\pi_{0}(\operatorname{Ig}_{b})$. Our proof ends with Section 5.2 where we
show that $J_{b}^{\prime}(\mathbb{Q}_{p})$ is generated by
$J_{b}^{\text{der}}(\mathbb{Q}_{p})$ and
$T_{1}(\mathbb{Q}_{p}),\cdots,T_{n}(\mathbb{Q}_{p})$ where
$T_{1},\cdots,T_{n}$ are (well-chosen) maximal tori of $J_{b}$.

In Section 6.1 we identify the quotient $H_{C}/H^{\prime}_{C}$ with
$G^{\text{ab}}(\mathbb{Z}_{p})$ for our particular choice of central leaf $C$,
and in Section 6.2 we state our main theorem in maximum generality.

##  2\. Shimura varieties and Igusa varieties

Let $(G,X)$ be a Shimura datum of Hodge type with $G^{\text{der}}$ simply
connected and let $p>2$ be a prime number such that $G_{\mathbb{Q}_{p}}$ is
quasi-split and splits over a tamely ramified extension. Let $U^{p}\subset
G(\mathbb{A}_{f}^{p})$ be a sufficiently small compact open subgroup and let
$U_{p}\subset G(\mathbb{Q}_{p})$ be a very special subgroup; in particular,
$U_{p}$ is hyperspecial when $G_{\mathbb{Q}_{p}}$ is unramified. Kisin
[kisin2010integral] and Kisin-Pappas [KisinPappas] (in the ramified case)
showed the existence of the canonical integral model of the Shimura variety of
level $U=U^{p}U_{p}$ over $\mathcal{O}_{E,v}$ for any prime $v\mid p$ of the
reflex field $E$ of $(G,X)$ 222In the ramified case, these are canonical in
the sense of [Pappas].. Let $\operatorname{Sh}_{G}$ denote the base change to
$\overline{\mathbb{F}}_{p}$ of this canonical model.

The canonical model $\operatorname{Sh}_{G}$ comes equipped with the Newton
stratification, indexed by the finite poset $B(G,\mu)$ which has a unique
minimal element called the _basic_ element. For the rest of this paper, we fix
a non-basic Newton stratum $\operatorname{Sh}_{G,b}$ on
$\operatorname{Sh}_{G}.$ Let $J_{b}$ denote the twisted centraliser of $b,$ an
algebraic group over $\mathbb{Q}_{p}$ defined by

$J_{b}(R)=\\{g\in
G(R\otimes_{\mathbb{Q}_{p}}\breve{\mathbb{Q}}_{p})|gb\sigma(g^{-1})=b\\},$

for any $\mathbb{Q}_{p}$-algebra $R$, where $\breve{\mathbb{Q}}_{p}$ is the
completion of a maximal unramified extension of $\mathbb{Q}_{p}$. The
algebraic group $J_{b}$ is an inner form of a Levi subgroup of $G$ and
therefore there is a morphism (where $G^{\text{ab}}$ is the maximal abelian
quotient of $G$)

(7) $\displaystyle J_{b}\to G^{\text{ab}}$

of algebraic groups over $\mathbb{Q}_{p}$, we let $J_{b}^{\prime}$ be its
kernel.

Let $(G,X)\hookrightarrow(\operatorname{GSp},S^{\pm})$ denote a choice of
embedding of $(G,X)$ into the Shimura datum for the symplectic group, chosen
as in [HamacherKim]*Section 2.3. This determines a finite morphism from
$\operatorname{Sh}_{G}$ to $\operatorname{Sh}_{\operatorname{GSp}}$, the
special fiber of the Shimura variety for $(\operatorname{GSp},S^{\pm})$ of
level $U^{\prime}=U^{\prime p}U^{\prime}_{p}$ with $U_{p}^{\prime}$
hyperspecial.333When $U_{p}$ is hyperspecial, the main result of
[xu2020normalization] tells us that this is a closed immersion. In particular,
this gives us a ‘universal’ abelian variety $\pi:A\to\operatorname{Sh}_{G}$.

###  2.1. Central leaves

Given a Newton stratum $\text{Sh}_{\operatorname{GSp},b}$ on
$\text{Sh}_{\operatorname{GSp}}$, Oort came up with the idea of studying
central leaves, which are loci where the $p$-divisible group associated to the
universal abelian variety $A$ over $\text{Sh}_{\operatorname{GSp}}$ is
geometrically fiberwise constant. To be more precise, given a point
$x\in\text{Sh}_{\operatorname{GSp}}(\overline{\mathbb{F}}_{p})$, the central
leaf $C(x)$ passing through $x$ is defined as (where $\simeq_{g}$ denotes an
isomorphism of $p$-divisible groups respecting the polarisations up to a
scalar)

$C(x)=\\{y\in\text{Sh}_{\operatorname{GSp}}(\overline{\mathbb{F}}_{p})\>|\;A_{y}[p^{\infty}]\simeq_{g}A_{y}[p^{\infty}]\\}.$

Oort proves that $C(x)$ is the set of $\overline{\mathbb{F}}_{p}$-points of a
locally closed subvariety of $\text{Sh}_{\operatorname{GSp},b}$, and that it
is in fact smooth and equidimensional ([oort2004foliations], [MantovanPEL]).
Central leaves are closed inside $\text{Sh}_{\operatorname{GSp},b}$ and give
rise to a ‘foliation’ by partitioning $\text{Sh}_{\operatorname{GSp},b}$ into
a disjoint union of central leaves of the same dimension related to each other
by isogeny correspondences.

This construction has been generalised to Shimura varieties of Hodge type by
Hamacher in [hamacher2019product]*§2.3 (see also
[hamacher2017almost]*Definition 5 and [KimLeaves]); basically, ones takes the
locally closed subvariety where the $p$-divisible group with Hodge tensors is
geometrically fiberwise constant. Corollary 5.3.1 of [KimLeaves] states that
these central leaves, which we still denote by $C(x)$, are smooth and
equidimensional.

###  2.2. Igusa varieties

Let $C(x)$ be a fixed central leaf in $\operatorname{Sh}_{G}$ and write
$\mathbb{X}=A_{x}[p^{\infty}]$ for the $p$-divisible group with additional
structures associated to $x$. Let $C^{\prime}$ denote the corresponding
central leaf in $\text{Sh}_{\operatorname{GSp}}.$

In the case of Siegel modular varieties, the perfect Igusa variety
$\operatorname{Ig}^{\prime}\to C^{\prime}$ is the scheme over $C^{\prime}$
whose set of $R$-points, for a test scheme $R/C^{\prime}$, consists of
isomorphisms

(8) $\displaystyle A_{R}[p^{\infty}]\simeq\mathbb{X}_{R}$

respecting the polarisations up to a scalar. By construction,
$\operatorname{Ig}^{\prime}$ has an action of the group
$\operatorname{Aut}(\mathbb{X})$, where $\mathbb{X}$ is considered as a
$p$-divisible group with polarisation. It follows from
[CaraianiScholze]*Section 4.3 that $\operatorname{Ig}^{\prime}$ actually has
an action of the group $\operatorname{QIsog}(\mathbb{X})$ of self quasi-
isogenies of $\mathbb{X}$ and that $\operatorname{Ig}^{\prime}$ does not
depend on $x$ but only on the Newton stratum containing $x$. Moreover, it is
proved in loc. cit. that $\operatorname{Ig}^{\prime}$ is a perfect scheme and
that the morphism

(9) $\displaystyle\alpha:\operatorname{Ig}^{\prime}\to C^{\text{perf}}$

is a pro-étale $\operatorname{Aut}(\mathbb{X})$-torsor. To be precise,
$\operatorname{Aut}(\mathbb{X})$ is a profinite group, and so there is an
affine group scheme $\underline{\operatorname{Aut}(\mathbb{X})}$ whose
topological space of connected components is homeomorphic to
$\operatorname{Aut}(\mathbb{X})$. The morphism $\alpha$ is a torsor for this
group scheme, which is an inverse limit of finite étale group schemes. There
is also a deperfection of the Igusa tower (c.f. [MantovanPEL]), but it is more
difficult to construct. Since the underlying topological space of a scheme is
the same as that of its perfection, we will ignore this difference here.

