injective for $i,j\in I$ and $j\mapsto i$ (replacing $A_{i}$ by its image in
$A_{j}$).

We assume that there is a perfectoid space
$\mathcal{X}=\mathrm{Spa}(A,A^{+})\sim\lim_{i}\mathcal{X}_{i}$. Then $A^{0}$
is the $p$-adic completion of $\mathrm{colim}_{i}A_{i}^{0}$ and $A^{0}$ is a
perfectoid $\mathcal{O}_{F}$-algebra: there is $\varpi\in A^{0}$ with
$\varpi^{p}\mid p$ and the Frobenius morphism
$\phi:A^{0}/\varpi^{p}\rightarrow A^{0}/\varpi^{p}$ is surjective. Moreover,
$A=A^{0}[1/p]$ and $A^{+}$ is the closure of $\mathrm{colim}A_{i}^{+}$ in $A$.
By approximation, we may assume that $\varpi\in A^{0}_{i}$ and by projection
we get an element $\varpi\in A_{i,e}$.

We need to see that the map
$\phi:\mathrm{colim}_{i}A_{i,e}^{0}/\varpi^{p}\rightarrow\mathrm{colim}_{i}A_{i,e}^{0}/\varpi^{p}$
is surjective. Let $x_{i,e}\in A_{i,e}/\varpi^{p}$. Since $A^{0}$ is
perfectoid, we see that there exists $j\mapsto i$ and $y_{j,e}\in
A_{j,e}/\varpi^{p}$ such that $y_{j,e}^{p}=x_{i,e}$. ∎

######  Lemma 4.4.7.

Let $\\{\mathcal{X}_{i}\\}_{i\in I}$ be a cofiltered inverse system of locally
of finite type separated adic spaces over a perfectoid field
$\mathrm{Spa}(F,\mathcal{O}_{F})$ with finite transition maps. Let $G$ be a
finite group acting on the inverse system via
$\mathrm{Spa}(F,\mathcal{O}_{F})$-morphisms. Let $\mathcal{X}$ be a perfectoid
space such that $\mathcal{X}\sim\lim_{i}\mathcal{X}_{i}$. Assume that for some
index $i$, we have a $G$-invariant covering of $\mathcal{X}_{i}$ by pregood
affinoids. Then the categorical quotient $\mathcal{Y}_{i}=\mathcal{X}_{i}/G$
is representable by an adic space for a cofinal subset of $I$, the categorial
quotient $\mathcal{Y}=\mathcal{X}/G$ is representable by a perfectoid space,
and $\mathcal{Y}\sim\lim_{i}\mathcal{Y}_{i}$.

###### Proof.

The quotient $\mathcal{X}_{i}/G$ exist by [Han19], thm. 1.3. We may reduce to
the affine case with $\mathcal{X}=\mathrm{Spa}(A,A^{+})$ and
$\mathcal{X}_{i}=\mathrm{Spa}(A_{i},A_{i}^{+})$. By [Han19], thm 1.4, $A^{G}$
is perfectoid. It is clear that $\mathrm{colim}A_{i}^{G}$ is dense in $A^{G}$
since we have a projector $A\rightarrow A^{G}$,
$a\mapsto\frac{1}{|G|}\sum_{g\in G}g.a$. ∎

####  4.4.8. Siegel Shimura varieties

We assume in this paragraph that $(G,X)$ is the Siegel Shimura datum
$(\mathrm{GSp}_{2g},\mathcal{H}_{g})$. Let $K=K^{p}K_{p}\subseteq
G(\mathbb{A}_{f})$ be a compact open subgroup. The reflex field is
$\mathbb{Q}$ and the Shimura variety $S_{K}$ is a moduli space of abelian
varieties $A$, with a level structure and polarization (prescribed by $K$).

The Shimura variety ${S}_{K}$ carries a right pro-étale
$G(\mathbb{Q}_{p})$-torsor. Namely, equip $\mathbb{Q}_{p}^{2g}$ with the
standard symplectic form and consider the torsor of isomorphisms
$\mathbb{Q}_{p}^{2g}\rightarrow\mathrm{H}_{1}(A,\mathbb{Q}_{p})$, respecting
the symplectic forms up to a similitude factor, where $A$ is the universal
abelian scheme (defined up to isogeny) and
$\mathrm{H}_{1}(A,\mathbb{Q}_{p})=V_{p}(A)$ is the rational Tate module of
$A$. After choosing a geometric point $\overline{x}\rightarrow S_{K}$, this
torsor corresponds to a representation of the algebraic fundamental group
$\pi_{1}(S_{K},\overline{x})\rightarrow G(\mathbb{Q}_{p})$. The image of this
morphism lies in the compact open subgroup $K_{p}\subset G(\mathbb{Q}_{p})$
and the corresponding $K_{p}$-torsor is realized geometrically by the tower of
Shimura varieties $\lim_{K^{\prime}_{p}\subseteq
K_{p}}S_{K^{p}K_{p}^{\prime}}$. By pullback to the adic space
$\mathcal{S}^{an}_{K}$, we get a pro-étale $G(\mathbb{Q}_{p})$-torsor
$\mathcal{G}^{an}_{pet,p}$. If $K_{p}\subseteq G(\mathbb{Z}_{p})$, this torsor
has a $G(\mathbb{Z}_{p})$-reduction of group structure that we denote by
$\mathcal{G}_{pet,p}$.