Hamacher and Kim [HamacherKim]*§5.1 generalise this construction to Shimura
varieties of Hodge type. The Igusa variety $\operatorname{Ig}_{b}$ over $C(x)$
is defined as the subscheme of
$\operatorname{Ig}^{\prime}\times_{{\operatorname{Sh}_{G}}^{\text{perf}}}\text{Sh}_{\operatorname{GSp}}^{\text{perf}}$
consisting of those isomorphisms that respects the crystalline Tate-tensors on
the $p$-divisible groups. In this case, $\operatorname{Ig}_{b}\to C(x)$ is a
pro-étale $H$-torsor, where $H:=\operatorname{Aut}(\mathbb{X})\cap
J_{b}(\mathbb{Q}_{p})$ is a compact open subgroup of
$J_{b}(\mathbb{Q}_{p})\subset\operatorname{QIsog}(\mathbb{X})$. The action of
$H$ extends to an action of $J_{b}(\mathbb{Q}_{p})$, but this is not
compatible with the projection $\operatorname{Ig}_{b}\to C(x)$. As before, the
scheme $\operatorname{Ig}_{b}$ with its action of $J_{b}(\mathbb{Q}_{p})$ does
not depend on the choice of $x$; however, $H$ does. We will often choose $x$
such that $C(x)$ is an Ekedahl-Oort stratum (or EKOR stratum when $G$ is
ramified, see [ShenYuZhang]), in which case $H$ will be a parahoric subgroup
of $J_{b}(\mathbb{Q}_{p})$.

##  3\. Geometric Monodromy

Assume that the adjoint group $G^{\text{ad}}$ of $G$ is $\mathbb{Q}$-simple
from now on. Let $b\in B(G,\mu)$ be non-basic and let
$C(x)\subset\operatorname{Sh}_{G,b}$ be a central leaf in the Newton stratum
associated to $b$. Let $\ell\not=p$ be a prime number such that
$U^{p}_{\ell}=G(\mathbb{Z}_{\ell})$ is hyperspecial and let
$\pi:A\to\operatorname{Sh}_{G}$ be the universal abelian variety. The local
system $R^{1}\pi_{\ast}\mathbb{Z}_{\ell}$ induces a pro-étale
$G(\mathbb{Z}_{\ell})$-torsor over $\operatorname{Sh}_{G}$, which is just the
tower of Shimura varieties over $\operatorname{Sh}_{G}$ obtained by shrinking
the level at $\ell$.

The induced $G^{\text{ab}}(\mathbb{Z}_{\ell})$-torsor is pulled back from the
discrete Shimura variety for $(G^{\text{ab}},X^{\text{ab}})$ via the natural
map and is therefore trivial. Therefore on each connected component of
$\operatorname{Sh}_{G}$, the image of monodromy is contained in
$G^{\text{der}}(\mathbb{Z}_{\ell})$. In fact the image is equal to
$G^{\text{der}}(\mathbb{Z}_{\ell})$ because the connected components of the
Shimura variety are naturally identified with the discrete Shimura variety for
$(G^{\text{ab}},X^{\text{ab}})$ for each choice of level at $\ell$. To proceed
we will to assume the following:

######  Assumption 3.0.1.

The natural map

(10) $\displaystyle C(x)\to\operatorname{Sh}_{G}$

induces a bijection on connected components for all choices of prime-to-$p$
level $U^{p}$.

If $C(x)$ is a distinguished central leaf and if moreover $G_{\mathbb{Q}_{p}}$
splits over an unramified extension, this assumption holds true by
[vanHoften]*Theorem 2. It also holds for distinguished central leaves in the
ramified case under certain technical assumptions; this is made precise in the
statement of our main results in Section 6.

Since we can apply the assumption after shrinking the level at $\ell$, it
follows that the $\ell$-adic monodromy of $R^{1}\pi_{\ast}\mathbb{Z}_{\ell}$,
over every connected component $C(x)^{\circ}$ of the leaf $C(x)$, is equal to
$G^{\text{der}}(\mathbb{Z}_{\ell})$ and therefore has Zariski closure equal to
$G^{\text{der}}$.

######  Notation 3.0.2.

We will write $J_{b}^{\text{der}}=J_{b,\text{an}}^{\text{der}}\times
J_{b,\text{iso}}^{\text{der}}$ with the first factor anisotropic and the
second factor isotropic, and use the same notations for subgroups of
$J_{b}^{\text{der}}$.

######  Proposition 3.0.3.

Suppose that Assumption 3.0.1 holds, then
$J_{b,\text{iso}}^{\text{der}}(\mathbb{Q}_{p})$ acts trivially on
$\pi_{0}(\operatorname{Ig}_{b})$.

Let $N$ be the image of $\pi_{1}^{\text{\'{e}t}}(C(x)^{\circ})\to H$ for a
connected component $C(x)^{\circ}$ of $C(x)$. We first prove the following
lemma:

######  Lemma 3.0.4.

The Zariski closure of $N$ inside $J_{b}(\overline{\mathbb{Q}}_{p})$ is equal
to $J_{b}^{\prime}$.

###### Proof.

As noted above, the assumption tells us that the Zariski closure of the
$\ell$-adic monodromy of $R^{1}\pi_{\ast}\mathbb{Q}_{\ell}$ is equal to
$G^{\text{der}}$. The central leaf $C(x)$ is smooth by [KimLeaves]*Corollary
5.3.1 (c.f. [MantovanPEL]*Proposition 1 in the PEL case and Proposition 2.6 of
[hamacher2017almost] in the unramified Hodge type case). Then
[d2020monodromy]*Theorem 1.2.1 shows that the geometric monodromy group of the
overconvergent isocrystal $\mathcal{E}^{{\dagger}}$ associated to $\pi:A\to
C(x)^{\circ}$ is equal to the geometric monodromy group of
$R^{1}\pi_{\ast}\mathbb{Q}_{\ell}$ and therefore equal to $G^{\text{der}}$.
Theorem 1.1.1 of [AdezzioII] shows that the geometric monodromy group of the
underlying isocrystal $\mathcal{E}$ is equal to the parabolic subgroup
$P(\lambda)\subset G^{\text{der}}$, where $\lambda$ is a cocharacter of
$G^{\text{der}}$ coming from the slope filtration on $\mathcal{E}$. The
$J_{b}(\mathbb{Q}_{p})$-torsor induced from the $H$-torsor given by the Igusa
variety, is the ‘associated graded’ of the slope filtration on $\mathcal{E}$,
and its geometric monodromy group is therefore isomorphic to the Levi subgroup
$N(\lambda)$ of $P(\lambda)$, which is the same thing as $J_{b}^{\prime}$. ∎

###### Proof of Proposition 3.0.3.

Because $J_{b}^{\text{der}}$ is semi-simple, it follows that the Lie algebra
of the $p$-adic Lie group $N^{\text{der}}$ agrees with the Lie-algebra of the
algebraic group $J_{b}^{\text{der}}$ (see [Borel]*Corollary 7.9). Therefore
$N^{\text{der}}$ contains a compact open subgroup of
$J_{b}^{\text{der}}(\mathbb{Q}_{p})$ using the $p$-adic exponential map. It
follows that $N^{\text{der}}$ is of finite index in $H^{\text{der}}$ because
they are both compact open subgroups of $J_{b}^{\text{der}}(\mathbb{Q}_{p})$.
Therefore, the action of $H^{\text{der}}$ on $\pi_{0}(\operatorname{Ig}_{b})$
has finite orbits.

Let $J_{b}^{\text{der}}=J_{b,\text{an}}^{\text{der}}\times
J_{b,\text{iso}}^{\text{der}}$ and let $\operatorname{Ig}_{b}^{\text{der}}$ be
the $J_{b,\text{iso}}^{\text{der}}(\mathbb{Q}_{p})$-orbit of a connected
component of $\operatorname{Ig}_{b}$, then
$\pi_{0}(\operatorname{Ig}_{b}^{\text{der}})$ is automatically compact because
$\pi_{0}(\operatorname{Ig}_{b})$ is compact Hausdorff and
$J_{b,\text{iso}}^{\text{der}}(\mathbb{Q}_{p})\times\\{1\\}\subset
J_{b}(\mathbb{Q}_{p})$ is closed. We claim that
$\pi_{0}(\operatorname{Ig}_{b}^{\text{der}})$ is finite. It suffices to show
that it is a union of finitely many $H^{\text{der}}_{\text{iso}}$-orbits
because $H^{\text{der}}$ has finite orbits as explained above. Let $Q\subset
J_{b,\text{iso}}^{\text{der}}(\mathbb{Q}_{p})$ be the stabiliser of a
connected component of $\operatorname{Ig}_{b}^{\text{der}}$, then the action
map induces a homeomorphism

(11) $\displaystyle\pi_{0}(\operatorname{Ig}_{b}^{\text{der}})\simeq
J_{b,\text{iso}}^{\text{der}}(\mathbb{Q}_{p})/Q.$

The quotient map $J_{b,\text{iso}}^{\text{der}}(\mathbb{Q}_{p})\to
J_{b,\text{iso}}^{\text{der}}(\mathbb{Q}_{p})/Q$ is open, hence
$H^{\text{der}}_{\text{iso}}$-orbits are open, and so there are finitely many
of them by compactness. Proposition 12.14 of [MalleTesterman] tells us that
$J_{b}^{\text{der}}$ is simply connected because $G^{\text{der}}$ is simply
connected and it follows that $J_{b,\text{iso}}^{\text{der}}$ is simply
connected. Therefore the group $J_{b,\text{iso}}^{\text{der}}(\mathbb{Q}_{p})$
has no nontrivial finite quotients by [PR]*Theorem 7.1, Theorem 7.5 . Since
$\pi_{0}(\operatorname{Ig}_{b})$ is finite, the stabiliser in
$J_{b,\text{iso}}^{\text{der}}(\mathbb{Q}_{p})$ of a point in
$\pi_{0}(\operatorname{Ig}_{b}^{\text{der}})$ is a finite index subgroup and
therefore equal to $J_{b,\text{iso}}^{\text{der}}(\mathbb{Q}_{p})$. ∎

This argument will not work for the anisotropic part of $J_{b}^{\text{der}}$,
because it is not true that $J_{b,\text{an}}^{\text{der}}(\mathbb{Q}_{p})$ has
no nontrivial finite quotients. In fact it is a profinite group (because it is
compact), and so it has many nontrivial finite quotients.

##  4\. Construction of a maximal torus

###  4.1. Prime-to-$p$ Hecke operators

By [HamacherKim]*Lemma 5.1.4, the prime-to-$p$ Hecke operators act on
$\operatorname{Ig}_{b}$ such that the projection
$\operatorname{Ig}_{b}\to\operatorname{Sh}_{G,b}$ is equivariant. To be
precise, the construction of $\operatorname{Ig}_{b}$ is compatible with
changing the level $U^{p}\subset G(\mathbb{A}_{f}^{p})$ and
$G(\mathbb{A}_{f}^{p})$ acts on the inverse limit over $U^{p}$ of the Igusa
variety. Let $\Sigma$ be the finite set consisting of $p,\infty$ and places of
$\mathbb{Q}$ where $G$ is anisotropic.

######  Proposition 4.1.1.

The prime-to-$\Sigma$ Hecke operators coming from $G^{\text{der}}$ act
trivially on $\pi_{0}(\operatorname{Ig}_{b})$

###### Proof.

This can be proven as in [LXXiao]*Section 4, making use of the fact that
$G^{\text{der}}(\mathbb{Q}_{\ell})$ has no finite quotients for
$\ell\not\in\Sigma$. ∎

For a point $y\in C(x),$ we let $I_{y}$ be the algebraic group over
$\mathbb{Q}$ of self-quasi isogenies of the abelian variety $A_{y}$ that
preserve the Hodge tensors. Choosing a quasi-isogeny between
$A_{y}[p^{\infty}]$ and $\mathbb{X}$, leads to an inclusion
$I_{y,\mathbb{Q}_{p}}\subset J_{b}$ . This is not necessarily an isomorphism,
but it is true that both groups have the same rank (see
[KisinPoints]*Corollary 2.1.7, [Zhou]*Corollary 9.5 and
[vanHoften]*Proposition A.4.3 in the ramified case). It follows from loc. cit.
that $I_{y}$ is an inner form of a subgroup of $G$, and therefore that there
is a morphism $I_{y}\to G^{\text{ab}}$; we let $I_{y}^{\prime}$ be its kernel.

######  Proposition 4.1.2.

For each $y\in C(x)^{\circ}$, the subgroup $I_{y}^{\prime}(\mathbb{Q})\subset
J_{b}(\mathbb{Q}_{p})$ acts trivially on $\pi_{0}(\operatorname{Ig}_{b})$.

###### Proof.

The point is that $I_{y}^{\prime}(\mathbb{Q})\subset
G^{\text{der}}(\mathbb{A}_{f}^{\Sigma})$ acts trivially on $C(x)$ while it
acts via $I_{y}(\mathbb{Q})\subset J_{b}(\mathbb{Q}_{p})$ on
$\operatorname{Ig}_{b}$. Since $G^{\text{der}}(\mathbb{A}_{f}^{\Sigma})$ acts
trivially on $\pi_{0}(\operatorname{Ig}_{b})$, the result follows. ∎

######  Corollary 4.1.3.

The closure of $I_{y}^{\prime}(\mathbb{Q})\subset J_{b}(\mathbb{Q}_{p})$ acts
trivially on $\pi_{0}(\operatorname{Ig}_{b})$.

###### Proof.

Let $z\in\pi_{0}(\operatorname{Ig}_{b})$ be a connected component with
stabiliser $Q_{z}$. Then $I_{y}^{\prime}(\mathbb{Q})\subset Q_{z}$ and the
corollary would follow if we could show that $Q_{z}$ was a closed subgroup.
There is a homeomorphism $\pi_{0}(\operatorname{Ig}_{b})\simeq
J_{b}(\mathbb{Q}_{p})/Q_{z}$ and since $\pi_{0}(\operatorname{Ig}_{b})$ is
Hausdorff, it follows that $Q_{z}$ is a closed subgroup of
$J_{b}(\mathbb{Q}_{p})$. ∎

######  Definition 4.1.4.

We call $y$ _weakly hypersymmetric_ if $I_{y,\mathbb{Q}_{p}}\simeq J_{b}$. If
in addition $I_{y}(\mathbb{Q})$ is dense in $I_{y}(\mathbb{Q}_{p})$, we call
$y$ _hypersymmetric_.

When $G_{\mathbb{Q}_{p}}$ splits over an unramified extension, weakly
hypersymmetric points are automatically hypersymmetric. Indeed, by
[PR]*Theorem 7.8 and the discussion in [PR]*Section 7.3, weak approximation at
a prime $p$ holds if the group is unramified there. Therefore in the PEL case,
this definition coincides with the definition given by Chai-Oort
[chai2006hypersymmetric]*Definition 6.4.

When the Newton stratum contains a hypersymmetric point, the arguments above
show that $J_{b}^{\prime}(\mathbb{Q}_{p})$ acts trivially on
$\operatorname{Ig}_{b}$, without using the results of Section 3.
Unfortunately, although Newton strata on Siegel modular varieties always
contain hypersymmetric points [chai2006hypersymmetric], they are sparse in
more general settings. For example, in the PEL case, not every Newton stratum
contains a hypersymmetric point [zong2008hypersymmetric]; they might not exist
even in the $\mu$-ordinary stratum [LXXiao]*Corollary 7.5.. Hence, all we can
say about $I_{y,\mathbb{Q}_{p}}$ in general is that it contains a maximal
torus of $J_{b}$. Using Honda-Tate theory, we will show that for every maximal
torus of $J_{b}$ we can find a point $y$ in $C(x)$ such that
$I_{y,\mathbb{Q}_{p}}$ contains that maximal torus, up to isomorphism of tori.

######  Remark 4.1.5.

Basic points in Shimura varieties of Hodge type are known to be hypersymmetric
[xiao2017cycles]*Lemma 7.2.14. It would be interesting to characterise all
Newton strata containing hypersymmetric points for Shimura varieties of Hodge
type.

######  Remark 4.1.6.

If we could show that $J_{b}^{\prime}(\mathbb{Q}_{p})$ is generated by the
closures of $I_{y}^{\prime}(\mathbb{Q})$ as $y$ ranges over points in $C(x)$,
then the main theorem would follow. This is closely related to Hida’s strategy
in [Hida], where he shows that the group $H$ is generated by
$T(\mathbb{Z}_{p})$ as $T$ ranges over isomorphism classes of maximal tori of
$J_{b}$, in the special case that $H=\operatorname{GL}_{n}(\mathcal{O}_{F})$
for some $p$-adic local field $F$. We were not able to make this strategy work
in general, which is why we need to make use of the result of Section 3.

###  4.2. Honda-Tate theory

Classical Honda-Tate theory describes isogeny classes of abelian varieties
over finite fields containing $\mathbb{F}_{p}$ in terms of $q$-Weil numbers.
Equivalently, we can describe isogeny classes of abelian varieties $A$ of
dimension $g$ by the characteristic polynomial of $\operatorname{Frob}_{p}$
acting on the $\ell$-adic Tate module of $A$ for some $\ell\not=p$ (this is an
element of $\mathbb{Z}[X]$). This gives us a (semisimple) conjugacy class of
matrices in $\operatorname{GL}_{2g}(\mathbb{Q})$ and its stabiliser is an
inner form of the group $I$ of self quasi-isogenies of $A$. Choosing a maximal
torus $T\subset I$ corresponds, roughly speaking, to choosing a CM lift of the
isogeny class.

For Shimura varieties of Hodge type like our $(G,X)$, things are best phrased
in terms of maximal tori $T\subset G$ such that some $h\in X$ factors through
$T_{\mathbb{R}}$, which makes $T$ into a special point. These special points
give rise to points on $\operatorname{Sh}_{G}$ via reduction mod $p$. The main
theorem of [KMPS] shows that every Newton stratum in our Shimura variety of
Hodge type contains many such points, and [vanHoften]*Theorem A.4.5.(2) shows
that in fact every isogeny class contains the reduction of a special point.