The (relative) Hodge-Tate filtration is the exact sequence of pro-étale
sheaves over $\mathcal{S}^{an}_{K}$:

$0\rightarrow\mathrm{Lie}(A)\otimes_{\mathscr{O}_{\mathcal{S}^{an}_{K}}}\hat{\mathscr{O}}_{\mathcal{S}^{an}_{K}}\rightarrow\mathrm{H}_{1}(A,\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\hat{\mathscr{O}}_{\mathcal{S}_{K}}\rightarrow\omega_{A^{t}}\otimes_{\mathscr{O}_{\mathcal{S}^{an}_{K}}}\hat{\mathscr{O}}_{\mathcal{S}^{an}_{K}}\rightarrow
0$

The groups $\mathcal{P}^{an}_{\mu}$, $\mathcal{G}^{an}$ and
$\mathcal{M}^{an}_{\mu}$ are naturally sheaves over the étale site of
$\mathcal{S}^{an}_{K}$. We extend them to sheaves on the pro-étale site of
$\mathcal{S}^{an}_{K}$ as follows: if $H$ is any of these groups, and if
$U\rightarrow\mathcal{S}^{an}_{K}$ is an object of the pro-étale site, we let

$H(U)=H(\hat{\mathscr{O}}_{\mathcal{S}^{an}_{K}}(U),\hat{\mathscr{O}}^{+}_{\mathcal{S}^{an}_{K}}(U)).$

######  Remark 4.4.9.

There is also an extension to the pro-étale site of $\mathcal{P}^{an}_{\mu}$,
$\mathcal{G}^{an}$ and $\mathcal{M}^{an}_{\mu}$ where one takes sections with
values in the uncompleted structure sheaf, but we will not need to consider
this extension.

The Hodge-Tate filtration gives a $\mathcal{P}^{an}_{\mu}$-reduction of
structure group $\mathcal{P}^{an}_{HT}$ of the $\mathcal{G}^{an}$-torsor
$\mathcal{G}^{an}_{pet,p}\times^{\mathcal{G}^{an}(\mathbb{Q}_{p})}\mathcal{G}^{an}$.
Namely, we consider trivializations of
$\mathrm{H}_{1}(A,\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\hat{\mathscr{O}}_{\mathcal{S}_{K}}$
which respect the filtration. This is a right $\mathcal{P}^{an}_{\mu}$-torsor.

We can consider the pushout
$\mathcal{P}^{an}_{HT}\times^{\mathcal{P}^{an}_{\mu}}\mathcal{M}^{an}_{\mu}:=\mathcal{M}^{an}_{HT}$.
This pro-étale torsor actually identifies canonically with (the pull back to
the pro-étale site of) the analytic torsor $\mathcal{M}^{an}_{dR}$, which is
the analytification of $M_{dR}$ (see section 4.1.1).

The pro-étale $K_{p}$-torsor $\lim_{K^{\prime}_{p}\subseteq
K_{p}}S_{K^{p}K^{\prime}_{p}}$ over $S_{K}$ extends to a pro-Kummer étale
$K_{p}$-torsor $\lim_{K^{\prime}_{p}\subseteq
K_{p}}S^{tor}_{K^{p}K^{\prime}_{p},\Sigma}$ over $S^{tor}_{K,\Sigma}$. We can
pull it back to the analytic space $\mathcal{S}_{K,\Sigma}^{tor}$ (see
[DLLZ19] for the definition of the pro-Kummer étale site). By pushout along
$K_{p}\rightarrow G(\mathbb{Q}_{p})$, we get a pro-Kummer étale
$G(\mathbb{Q}_{p})$-torsor over $\mathcal{S}^{tor}_{K,\Sigma}$ extending
$\mathcal{G}^{an}_{pet,p}$, which we also denote by
$\mathcal{G}^{an}_{pet,p}$. If $K_{p}\subseteq G(\mathbb{Z}_{p})$ we also have
the pro-Kummer étale $G(\mathbb{Z}_{p})$-torsor $\mathcal{G}_{pet,p}$ over
$\mathcal{S}^{tor}_{K,\Sigma}$.