######  Proposition 4.2.1.

For each maximal torus $T^{\prime}\subset J_{b}^{\prime}$, there is a point
$y$ in the leaf $C(x)$ such that $I_{y,\mathbb{Q}}^{\prime}$ contains a torus
$\mathscr{T}^{\prime}$ with $\mathscr{T}^{\prime}_{\mathbb{Q}_{p}}$ isomorphic
to $T^{\prime}$ under $I_{y,\mathbb{Q}_{p}}^{\prime}\subset J_{b}^{\prime}$.
Moreover, we can choose $\mathscr{T}^{\prime}$ such that it has weak
approximation, i.e., such that $\mathscr{T}^{\prime}(\mathbb{Q})$ is dense in
$\mathscr{T}^{\prime}(\mathbb{Q}_{p})$.

###### Proof.

First of all we can uniquely extend $T^{\prime}$ to a maximal torus $T\subset
J_{b}$, and since $G_{\mathbb{Q}_{p}}$ is quasi-split, we can transfer $T$ to
$G$. Then [KMPS]*Proposition 1.2.5 tells us that we can find a maximal torus
$\mathscr{T}\subset G$ such that: There is an $h\in X$ that factors through
$\mathscr{T}_{\mathbb{R}}$, i.e., $(\mathscr{T},h)\subset(G,X)$ is a special
point, and this special point induces an isogeny class
$\mathscr{I}\subset\operatorname{Sh}_{G,b}(\overline{\mathbb{F}}_{p})$ with
automorphism group $I_{\mathbb{Q}}$ containing $\mathscr{T}$ as a maximal
torus, and such that $\mathscr{T}_{\mathbb{Q}_{p}}$ is
$G(\mathbb{Q}_{p})$-conjugate to $T$ and in particular
$\mathbb{Q}_{p}$-isomorphic to $T$. After basechange to $\mathbb{Q}_{p}$, the
inclusions

(12) $\displaystyle\mathscr{T}_{\mathbb{Q}_{p}}\subset
I_{\mathbb{Q}_{p}}\subset J_{b}$

then gives us a maximal torus of $J_{b}$, and thus a torus of $J_{b}^{\prime}$
which is $\mathbb{Q}_{p}$-isomorphic to $T^{\prime}$.

The construction of $\mathscr{T}$ in the proof of loc. cit. is quite flexible.
They start by choosing any maximal torus $\mathscr{T}_{0}$ in $G$ such that
there is an $h\in X$ factoring through $\mathscr{T}_{0}$ and such that
$\mathscr{T}_{0,\mathbb{Q}_{p}}$ is conjugate to $T$. Next, they choose $g\in
G^{\text{der}}(\overline{\mathbb{Q}})$ so that the cocycle $\sigma\mapsto
g\sigma(g)^{-1}$ lies in $\mathscr{T}_{0}^{\prime}(\overline{\mathbb{Q}})$,
where $\mathscr{T}^{\prime}_{0}=\mathscr{T}_{0}\cap G^{\text{der}}$. By Lemma
1.2.1 of loc. cit., this cocycle is chosen so that its cohomology class is
trivial in $H^{1}(\mathbb{Q}_{p},\mathscr{T}_{0}^{\prime})$, and in fact the
proof of the lemma shows that we can choose this cocycle such that it is
trivial in $H^{1}(\mathbb{Q}_{\ell},\mathscr{T}_{0}^{\prime})$ for finitely
many $\ell$’s. Then $\mathscr{T}$ arises as
$\text{int}(g^{-1})(\mathscr{T}_{0,\overline{\mathbb{Q}}}),$ which is defined
over $\mathbb{Q}.$

By weak approximation for the variety of tori for $G$, which is isomorphic to
the variety of tori for $G^{\text{der}}$, we can choose $\mathscr{T}_{0}$ with
specified $G^{\text{der}}(\mathbb{Q}_{\ell})$-conjugacy class of
$\mathscr{T}_{0}^{\prime}$ for finitely many $\ell$’s. By the discussion
above, we can choose $g$ such that $\mathscr{T}_{0}^{\prime}$ has the same
specified $G^{\text{der}}(\mathbb{Q}_{\ell})$-conjugacy classes. It now
follows from [PrasadRapinchuk]*Theorem 1.(i) and the result of V.E.
Voskresenskii mentioned afterwards that this gives us enough flexibility to
choose $\mathscr{T}^{\text{der}}$ to have weak approximation.

The proposition now follows from the fact that our isogeny class $\mathscr{I}$
intersects every central leaf in the Newton stratum $\operatorname{Sh}_{G,b}$,
which follows from Rapoport-Zink uniformisation of isogeny classes
([KisinPoints]*Proposition 2.1.3 and [vanHoften]*Theorem A.4.5.(1) in the
ramified case). To be precise, every isogeny class receives a map from a
Rapoport-Zink space, and these Rapoport Zink spaces contain every isomorphism
class of $p$-divisible groups with extra structure in the given Newton
stratum. In other words, they intersect every central leaf in the Newton
stratum. ∎

######  Corollary 4.2.2.

For every maximal torus $T^{\prime}\subset J_{b}^{\prime}$, we can find a
maximal torus $\hat{T}\subset J_{b}^{\prime}$ such that $\hat{T}\simeq
T^{\prime}$ as algebraic groups and such that $\hat{T}(\mathbb{Q}_{p})$ acts
trivially on $\pi_{0}(\operatorname{Ig}_{b})$.

###### Proof.

This is a direct consequence of Proposition 4.1.2 and Proposition 4.2.1. ∎

##  5\. Anisotropic groups

###  5.1. Action of the anisotropic part

It follows from [PR]*Theorem 6.5 that a semi-simple simply connected
anisotropic group over $\mathbb{Q}_{p}$ is isomorphic to a product of groups
of the form $\operatorname{Res}_{K/\mathbb{Q}_{p}}\operatorname{SL}_{1}(D)$,
where $D/K$ is a central division algebra with $K/\mathbb{Q}_{p}$ a finite
extension and $\operatorname{SL}_{1}(D)$ is the kernel of the norm
homomorphism $D^{\times}\to\mathbb{G}_{m,K}$.

Choose such a $K$ and $D$, and assume that the rank of $D$ is $n^{2}$. Maximal
tori of $D^{\times}$ correspond to subfields of degree $n$ of $D$, and each
degree $n$ field extension of $K$ is such a subfield (see [milneCFT]*Remark
IV.4.4.(c)). Let $L$ be an unramified subfield of $D$ degree $n$ and $F$ a
totally ramified subfield of $D$ degree $n$.

######  Lemma 5.1.1.

Suppose that $p$ is coprime to $n$. Then the group $\mathcal{O}_{D}^{\times}$
is (topologically) generated by $\mathcal{O}_{L}^{\times}$ and
$\mathcal{O}_{F}^{\times}$.

###### Proof.

The quotient $\mathcal{O}_{D}^{\times}/(1+\pi\mathcal{O}_{D})$, where $\pi$ is
a uniformiser of $D$, is isomorphic to $k_{L}^{\times}$, where $k_{L}$ is the
residue field of $L$. Because $\mathcal{O}_{L}^{\times}$ surjects onto
$k_{L}^{\times}$, it suffices to show that the group (topologically) generated
by $\mathcal{O}_{L}^{\times}$ and $\mathcal{O}_{F}^{\times}$ contains
$1+\pi\mathcal{O}_{D}$. For this it is enough to show that the Lie algebra
(topologically) generated by $\mathcal{O}_{L}$ and $\mathcal{O}_{F}$ is
isomorphic to $\mathcal{O}_{D}$, because the exponential map converges on
$\pi\mathcal{O}_{D}\subset\mathcal{O}_{D}=\operatorname{Lie}\mathcal{O}_{D}^{\times}$.