Let $A_{\Sigma}$ be the semi-abelian scheme over
$\mathcal{S}^{tor}_{K,\Sigma}$. The Hodge-Tate exact sequence extends to a
sequence over the pro-Kummer étale site

$0\rightarrow\mathrm{Lie}(A_{\Sigma})\otimes_{\mathscr{O}_{\mathcal{S}^{tor}_{K,\Sigma}}}\hat{\mathscr{O}}_{\mathcal{S}^{tor}_{K,\Sigma}}\rightarrow\mathrm{H}_{1}(A_{\Sigma},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\hat{\mathscr{O}}_{\mathcal{S}^{tor}_{K,\Sigma}}\rightarrow\omega_{A_{\Sigma}^{t}}\otimes_{\mathscr{O}_{\mathcal{S}^{tor}_{K,\Sigma}}}\hat{\mathscr{O}}_{\mathcal{S}^{an}_{K}}\rightarrow
0$

Therefore, the torsors $\mathcal{P}^{an}_{HT}$ and $\mathcal{M}^{an}_{HT}$
extend over $\mathcal{S}^{tor}_{K,\Sigma}$, and as usual, we continue to use
the same notation for the extensions. Moreover, again by construction, the
torsors $\mathcal{M}^{an}_{HT}$ and $\mathcal{M}^{an}_{dR}$ are canonically
identified.

####  4.4.10. Perfectoid Siegel Shimura varieties

By [Sch15], thm. III.3.17 there is a perfectoid space
$\mathcal{S}^{\star}_{{K^{p}}}\sim\lim_{K_{p}}\mathcal{S}^{\star}_{{K^{p}K_{p}}}$.

By [Sch15], we have a $G(\mathbb{Q}_{p})$-equivariant map
$\pi_{HT}:\mathcal{S}^{\star}_{{K^{p}}}\rightarrow\mathcal{FL}_{G,\mu}$.
Moreover, there exists an affinoid covering
$\mathcal{FL}_{G,\mu}=\cup_{i}V_{i}$ such that for each $i$,
$\pi_{HT}^{-1}(V_{i})$ is a good affinoid perfectoid open subset of
$\mathcal{S}^{\star}_{{K^{p}}}$ (see definition 4.4.3).

The construction of the map
$\pi_{HT}:\mathcal{S}^{\star}_{{K^{p}}}\rightarrow\mathcal{FL}_{G,\mu}$ is
delicate at the boundary, but over the complement of the boundary
$\mathcal{S}^{an}_{{K^{p}}}\sim\lim_{K_{p}}\mathcal{S}^{an}_{K^{p}K_{p}}$ it
has a simple description which is given below.

By [PS16], thm. 0.4, for any cone decomposition $\Sigma$, there is also a
perfectoid space
$\mathcal{S}^{tor}_{{K^{p}},\Sigma}\sim\lim_{K_{p}}\mathcal{S}^{tor}_{{K^{p}K_{p}},\Sigma}$
(it is important that the cone decomposition does not depend on $K_{p}$ for
the limit to be perfectoid) and we have a map
$\mathcal{S}^{tor}_{{K^{p}},\Sigma}\rightarrow\mathcal{S}^{\star}_{{K^{p}}}$
of perfectoid spaces induced by the maps at finite level $K_{p}$.

The torsor $\mathcal{G}^{an}_{pet,p}$ becomes trivial over
$\mathcal{S}^{tor}_{{K^{p}},\Sigma}$ and we therefore get a Hodge-Tate period
map
$\pi^{tor}_{HT}:\mathcal{S}^{tor}_{{K^{p}},\Sigma}\rightarrow\mathcal{FL}_{G,\mu}$.
Let us explain very concretely how this map is defined. Let
$\mathrm{Spa}(R,R^{+})$ be a perfectoid affinoid open subset of
$\mathcal{S}^{tor}_{{K^{p}},\Sigma}$. We can evaluate the sequence