Remark IV4.4.(b) of [milneCFT] tells us that $D$ can be described as the
(noncommutative) algebra $L\\{\pi\\}$ where $\pi^{n}=\varpi_{L}$ is a
uniformiser in $L$ and where

(13) $\displaystyle\pi\cdot a=\sigma^{k}(a)\cdot\pi,$

for some $1\leq k\leq n$ coprime to $n$ with $\sigma$ is the Frobenius
automorphism of $L/K$. It follows in the usual way that $\mathcal{O}_{D}$ is
generated as an $\mathcal{O}_{K}$-algebra by $\mathcal{O}_{L}$ and $\pi$. We
then have that, as an $\mathcal{O}_{L}$ module,

(14)
$\displaystyle\mathcal{O}_{D}=\bigoplus_{j=0}^{n-1}\mathcal{O}_{L}\pi^{j}.$

The Lie bracket is given by the commutator, which means that

(15) $\displaystyle[a,\pi^{j}]=(a-\sigma^{jk}(a))\pi.$

Therefore, the $\mathcal{O}_{K}$-linear Lie algebra generated by
$\mathcal{O}_{L}$ and $\mathcal{O}_{F}$ contains expressions of the form

(16) $\displaystyle a-\sigma^{jk}(a),b\pi^{jk}$

for $b\in\mathcal{O}_{K}$ and $a\in\mathcal{O}_{L}$, and so it suffices to
show that $\mathcal{O}_{L}$ is generated by

(17)
$\displaystyle\\{a-\sigma^{jk}(a)\;|\;a\in\mathcal{O}_{l}\\}\cup\mathcal{O}_{K}.$

We note that the $\mathcal{O}_{K}$ module $N$ generated by
$\\{a-\sigma^{jk}(a)\;|\;a\in\mathcal{O}_{l}\\}$ sits in a short exact
sequence

(18) $\displaystyle 0\to
N\to\mathcal{O}_{L}\to\left(\mathcal{O}_{L}\right)_{\operatorname{Gal}(L/K)}\to
0,$

where the subscript indicates taking coinvariants for the action of the Galois
group given by the $jk$-th power of the natural action; this is well defined
since $\operatorname{Gal}(L/K)\simeq\mathbb{Z}/n\mathbb{Z}$. If $p\nmid n$,
then there is an isomorphism from invariants to coinvariants

(19)
$\displaystyle\mathcal{O}_{K}=(\mathcal{O}_{L})^{\operatorname{Gal}(L/K)}\simeq\left(\mathcal{O}_{L}\right)_{\operatorname{Gal}(L/K)},$

which gives a direct sum decomposition
$\mathcal{O}_{L}=N\oplus\mathcal{O}_{K}$, where the inclusion of
$\mathcal{O}_{K}\to\mathcal{O}_{L}$ is the natural one up to multiplication by
$n$. ∎

######  Lemma 5.1.2.

Suppose that $p$ is coprime to $n$, then the group
$\operatorname{SL}_{1}(\mathcal{O}_{D})$ is generated by

(20) $\displaystyle\operatorname{SL}_{1}(\mathcal{O}_{L})\text{ and
}\operatorname{SL}_{1}(\mathcal{O}_{F}),$

with $L$ and $F$ as above.

###### Proof.

It is enough to show that the Lie algebra of
$\operatorname{SL}_{1}(\mathcal{O}_{D})$ is generated by the Lie algebras of
$\operatorname{SL}_{1}(\mathcal{O}_{L})$ and
$\operatorname{SL}_{1}(\mathcal{O}_{F})$ by the same argument as above. The
short exact sequence of $p$-adic Lie groups

(21) $\displaystyle
0\to\operatorname{SL}_{1}(\mathcal{O}_{D})\to\mathcal{O}_{D}^{\times}\to\mathcal{O}_{K}^{\times}\to
0$

induces a short exact sequence of Lie algebras

(22) $\displaystyle
0\to\mathcal{O}_{D}^{\operatorname{Tr}_{D/K}=0}\to\mathcal{O}_{D}\to\mathcal{O}_{K}\to
0.$

This sequence is split exact if $p$ is coprime to $n$; a section of the trace
map $\mathcal{O}_{D}\to\mathcal{O}_{K}$ is given by $1/n$ times the usual
inclusion $\mathcal{O}_{K}\subset\mathcal{O}_{D}$. It follows that

(23)
$\displaystyle\mathcal{O}_{D}\simeq\mathcal{O}_{K}\oplus\mathcal{O}_{D}^{\operatorname{Tr}_{D/K}=0}$

and similarly that

(24) $\displaystyle\mathcal{O}_{L}$
$\displaystyle\simeq\mathcal{O}_{L}^{\operatorname{Tr}_{L/K}=0}\oplus\mathcal{O}_{K}$
(25) $\displaystyle\mathcal{O}_{F}$
$\displaystyle\simeq\mathcal{O}_{F}^{\operatorname{Tr}_{F/K}=0}\oplus\mathcal{O}_{K}.$

Now the fact that $\mathcal{O}_{D}$ is generated by $\mathcal{O}_{L}$ and
$\mathcal{O}_{F}$ implies the analogous result for the trace zero part,
concluding the proof. ∎

######  Proposition 5.1.3.

$J_{b,\text{an}}^{\text{der}}$ acts trivially on
$\pi_{0}(\operatorname{Ig}_{b})$.

###### Proof.

Since $\operatorname{SL}_{1}(\mathcal{O}_{D})=\operatorname{SL}_{1}(D)$, Lemma
5.1.2 implies that the $\mathbb{Q}_{p}$-points of each simple component of
$J_{b,\text{an}}^{\text{der}}$ is generated by two maximal tori, which can be
specified up to isomorphism. By Corollary 4.2.2, these tori may be chosen to
act trivially on $\pi_{0}(\operatorname{Ig}_{b})$. ∎

In the last part of this section, we will collect a lemma for later use:

######  Lemma 5.1.4.

Let $Z$ be the center of $\operatorname{SL}_{1}(D)$. Then any cohomology class
$\alpha\in H^{1}(K,Z)$ maps to zero in either
$H^{1}(K,\operatorname{SL}_{1}(L))$ or $H^{1}(K,\operatorname{SL}_{1}(F))$.

###### Proof.

We know that $Z\simeq\mu_{n,K}$ and therefore
$H^{1}(K,Z)=K^{\times}/K^{\times,n}$. A standard long exact sequence argument
shows that
$H^{1}(K,\operatorname{SL}_{1}(L))=K^{\times}/\operatorname{Nm}_{L/K}L^{\times}$
and
$H^{1}(K,\operatorname{SL}_{1}(F))=K^{\times}/\operatorname{Nm}_{F/K}F^{\times}$.
There are inclusions

(26) $\displaystyle
K^{\times,n}\subset\operatorname{Nm}_{L/K}L^{\times},\operatorname{Nm}_{F/K}F^{\times}$

which give rise to the natural maps $H^{1}(K,Z)\to
H^{1}(K,\operatorname{SL}_{1}(F))$ and $H^{1}(K,Z)\to
H^{1}(K,\operatorname{SL}_{1}(L))$. The result follows because the group
generated by $\operatorname{Nm}_{L/K}L^{\times}$ and
$\operatorname{Nm}_{F/K}F^{\times}$ is equal to $K^{\times}$. ∎

###  5.2. The product is surjective

So far we have seen that $J_{b}^{\text{der}}(\mathbb{Q}_{p})$ acts trivially
on $\pi_{0}(\operatorname{Ig}_{b})$ and that we can find maximal tori of
$J_{b}^{\prime}(\mathbb{Q}_{p})$ which act trivially on
$\operatorname{Ig}_{b}$. In this section we show that this implies that
$J_{b}^{\prime}(\mathbb{Q}_{p})$ acts trivially, too. As a warm-up, we will
deal with the case of quasi-split groups:

######  Lemma 5.2.1.

Suppose that $J_{b}$ is quasi-split, then $J_{b}^{\prime}(\mathbb{Q}_{p})$ is
generated by $J_{b}^{\text{der}}(\mathbb{Q}_{p})$ and $T(\mathbb{Q}_{p})$ for
a choice of maximal torus $T$.

###### Proof.

Let $T^{\prime}$ be the centraliser of a maximal split torus $T$. The Galois
module $X_{\ast}(T^{\text{der}})$ is induced by [BTII]*Proposition 4.4.16 and
the short exact sequence

(27) $\displaystyle 1\to T^{\text{der}}\to J_{b}^{\text{der}}\times
T^{\prime}\to J_{b}^{\prime}\to 1$

induces a short exact sequence on $\mathbb{Q}_{p}$ points because
$H^{1}(\mathbb{Q}_{p},T^{\text{der}})$ vanishes. ∎

######  Proposition 5.2.2.

For a general $J_{b}^{\prime}$, not necessarily quasi-split, the group
$J_{b}^{\prime}(\mathbb{Q}_{p})$ is generated by
$J_{b}^{\text{der}}(\mathbb{Q}_{p})$ and
$T_{1}(\mathbb{Q}_{p}),\cdots,T_{n}(\mathbb{Q}_{p}),$ for finitely many
maximal tori of $J_{b}^{\prime}$ (which can be specified up to isomorphism).

The proof goes via a series of lemmas:

######  Lemma 5.2.3.

The subgroup of $J_{b}^{\prime}(\mathbb{Q}_{p})$ generated by
$J_{b}^{\text{der}}(\mathbb{Q}_{p})$ and $T^{\prime}(\mathbb{Q}_{p})$, for a
certain choice of $T^{\prime}$, contains the $\mathbb{Z}_{p}$-points of every
parahoric subgroup of $J_{b}^{\prime}(\mathbb{Q}_{p})$.