$0\rightarrow\mathrm{Lie}(A_{\Sigma})\otimes_{\mathscr{O}_{\mathcal{S}^{tor}_{K,\Sigma}}}\hat{\mathscr{O}}_{\mathcal{S}^{tor}_{K,\Sigma}}\rightarrow\mathrm{H}_{1}(A_{\Sigma},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\hat{\mathscr{O}}_{\mathcal{S}^{tor}_{K,\Sigma}}\rightarrow\omega_{A_{\Sigma}^{t}}\otimes_{\mathscr{O}_{\mathcal{S}^{tor}_{K,\Sigma}}}\hat{\mathscr{O}}_{\mathcal{S}^{an}_{K}}\rightarrow
0$

on $(R,R^{+})$ (viewed as an object of the pro-Kummer-étale site of
$\mathcal{S}^{tor}_{K,\Sigma}$) and use the trivialization
$\mathbb{Q}_{p}^{2g}\simeq\mathrm{H}_{1}(A_{\Sigma},\mathbb{Q}_{p})$ to get an
exact sequence:

$0\rightarrow\mathrm{Lie}(A_{\Sigma})\otimes R\rightarrow
R^{2g}\rightarrow\omega_{A_{\Sigma}^{t}}\otimes R\rightarrow 0$

After localizing, we may even assume that $\mathrm{Lie}(A_{\Sigma})\otimes R$
and $\omega_{A_{\Sigma}^{t}}\otimes R$ are free $R$-modules. Now let
$0\rightarrow R^{g}\rightarrow R^{2g}\rightarrow R^{g}\rightarrow 0$ be the
(polarized) chain with automorphism group $P_{\mu}(R)$. We have that
$\mathcal{P}_{HT}^{an}(R,R^{+})=$

$\mathrm{Isom}_{\mathrm{symp}}(0\rightarrow R^{g}\rightarrow R^{2g}\rightarrow
R^{g}\rightarrow 0,0\rightarrow\mathrm{Lie}(A_{\Sigma})\otimes R\rightarrow
R^{2g}\rightarrow\omega_{A_{\Sigma}^{t}}\otimes R\rightarrow 0)$

and
$\mathcal{P}_{HT}^{an}(R,R^{+})\subseteq\mathrm{Isom}_{\mathrm{symp}}(R^{2g})=\mathrm{GSp}_{2g}(R).$
This is a right $P_{\mu}(R)$-torsor and there is an element $x\in G(R)$ such
that $\mathcal{P}_{HT}^{an}(R,R^{+})=xP_{\mu}(R)$. The automorphism group of
$0\rightarrow\mathrm{Lie}(A_{\Sigma})\otimes R\rightarrow
R^{2g}\rightarrow\omega_{A_{\Sigma}^{t}}\otimes R\rightarrow 0$ is
$xP_{\mu}(R)x^{-1}$. Finally we let
$\pi_{HT}^{tor}(\mathrm{Spa}(R,R^{+}))=x^{-1}\in FL_{G,\mu}(R)$.

######  Remark 4.4.11.

We are forced to use $x^{-1}$ above because $FL_{G,\mu}=P_{\mu}\backslash G$.
Note that taking the quotient by the left action of $P_{\mu}$ is natural
because right translation on $G$ defines a right $G$-action of $FL_{G,\mu}$
and the map
$\pi_{HT}:\mathcal{S}^{\star}_{{K^{p}}}\rightarrow\mathcal{FL}_{G,\mu}$ is
equivariant for the right $G(\mathbb{Q}_{p})$-action. We chose to define
$\mathcal{P}_{HT}^{an}$ as a right $\mathcal{P}_{\mu}^{an}$-torsor, because we
want to identify the torsors $\mathcal{M}_{HT}^{an}$ and
$\mathcal{M}_{dR}^{an}$. But in the classical theory, $M_{dR}$ is a right
torsor. It means that in our convention, the torsor $\mathcal{P}_{HT}^{an}$ is
pulled back via $\pi_{HT}^{tor}$ from the torsor
$\mathcal{G}^{an}\rightarrow\mathcal{FL}_{G,\mu}$, $x\mapsto x^{-1}$.

The maps $\pi^{tor}_{HT}$ and $\pi_{HT}$ coincide by construction on the open
subset $\mathcal{S}^{an}_{{K^{p}}}$. We deduce that we have a commutative
diagram:

The key properties of this diagram that we will use are:

  * •

The pull back of the torsor
$\mathcal{G}^{an}/U_{\mathcal{P}_{\mu}^{an}}\rightarrow\mathcal{FL}_{G,\mu}$
via $\pi_{HT}^{tor}$ is $\mathcal{M}_{HT}^{an}$ and this is canonically
identified with the pull back via
$\mathcal{S}^{tor}_{{K^{p}},\Sigma}\rightarrow\mathcal{S}^{tor}_{{K^{p}K_{p}},\Sigma}$
of $\mathcal{M}_{dR}^{an}$.

  * •

The map $\pi_{HT}$ is affine.

####  4.4.12. Formal models of perfectoid Siegel Shimura varieties

We need to consider formal models of the perfectoid Siegel Shimura varieties
in order to be able to use the vanishing result below (theorem 4.4.23). We
first recall a number of statements from [PS16]. Let $K=K^{p}K_{p}$. For
$K_{p}=\mathrm{GSp}_{2g}(\mathbb{Z}_{p})$, we have natural models over
$\mathrm{Spec}~{}\mathbb{Z}_{p}$ for $S_{K^{p}K_{p}}$,
$S^{\star}_{K^{p}K_{p}}$, and $S^{tor}_{K^{p}K_{p},\Sigma}$ that we denote
$\mathbf{S}_{K^{p}K_{p}}$, $\mathbf{S}^{\star}_{K^{p}K_{p}}$, and
$\mathbf{S}^{tor}_{K^{p}K_{p},\Sigma}$. We can also consider the corresponding
$p$-adic formal schemes $\mathfrak{S}_{K^{p}K_{p}}$,
$\mathfrak{S}^{\star}_{K^{p}K_{p}}$, and
$\mathfrak{S}^{tor}_{K^{p}K_{p},\Sigma}$. We denote by $A$ the semi-abelian
scheme over $\mathfrak{S}^{tor}_{K^{p}K_{p},\Sigma}$ (it is defined up to
prime-to-$p$ isogeny by our choice of level structure). By the construction of
the minimal compactification, the line bundle $\det\omega_{A}$ on
$\mathfrak{S}_{K^{p}K_{p}}$ extends to a line bundle that we also denote by
$\det\omega_{A}$ on $\mathfrak{S}^{\star}_{K^{p}K_{p}}$, and whose pullback to
$\mathfrak{S}^{tor}_{K^{p}K_{p},\Sigma}$ agrees with the extension defined by
the semi-abelian scheme $A$.

Now let $K_{p}\subseteq\mathrm{GSp}_{2g}(\mathbb{Z}_{p})$ be an open subgroup.
We can define $\mathfrak{S}_{K^{p}K_{p}}$,
$\mathfrak{S}^{\star}_{K^{p}K_{p}}$, and
$\mathfrak{S}^{tor}_{K^{p}K_{p},\Sigma}$ as the normalizations of
$\mathfrak{S}_{K^{p}\mathrm{GSp}_{2g}(\mathbb{Z}_{p})}$,
$\mathfrak{S}^{\star}_{K^{p}\mathrm{GSp}_{2g}(\mathbb{Z}_{p})}$, and
$\mathfrak{S}^{tor}_{K^{p}\mathrm{GSp}_{2g}(\mathbb{Z}_{p}),\Sigma}$ in
$\mathcal{S}_{K^{p}K_{p}}$, $\mathcal{S}^{\star}_{K^{p}K_{p}}$, and
$\mathcal{S}^{tor}_{K^{p}K_{p},\Sigma}$ respectively.

Let us denote by
$K_{p,n}=\\{M\in\mathrm{GSp}_{2g}(\mathbb{Z}_{p}),~{}M=1~{}\mod p^{n}\\}$.
This is the principal level $p^{n}$ subgroup. Consider the module
$\mathbb{Z}^{2g}$ with canonical basis $e_{1},\cdots,e_{2g}$, equipped with
the standard symplectic form $\langle,\rangle$ given by $\langle
e_{i},e_{2g-i+1}\rangle=1$ for $1\leq i\leq g$, and $\langle
e_{i},e_{j}\rangle=0$ if $i+j\neq 2g+1$. Over $\mathcal{S}_{K^{p}K_{p,n}}$ we
have a symplectic isomorphism $(\mathbb{Z}/p^{n}\mathbb{Z})^{2g}\rightarrow
A[p^{n}]$ (up to a similitude factor), and it extends to a morphism of group
schemes over $\mathfrak{S}_{K^{p}K_{p,n}}$,
$(\mathbb{Z}/p^{n}\mathbb{Z})^{2g}\rightarrow A[p^{n}]$. We have the Hodge-
Tate morphism $\mathrm{HT}_{n}:A[p^{n}]\rightarrow\omega_{A^{t}}/p^{n}$. Using
the prime-to-$p$ polarization, we can identify $\omega_{A^{t}}$ and
$\omega_{A}$. We therefore have sections
$\mathrm{HT}_{n}(e_{i})\in\mathrm{H}^{0}(\mathfrak{S}_{K^{p}K_{p,n}},\omega_{A}/p^{n})$.
We also have a map
$\Lambda^{g}\mathrm{HT}_{n}:\Lambda^{g}(\mathbb{Z}/p^{n}\mathbb{Z})^{2g}\rightarrow\det\omega_{A}/p^{n}$.
Let $r=$${g}\choose{2g}$. Let $f_{1},\cdots,f_{r}$ be a basis of
$\Lambda^{g}\mathbb{Z}^{2g}$ obtained by taking exterior products of
$e_{1},\cdots,e_{2g}$. We get sections
$\Lambda^{g}\mathrm{HT}_{n}(f_{i})\in\mathrm{H}^{0}(\mathfrak{S}_{K^{p}K_{p,n}},\det\omega_{A}/p^{n})$.