###### Proof.

We can take a torus $T^{\prime}$ that becomes the centraliser of a maximal
split torus over the maximal unramified extension of $\mathbb{Q}_{p}$ (these
exist by [Tits]*1.10). Remark 16 of the appendix to [PappasRapoportAffineFlag]
tells us that the subgroup of $J_{b}^{\prime}(\mathbb{Q}_{p})$ generated by
$J_{b}^{\text{der}}$ and $T^{\prime}(\mathbb{Z}_{p})$ is equal to the group
generated by all parahoric subgroups of $J_{b}^{\prime}(\mathbb{Q}_{p})$. ∎

###### Proof of Proposition 5.2.2.

The Iwasawa decomposition ([Tits]*3.3.3) tells us that

(28) $\displaystyle J_{b}^{\prime}(\mathbb{Q}_{p})=KM(\mathbb{Q}_{p})K$

where $K$ is a special maximal parahoric subgroup and $M$ is a minimal
$\mathbb{Q}_{p}$-rational Levi subgroup of $J_{b}^{\prime}$. The group $M$ is
the centraliser of a maximal split torus $S$ in $J_{b}^{\prime}$, and we know
that $M$ is an anisotropic group. The group generated by
$J_{b}^{\text{der}}(\mathbb{Q}_{p})$ and $T^{\prime}(\mathbb{Q}_{p})$ is
normal, so if we choose tori that contain a maximal split torus of
$J_{b}^{\prime}$ (which is something well defined up to isomorphism of tori),
then we can conjugate them to contain $S$ and therefore lie in $M$. The result
now follows from Lemma 5.2.4. ∎

######  Lemma 5.2.4.

The group $M(\mathbb{Q}_{p})$ is generated by $M^{\text{der}}(\mathbb{Q}_{p})$
and $T_{1}(\mathbb{Q}_{p}),T_{2}(\mathbb{Q}_{p})$ for maximal tori
$T_{1},T_{2}$ of $M$.

###### Proof.

The derived subgroup $M^{\text{der}}$ is simply connected because $M$ is a
Levi subgroup of $J_{b}^{\prime}$ and $J_{b}^{\text{der}}$ is simply connected
(see [MalleTesterman]*Proposition 12.14). Therefore there is a short exact
sequence

(29) $\displaystyle 1\to M^{\text{der}}(\mathbb{Q}_{p})\to
M(\mathbb{Q}_{p})\to M^{\text{ab}}(\mathbb{Q}_{p})\to 1.$

We see that it suffices to show that we can choose maximal tori such that the
group generated by $T_{1}(\mathbb{Q}_{p}),\cdots,T_{n}(\mathbb{Q}_{p})$
surjects onto $M^{\text{ab}}(\mathbb{Q}_{p})$. Let $Z_{M}$ and
$Z_{M^{\text{der}}}$ denote the centraliser of $M$ and $M^{\text{der}}$,
respectively. If $T$ is any maximal torus of $M$ then there are short exact
sequences

(30)
${1}$${Z_{M^{\text{der}}}}$${Z_{M}}$${G^{\text{ab}}}$${1}$${1}$${T^{\text{der}}}$${T}$${G^{\text{ab}}}$${1}$

which give rise to long exact sequences

(31)
${1}$${Z_{M^{\text{der}}}(\mathbb{Q}_{p})}$${Z_{M}(\mathbb{Q}_{p})}$${G^{\text{ab}}(\mathbb{Q}_{p})}$${H^{1}(\mathbb{Q}_{p},Z_{M^{\text{der}}})}$${1}$${1}$${T^{\text{der}}(\mathbb{Q}_{p})}$${T(\mathbb{Q}_{p})}$${G^{\text{ab}}(\mathbb{Q}_{p})}$${H^{1}(\mathbb{Q}_{p},T^{\text{der}})}$${H^{1}(\mathbb{Q}_{p},T).}$

We see that it is enough to show that for every $\alpha\in
H^{1}(\mathbb{Q}_{p},Z_{M}^{\text{der}})$, there is a maximal torus
$T_{\alpha}$ of $M$ such that $\alpha$ maps to zero in
$H^{1}(\mathbb{Q}_{p},T_{\alpha}^{\text{der}})$. This is something we can
solve entirely inside $M^{\text{der}}$, which is a product of restrictions of
scalars of groups of the form $\operatorname{SL}_{1}(D)/K$, as in Section 5.1.
For those groups, the existence of these maximal tori is proven in Lemma
5.1.4. ∎

##  6\. Proof of the main result

###  6.1. Identification of the quotient

In this section we identify the quotient $H/H^{\prime}$ where $H=H_{C}$ and
$C$ is a distinguished central leaf; we will show that $H/H^{\prime}\simeq
G^{\text{ab}}$.

After choosing a basis of the Dieudonné module of $A_{x}$, the Frobenius
corresponds to an element $b\in
G(\breve{\mathbb{Q}}_{p})\subset\operatorname{GL}_{V}(\breve{\mathbb{Q}}_{p})$.
The group of automorphisms of $A_{x}[p^{\infty}]$ is precisely the twisted
centraliser of this element $b$ intersected with
$\operatorname{GL}_{V}(\breve{\mathbb{Z}}_{p})$. Similarly, the group $H$ is
the intersection of $J_{b}(\mathbb{Q}_{p})\subset
G(\breve{\mathbb{Q}}_{p})\subset\operatorname{GL}_{V}(\breve{\mathbb{Q}}_{p})$
with $\mathcal{G}(\breve{\mathbb{Z}}_{p})$, where $\mathcal{G}/\mathbb{Z}_{p}$
is the parahoric group scheme such that $\mathcal{G}(\mathbb{Z}_{p})=U_{p}$.
It follows that $H$ is precisely the stabiliser in $J_{b}(\mathbb{Q}_{p})$ of
the element $1$ in the affine Deligne-Lusztig variety $X(\mu,b)$ for $G$. If
we have a different point $x^{\prime}$ such that $b^{\prime}$ is
$\sigma$-conjugate to $b$, then the stabiliser $H_{x^{\prime}}$ corresponds to
the stabiliser of some element in $X(\mu,b)$, and every such stabiliser occurs
as an $H_{x^{\prime}}$.

If we choose $b^{\prime}$ to be a $\sigma$-straight element, then the
corresponding stabiliser group is a parahoric subgroup of
$J_{b}(\mathbb{Q}_{p})$ (see the proof of Proposition 3.1.4 of [ZhouZhu]) and
the corresponding central leaf $C$ is called a _distinguished central leaf_
because it is also equal to an Ekedahl-Oort stratum (see the proof of Theorem
E of [ShenYuZhang]).

This means that there is a smooth affine group scheme
$\mathcal{J}/\mathbb{Z}_{p}$ with connected special fiber such that
$\mathcal{J}(\mathbb{Z}_{p})=H$. This group scheme $\mathcal{J}$ is the
connected component of the identity of a Bruhat-Tits stabiliser group scheme
$\tilde{\mathcal{J}}$. Proposition 2.4.9 of [KisinZhou] tells us that the
natural map $\tilde{\mathcal{J}}^{\prime}\to\tilde{\mathcal{J}}$ of Bruhat-
Tits stabiliser group schemes induced by $\beta:J_{b}^{\prime}\to J_{b}$ is a
closed immersion (the assumptions are satisfied because $J_{b}$ splits over a
tamely ramified extension). We define $\mathcal{J}^{\prime}_{\beta}$ to be the
inverse image of $\mathcal{J}$ under
$\tilde{\mathcal{J}}^{\prime}\to\tilde{\mathcal{J}}$, or equivalently the
Zariski closure of $J_{b}^{\prime}$ in $\mathcal{J}$. This tells us that
$\mathcal{J}^{\prime}_{\beta}$ is a smooth affine group scheme with
$\mathcal{J}^{\prime}_{\beta}(\mathbb{Z}_{p})=H^{\prime}$.

######  Lemma 6.1.1.

There is a short exact sequence

(32) $\displaystyle
1\to\mathcal{J}^{\prime}_{\beta}\to\mathcal{J}\to\mathcal{Z}\to 1,$

where $\mathcal{Z}$ is the connected Néron model of $G^{\text{ab}}$.

###### Proof.

There is a natural map $\tilde{\mathcal{J}}\to\tilde{\mathcal{Z}}$, where
$\tilde{\mathcal{Z}}$ is the finite type Néron model of $G^{\text{ab}}$. This
map induces a map $\mathcal{J}\to\mathcal{Z}$ of identity components, which we
claim to be surjective. To prove this claim, it suffices to prove that
$\mathcal{T}\to\mathcal{Z}$ is surjective, where
$\mathcal{T}\subset\mathcal{J}$ is the connected Néron model of some maximal
torus $T$ of $J_{b}$. Such a $\mathcal{T}$ exists and in fact we can take $T$
to be tamely ramified. In that case, [KisinPappas]*Proposition 1.1.4 gives the
surjectivity of $\mathcal{T}\to\mathcal{Z}$.

We are left to show that $\mathcal{J}_{\beta}^{\prime}$ is the kernel of the
map $\mathcal{J}\to\mathcal{Z}$. This can be checked on
$\breve{\mathbb{Z}}_{p}$-points, and there we consider the following diagram

(33)
${\mathcal{J}^{\prime}_{\beta}(\breve{\mathbb{Z}}_{p})}$${\mathcal{J}(\breve{\mathbb{Z}}_{p})}$${\mathcal{Z}(\breve{\mathbb{Z}}_{p})}$${1}$${1}$${J_{b}^{\prime}(\breve{\mathbb{Q}}_{p})}$${J_{b}(\breve{\mathbb{Q}}_{p})}$${G^{\text{ab}}(\breve{\mathbb{Q}}_{p})}$${1.}$

The exactness of the left of the top row follows from the fact that the
leftmost square is Cartesian, and so we are done. ∎

By construction, the connected components of the special fiber of
$\mathcal{J}_{\beta}^{\prime}$ are given by the kernel of

(34)
$\displaystyle\pi_{1}(J_{b}^{\prime})_{I,\text{tors}}\to\pi_{1}(J_{b})_{I,\text{tors}}.$

######  Lemma 6.1.2.

The special fiber of $\mathcal{J}^{\prime}_{\beta}$ is connected.

###### Proof.

We know that $J_{b}$ is an inner form of a standard Levi $M_{b}\subset G$ and
similarly that $J_{b}^{\prime}$ is an inner form of a standard Levi
$M_{b}^{\prime}\subset G^{\text{der}}$, we can arrange things such that
$M_{b}^{\prime}\subset M_{b}$. Using the isomorphism
$\pi_{1}(J_{b})\simeq\pi_{1}(M_{b})$ and
$\pi_{1}(J_{b}^{\prime})\simeq\pi_{1}(M_{b}^{\prime})$ it suffices to show
that the kernel of

(35)
$\displaystyle\pi_{1}(M_{b}^{\prime})_{I,\text{tors}}\to\pi_{1}(M_{b})_{I,\text{tors}}$

is trivial. Because $M_{b}$ is a standard Levi of $G$, it contains the
centraliser $T$ of a maximal split torus $S$ of $G$ and moreover $T$ is also
the centraliser of $S$ in $M_{b}$. Let $T^{\prime},T^{\text{der}}$ be the
intersections of $T$ with $M_{b}^{\prime}$ and $M_{b}^{\text{der}}$
respectively, then both $X_{\ast}(T^{\prime})$ and $X_{\ast}(T^{\text{der}})$
are induced Galois modules by [BTII]*Proposition 4.4.16 (recall that
$M_{b}^{\text{der}}$ is simply connected by [MalleTesterman]*Proposition
12.14). This means that both $X_{\ast}(T^{\text{der}})_{I}$ and
$X_{\ast}(T^{\prime})_{I}$ are torsion free, and so there is a diagram of
short exact sequences

(36)
${0}$${X_{\ast}(T^{\text{der}})_{I}}$${X_{\ast}(T^{\prime})_{I}}$${\pi_{1}(M_{b}^{\prime})_{I}}$${0}$${0}$${X_{\ast}(T^{\text{der}})_{I}}$${X_{\ast}(T)_{I}}$${\pi_{1}(M_{b})_{I}}$${0.}$$\scriptstyle{b}$$\scriptstyle{c}$

The snake lemma gives us an isomorphism $\ker c\simeq\ker b$ and since
$X_{\ast}(T^{\prime})_{I}$ is torsion free it follows that there is no torsion
in the kernel of

(37) $\displaystyle\pi_{1}(M_{b}^{\prime})_{I}\to\pi_{1}(M_{b})_{I}.$

Hence, the kernel of

(38)
$\displaystyle\pi_{1}(M_{b}^{\prime})_{I,\text{tors}}\to\pi_{1}(M_{b})_{I,\text{tors}}$

is trivial. ∎

######  Corollary 6.1.3.

The natural map $H\to
G^{\text{ab}}(\mathbb{Z}_{p}):=\mathbb{Z}(\mathbb{Z}_{p})$ induces an
isomorphism $H/H^{\prime}\simeq G^{\text{ab}}(\mathbb{Z}_{p})$.

###### Proof.

This follows by taking $\mathbb{Z}_{p}$-points of the short exact sequence
(32) and using Lang’s Lemma to deduce that the cohomology of
$\mathcal{J}_{\beta}^{\prime}$ vanishes. ∎

###  6.2. Main theorems

In this section we will state the main theorems in full generality, giving
Theorems 1 and 2 as special cases. The assumptions H1 and H2 below come into
play because they are necessary for applying Theorem 2 and Theorem 5.4.3 of
[vanHoften], respectively.

Let $(G,X)$ be a Shimura datum of Hodge type and let $p>2$ be a prime such
that $G_{\mathbb{Q}_{p}}$ is quasi-split and splits over a tamely ramified
extension. Suppose moreover that $G^{\text{der}}$ is simply connected and that
$G^{\text{ad}}$ is $\mathbb{Q}$-simple. Let $b\in B(G,\mu)$ be a non-basic
element. Let $\operatorname{Ig}_{b}$ the Igusa variety associated to it,
$J_{b}$ its twisted centraliser and write its derived group as
$J_{b}^{\text{der}}=J_{b,\text{an}}^{\text{der}}\times
J_{b,\text{iso}}^{\text{der}}$ with $J_{b,\text{an}}^{\text{der}}$ anisotropic
and $J_{b,\text{iso}}^{\text{der}}$ isotropic. As before, the classification
of reductive groups over $\mathbb{Q}_{p}$ tells us that

(39) $\displaystyle
J_{b,\text{an}}^{\text{der}}\simeq\prod_{i=1}^{k}\operatorname{Res}_{K_{i}/\mathbb{Q}_{p}}\operatorname{SL}_{1}(D_{i}),$

where $K_{i}/\mathbb{Q}_{p}$ is a finite extension and $D_{i}/K_{i}$ is a
central division algebra of rank $n_{i}^{2}$. We will assume that $p$ is
coprime to $n_{i}$ for all $i$. This is true if $p$ does not divide the order
of $W(G)$, because then $p$ does not divide the order of $W(J(b))$ since it is
an inner form of a Levi and hence does not divide the order of the Weyl group
of $J_{b,\text{an}}^{\text{der}}$, which is $n!$. Now consider the following
two sets of hypotheses, where $\tau$ is the unique basic element of
$B(G,\mu)$:

  1. H1

The group $G_{\mathbb{Q}_{p}}$ splits over an unramified extension.

  2. H2

The group $J_{\tau}^{\text{ad}}$ has no anisotropic factors and either
$\operatorname{Sh}_{U}$ is proper or [vanHoften]*Conjecture 3.7.5 holds.

We can now state our main theorem:

######  Theorem 6.2.1.

Let $(G,X)$ be as above and suppose that either H1 or H2 holds. Then there is
a bijection

(40)
$\displaystyle\pi_{0}(\operatorname{Ig}_{b})\simeq\pi_{0}(\operatorname{Sh}_{G})\times
G^{ab}(\mathbb{Z}_{p}),$

where $G^{ab}(\mathbb{Z}_{p})$ is the group of $\mathbb{Z}_{p}$-points of the
connected Néron model of $G^{\text{ab}}(\mathbb{Q}_{p})$.

###### Proof of Theorem 6.2.1.

Theorem E of [ShenYuZhang] tells us that we can arrange $C(x)$ such that it is
an EKOR stratum and Theorem 2 (under assumption H1) and [vanHoften]*Theorem
5.4.3 (under assumption H2) tell us that the natural map

(41) $\displaystyle\pi_{0}(C(x))\to\pi_{0}(\operatorname{Sh}_{G})$

is a bijection. Proposition 3.0.3 and Proposition 5.1.3 tell us that
$J_{b}^{\text{der}}(\mathbb{Q}_{p})$ acts trivially on
$\pi_{0}(\operatorname{Ig}_{b}).$ Moreover, it follows from Proposition 5.2.2
that $J_{b}^{\prime}(\mathbb{Q}_{p})$ acts trivially on
$\pi_{0}(\operatorname{Ig}_{b})$ and hence
$\pi_{0}(\operatorname{Ig}_{b})\simeq H/H^{\prime}$. We conclude by applying
the fact that $H/H^{\prime}\simeq G^{\text{ab}}(\mathbb{Z}_{p})$, as proven in
Section 6.1 ∎

######  Corollary 6.2.2.

There is a bijection

(42)
$\displaystyle\pi_{0}(C(x))\simeq\pi_{0}(\operatorname{Sh}_{G,b})\times\frac{G^{\text{ab}}(\mathbb{Z}_{p})}{H_{C}^{\text{ab}}},$

where $H_{C}^{\text{ab}}$ is the image of $H_{C}$ in
$G^{\text{ab}}(\mathbb{Z}_{p})$ via the natural map $J_{b}\to G^{\text{ab}}$.
The group $H_{C}^{\text{ab}}$ is of finite index in
$G^{\text{ab}}(\mathbb{Z}_{p})$ because it always contains the image of
$Z_{G}(\mathbb{Z}_{p})$.