By [PS16], proposition 1.7 and corollary 1.7, the sections
$\mathrm{HT}_{n}(e_{i})$ extend to sections
$\mathrm{HT}_{n}(e_{i})\in\mathrm{H}^{0}(\mathfrak{S}^{tor}_{K^{p}K_{p,n},\Sigma},\omega_{A}/p^{n})$
and the sections $\Lambda^{g}\mathrm{HT}_{n}(f_{i})$ extend to sections
$\Lambda^{g}\mathrm{HT}_{n}(f_{i})\in\mathrm{H}^{0}(\mathfrak{S}^{\star}_{K^{p}K_{p,n}},\det\omega_{A}/p^{n})$.

It follows that over $\mathfrak{S}^{\star}_{K^{p}K_{p,n}}$ we have a morphism
$\Lambda^{g}\mathrm{HT}_{n}:\Lambda^{g}(\mathbb{Z}/p^{n}\mathbb{Z})^{2g}\rightarrow\det\omega_{A}/p^{n}$.
These morphisms satisfy the natural compatibilities as $n$ varies. The
cokernel of this morphism is killed by $p^{\frac{g}{p-1}}$ (resp. $4^{g}$ if
$p=2$) by [Far10], theorem 7. We let $\det\omega_{A}^{mod,n}$ be the subsheaf
of $\det\omega_{A}$ which is the inverse image of

$\mathrm{Im}(\Lambda^{g}\mathrm{HT}_{n}(\Lambda^{g}(\mathbb{Z}/p^{n}\mathbb{Z})^{2g})\otimes\mathscr{O}_{\mathfrak{S}^{\star}_{K_{p,n}K^{p}}}\rightarrow\det\omega_{A}/p^{n})$

in $\det\omega_{A}$. We have
$p^{\frac{g}{p-1}}\det\omega_{A}\subseteq\det\omega_{A}^{mod,n}\subseteq\det\omega_{A}$
(resp.
$4^{{g}}\det\omega_{A}\subseteq\det\omega_{A}^{mod,n}\subseteq\det\omega_{A}$
if $p=2$) .

Let
$f:\mathfrak{S}^{\star}_{K^{p}K_{p,n+1}}\rightarrow\mathfrak{S}^{\star}_{K^{p}K_{p,n}}$
be the projection. We clearly have a map
$\det\omega_{A}^{mod,n+1}\rightarrow\mathrm{Im}(f^{\star}\det\omega_{A}^{mod,{n}}\rightarrow\det\omega_{A})$.
This map is an isomorphism if $n\geq\frac{g}{p-1}$ (resp. $n\geq 2g$ if
$p=2$). We now assume that $n$ is larger than $n_{0}=\frac{g}{p-1}$ (resp.
$n_{0}=2g$ if $p=2$). We simply denote $\det\omega_{A}^{mod,n}$ by
$\det\omega_{A}^{mod}$, this is a subsheaf of $\det\omega_{A}$. The sheaf
$\det\omega_{A}^{mod}$ is not locally free. We can perform a blow up to make
it locally free. Let us start with a definition:

######  Definition 4.4.13.

Let $\mathfrak{X}$ be a $p$-adic formal scheme, locally of finite type over
$\mathrm{Spf}~{}\mathbb{Z}_{p}$, and let $\mathcal{I}$ be a coherent sheaf of
ideals such that $p\in\sqrt{\mathcal{I}}$. We let
$\mathrm{BL}_{\mathcal{I}}(\mathfrak{X})$ be the $p$-adic formal scheme
obtained by taking the admissible blow-up of $\mathfrak{X}$ at $\mathcal{I}$.
We let $\mathrm{NBL}_{\mathcal{I}}(\mathfrak{X})$ be the normalization of
$\mathrm{BL}_{\mathcal{I}}(\mathfrak{X})$, this is the normalized blow-up.

######  Remark 4.4.14.

The normalization of a formal scheme is well defined in our context by
[Con99], coro. 1.2.3.

Let $\mathcal{I}_{n}$ be the sheaf of ideals of
$\mathscr{O}_{\mathfrak{S}^{\star}_{K^{p}K_{p,n}}}$ given by
$\mathcal{I}_{n}=\\{a\in\mathscr{O}_{\mathfrak{S}^{\star}_{K^{p}K_{p,n}}},a\det\omega_{A}\subseteq\det\omega_{A}^{mod}\\}$.
We let
${\mathfrak{S}}^{\star,mod}_{K^{p}K_{p,n}}=\mathrm{NBL}_{\mathcal{I}_{n}}(\mathfrak{S}^{\star}_{K^{p}K_{p,n}})$.
Over ${\mathfrak{S}}^{\star,mod}_{K^{p}K_{p,n}}$, the sheaf
$\det\omega_{A}^{mod}$ is locally free and moreover the map

$\Lambda^{g}\mathrm{HT}_{n}:\Lambda^{g}(\mathbb{Z}/p^{n}\mathbb{Z})^{2g}\rightarrow\det\omega_{A}/p^{n}$

induces a surjective map

$\Lambda^{g}\mathrm{HT}^{\prime}_{n}:\Lambda^{g}(\mathbb{Z}/p^{n}\mathbb{Z})^{2g}\rightarrow\det\omega^{mod}_{A}/p^{n-\frac{g}{p-1}}$

(resp.
$\Lambda^{g}\mathrm{HT}^{\prime}_{n}:\Lambda^{g}(\mathbb{Z}/p^{n}\mathbb{Z})^{2g}\rightarrow\det\omega^{mod}_{A}/p^{n-2{g}}$
if $p=2$).

######  Lemma 4.4.15.

Let $n\geq n_{0}$. We have a commutative diagram

where the vertical maps are finite. Moreover,
$h^{\star}\det\omega_{A}^{mod}=\det\omega_{A}^{mod}$ and
$h^{\star}\Lambda^{g}\mathrm{HT}^{\prime}_{n}(f_{i})=\Lambda^{g}\mathrm{HT}^{\prime}_{n+1}(f_{i})\in\mathrm{H}^{0}(\mathfrak{S}^{\star,mod}_{K^{p}K_{p,n+1}},\det\omega_{A}^{mod}/p^{n-\frac{g}{p-1}})$
(resp.
$\in\mathrm{H}^{0}(\mathfrak{S}^{\star,mod}_{K^{p}K_{p,n+1}},\det\omega_{A}^{mod}/p^{n-2g})$
if $p=2$).

###### Proof.

We first observe that
$\mathrm{Im}(f^{\star}\mathcal{I}_{n}\rightarrow\mathscr{O}_{\mathfrak{S}^{\star}_{K^{p}K_{p,n+1}}})=\mathcal{I}_{n+1}$.
It follows that we have maps
$\mathrm{BL}_{\mathcal{I}_{n}}{\mathfrak{S}}^{\star}_{K^{p}K_{p,n}}\times_{\mathfrak{S}^{\star,mod}_{K^{p}K_{p,n}}}\mathfrak{S}^{\star}_{K^{p}K_{p,n+1}}\rightarrow\mathrm{BL}_{\mathcal{I}_{n+1}}{\mathfrak{S}}^{\star}_{K^{p}K_{p,n+1}}\rightarrow\mathrm{BL}_{\mathcal{I}_{n}}{\mathfrak{S}}^{\star}_{K^{p}K_{p,n}}$.
The map
$\mathrm{BL}_{\mathcal{I}_{n+1}}{\mathfrak{S}}^{\star}_{K^{p}K_{p,n+1}}\rightarrow\mathrm{BL}_{\mathcal{I}_{n}}{\mathfrak{S}}^{\star}_{K^{p}K_{p,n}}$
is therefore finite, and so is the map
$h:{\mathfrak{S}}^{\star,mod}_{K^{p}K_{p,n+1}}\rightarrow{\mathfrak{S}}^{\star,mod}_{K^{p}K_{p,n}}$.
We have a surjective map
$h^{\star}\det\omega_{A}^{mod}\rightarrow\det\omega_{A}^{mod}$ of invertible
sheaves. This map is therefore an isomorphism. The last compatibility follows
from the property that
$f^{\star}\Lambda^{g}\mathrm{HT}_{n}(f_{i})=\Lambda^{g}\mathrm{HT}_{n+1}(f_{i})\in\mathrm{H}^{0}(\mathfrak{S}^{\star}_{K^{p}K_{p,n+1}},\det\omega_{A}/p^{n})$.
∎

We can similarly define $\mathfrak{S}^{\star,mod}_{K^{p}K_{p}}$ for any open
$K_{p}\subseteq K_{p,n_{0}}$. The ideal $\mathcal{I}_{n_{0}}$ pulls back to an
ideal $\mathcal{I}_{K_{p}}$ of $\mathfrak{S}^{\star}_{K^{p}K_{p}}$ and we let
$\mathfrak{S}^{\star,mod}_{K^{p}K_{p}}=\mathrm{NBL}_{\mathcal{I}_{K_{p}}}(\mathfrak{S}^{\star}_{K^{p}K_{p}})$.
If $K_{p}=K_{p,n}$ for $n\geq n_{0}$, we recover the previous definition.