######  Remark 6.2.3.

It follows from the proof of Lemma 8.1.1 of [KretShin] that there is a maximal
torus $T\subset J_{b}(\mathbb{Q}_{p})$ such that $T(\mathbb{Q}_{p})\cap H$
acts transitively on $H/H^{\prime}$, confirming an assumption of Eischen-
Mantovan, see of [eischen2017p]*Remark 3.3.4.

Now we state the generalisation of Theorem 2.

######  Theorem 6.2.4.

Let $(G,X)$ be as above and suppose that either H1 or H2 holds, then the
natural map

(43)
$\displaystyle\pi_{0}(\operatorname{Sh}_{G,b})\to\pi_{0}(\operatorname{Sh}_{G})$

is an isomorphism. Moreover the number of irreducible components of each
connected component of $\operatorname{Sh}_{G,b}$ is given by the
representation theoretic constant

(44)
$\displaystyle\operatorname{Dim}V^{\hat{H}}_{\underline{\mu}}(\lambda_{b})_{\text{rel}},$

introduced in the appendix of [ZhouZhu].

###### Proof.

Let $X_{\mu}(b)$ be the affine Deligne-Lusztig variety for $G$ associated to
$b$ and $\mu$, this is a perfect scheme over $\overline{\mathbb{F}}_{p}$. The
Mantovan product formula tells us that there is a morphism

(45) $\displaystyle
X_{\mu}(b)\times\operatorname{Ig}_{b}\to\operatorname{Sh}_{G,b}^{\text{perf}},$

which is a $J_{b}(\mathbb{Q}_{p})$-torsor. Therefore, in order to determine
connected or irreducible components of $\operatorname{Sh}_{G,b}$, it suffices
to determine the $J_{b}(\mathbb{Q}_{p})$-orbits of connected or irreducible
components of $X_{\mu}(b)\times\operatorname{Ig}_{b}$.

Theorem 3.1.1 of [ZhouZhu] tells us that the stabiliser in
$J_{b}(\mathbb{Q}_{p})$ of an irreducible component of $X_{\mu}(b)$ is a
parahoric subgroup. If $g\in J_{b}(\mathbb{Q}_{p})$ stabilises an irreducible
component $Z$ of $X_{\mu}(b)$, then it also stabilises the connected component
that $Z$ lies in. Thus, the stabiliser of a connected component contains a
parahoric subgroup.

By Theorem A.1.4. of [vanHoften], $J_{b}(\mathbb{Q}_{p})$ acts transitively on
$\pi_{0}(X_{\mu}(b))$. The stabiliser of any connected component surjects onto
$G^{\text{ab}}(\mathbb{Z}_{p})$ because it contains a parahoric subgroup, and
therefore $J_{b}(\mathbb{Q}_{p})$ acts transitively on the fibers of

(46)
$\displaystyle\pi_{0}\left(X_{\mu}(b)\times\operatorname{Ig}_{b}\right)\to\pi_{0}(\operatorname{Sh}_{G}).$

Since the natural map
$\pi_{0}(\operatorname{Sh}_{G,b})\to\pi_{0}(\operatorname{Sh}_{G})$ is
surjective by the proof of Axiom 5 in [Zhou], the result on $\pi_{0}$ follows.

Theorem A.3.1 of [ZhouZhu] tells us that the number of
$J_{b}(\mathbb{Q}_{p})$-orbits of irreducible components in $X_{\mu}(b)$ is
given by

(47) $\displaystyle
N:=\operatorname{Dim}V^{\hat{H}}_{\underline{\mu}}(\lambda_{b})_{\text{rel}}.$

Since the stabiliser of each irreducible component is a parahoric subgroup
which acts transitively on the fibers of
$\pi_{0}(\operatorname{Ig}_{b})\to\pi_{0}(\operatorname{Sh}_{G})$, we find
that the irreducible components of $\operatorname{Sh}_{G,b}$ are in bijection
with

(48) $\displaystyle N\times\pi_{0}(\operatorname{Sh}_{G})\simeq
N\times\pi_{0}(\operatorname{Sh}_{G,b})$

and the result follows. ∎

###  6.3. A counterexample to the discrete Hecke orbit conjecture (due to Rong
Zhou)

Our counterexample is a unitary Shimura variety of PEL type. Let $F=F^{+}E$ be
a CM field where $F^{+}$ is totally real of degree $4$ and $E$ is an imaginary
quadratic field. Let $V$ be a Hermitian $F$ vector space of rank $2$ with
signature $(1,1)$ at all infinite places of $F^{+}$ and let
$(G,X)=(\operatorname{GU}_{V},X_{V})$ be the corresponding Shimura datum of
PEL type.

Let $p$ be a prime which splits in $E$ and which splits as
$p=\mathfrak{p}_{1}\mathfrak{p}_{2}$ in $F^{+}$ where both $\mathfrak{p}_{1}$
and $\mathfrak{p}_{2}$ have residual degree $2$. Then

(49) $\displaystyle
G_{\mathbb{Q}_{p}}\simeq\operatorname{Res}_{K/\mathbb{Q}_{p}}\operatorname{GL}_{2}\times\operatorname{Res}_{K/\mathbb{Q}_{p}}\operatorname{GL}_{2}\times\mathbb{G}_{m},$

where $K/\mathbb{Q}_{p}$ is the unique unramified quadratic extension. Let $b$
be the element which is $\mu_{1}$-ordinary in the first factor and
$\mu_{2}$-basic in the second factor and $\mu_{3}$-basic in the third factor.
Here $\mu=(\mu_{1},\mu_{2},\mu_{3})$ according to the product decomposition of
$G_{\mathbb{Q}_{p}}$.

We have seen above that $H=H_{x}$ is the precisely the stabiliser in
$J_{b}(\mathbb{Q}_{p})$ of an $\overline{\mathbb{F}}_{p}$-point $x$ of the
affine Deligne-Lusztig variety $X(\mu,b)$. By the product decomposition of $G$
we get a corresponding product decomposition of $X(\mu,b)$ and therefore

(50) $\displaystyle H_{x}=H_{1}\times H_{2}\times H_{3}.$

It suffices to show that $H_{2}$ does not surject onto the maximal abelian
quotient of $\operatorname{Res}_{K/\mathbb{Q}_{p}}\operatorname{GL}_{2}$. In
this case
$J_{b_{2}}=\operatorname{Res}_{K/\mathbb{Q}_{p}}\operatorname{GL}_{2}$ and
$X(\mu_{2},b_{2})$ is equidimensional of dimension $1$.

######  Lemma 6.3.1.

The irreducible components of $X(\mu_{2},b_{2})$ are isomorphic to
$\mathbb{P}^{1}$. The stabiliser of an irreducible component is a hyperspecial
subgroup of $\operatorname{GL}_{2}(K)$ isomorphic to
$\operatorname{GL}_{2}(\mathcal{O}_{K})$ which acts on $\mathbb{P}^{1}$ via
the natural map
$\operatorname{GL}_{2}(\mathcal{O}_{K})\to\operatorname{GL}_{2}(\mathbb{F}_{q^{2}})$.

###### Proof.

It follows from the main result of [GortzHeNie] that $X(\mu_{2},b_{2})$ is a
union of two one-dimensional Ekedahl-Oort strata and one zero-dimensional
Ekedahl-Oort stratum. To prove the lemma, it suffices to work with the one-
dimensional Ekedahl-Oort strata individually. It follows from paragraph 5.10
of [GortzHeNie] that these are each unions of (closures of) classical Deligne-
Lusztig varieties for for the group
$\operatorname{Res}_{\mathbb{F}_{p^{2}}/\mathbb{F}_{p}}$. Therefore their
irreducible components are isomorphic to $\mathbb{P}^{1}$ and the action of
$\operatorname{GL}_{2}(\mathcal{O}_{K})$ factors through the natural action of
$\operatorname{GL}_{2}(\mathbb{F}_{q^{2}})$. ∎

A direct computation shows that we can find a point
$x\in\mathbb{P}^{1}(\overline{\mathbb{F}}_{p})$ such that its stabiliser in
$\operatorname{GL}_{2}(\mathbb{F}_{q^{2}})$ is given by the group of scalar
matrices, and therefore does not surject onto $\mathbb{F}_{p^{2}}$ via the
determinant map. This implies that the stabiliser of $x$ in
$\operatorname{GL}_{2}(\mathcal{O}_{K})$ and therefore in
$J_{b}(\mathbb{Q}_{p})$ does not surject onto $\mathcal{O}_{K}^{\times}$ via
the determinant map.

###  6.4. Acknowledgements

We would like to thanks Marco D’Addezio for answering our mathoverflow
question and making his work in progress [d2020monodromy] available to us. We
would also like to thank Andrew Graham, Xuhua He, João Lourenço, Ambrus Pal,
Syed Waqar Ali Shah, Sug-Woo Shin, Jize Yu and Rong Zhou for helpful
discussions and correspondences. We are grateful to James Newton for comments
on an earlier version of this work.

## References