We finally let
${\mathfrak{S}}^{\star,mod}_{K^{p}}=\lim_{n}{\mathfrak{S}}^{\star,mod}_{K^{p}K_{p,n}}$
where the inverse limit is taken in the category of $p$-adic formal schemes.
This inverse limit exists because the transition morphisms are affine. In the
limit we have a map
$\Lambda^{g}\mathrm{HT}:\Lambda^{g}\mathbb{Z}_{p}^{2g}\rightarrow\det\omega_{A}^{mod}$
whose linearization is surjective. It follows that we have a morphism
$\pi_{HT}:{\mathfrak{S}}^{\star,mod}_{K^{p}}\rightarrow\mathbb{P}^{r-1}$. Let
$X_{1},\cdots,X_{r}$ be the homogeneous coordinates on $\mathbb{P}^{r-1}$. For
all $1\leq i\leq r$, let $\mathfrak{U}_{i}$ be the formal open subscheme
defined by the condition $X_{i}\neq 0$. Let $\mathcal{U}_{i}$ be its generic
fiber.

######  Proposition 4.4.16.

  1. (1)

The formal scheme ${\mathfrak{S}}^{\star,mod}_{K^{p}}$ is integral perfectoid
and its generic fiber is the perfectoid space $\mathcal{S}^{\star}_{K^{p}}$.

  2. (2)

The Hodge-Tate map factors through a map
$\pi_{HT}:\mathfrak{S}^{\star,mod}_{K^{p}}\rightarrow\mathfrak{FL}_{G,\mu}$
and it induces the Hodge-Tate period map
$\pi_{HT}:\mathcal{S}^{\star}_{K^{p}}\rightarrow\mathcal{FL}_{G,\mu}$ of
section 4.4.10 on the generic fiber.

  3. (3)

For all $1\leq i\leq r$, we have that $\pi_{HT}^{-1}(\mathcal{U}_{i})$ is a
good open affinoid subset of $\mathcal{S}^{\star}_{K^{p}}$.

###### Proof.

See [PS16], thm 1.18. This relies on the main theorems of [Sch15]. ∎

######  Remark 4.4.17.

For $n$ large enough, the open subsets $\pi_{HT}^{-1}(\mathcal{U}_{i})$ come
from open subsets
$\pi_{HT}^{-1}(\mathcal{U}_{i})_{K_{p,n}}\hookrightarrow\mathcal{S}_{K^{p}K_{p,n}}^{\star}$.
One can define ([Sch15], p. 72) a formal model
$\mathfrak{S}_{K^{p}K_{p,n}}^{\star-HT}$ by gluing the formal schemes
$\mathrm{Spf}~{}\mathrm{H}^{0}(\pi_{HT}^{-1}(\mathcal{U}_{i})_{K_{p,n}},\mathscr{O}_{\mathcal{S}_{K^{p}K_{p,n}}}^{+})$.
We will not use this formal model.

We can perform similar constructions with the toroidal compactification at
level $K_{p}\subseteq K_{p,n_{0}}$. Namely, the ideal $\mathcal{I}_{K_{p}}$
pulls back to an ideal $\mathcal{J}_{K_{p}}$ of
$\mathfrak{S}^{tor}_{K^{p}K_{p},\Sigma}$. We denote by
$\mathfrak{S}^{tor,mod}_{K^{p}K_{p},\Sigma}=\mathrm{NBL}_{\mathcal{J}_{K_{p}}}({\mathfrak{S}}^{tor}_{K^{p}K_{p},\Sigma})$.
We have natural morphisms
$\mathfrak{S}^{tor,mod}_{K^{p}K_{p},\Sigma}\rightarrow\mathfrak{S}^{\star,mod}_{K^{p}K_{p}}$.

######  Remark 4.4.18.

The space $\mathfrak{S}^{tor,mod}_{K^{p}K_{p},\Sigma}$ is not exactly the
space considered in [PS16]. Namely, in that reference, we considered further
blow ups in order to make the sheaf denoted $\omega_{A}^{mod}$ in _loc. cit._
(the subsheaf of $\omega_{A}$ generated by the image of the Hodge-Tate period
map) locally free.

We let
${\mathfrak{S}}^{tor,mod}_{K^{p},\Sigma}=\lim_{n}{\mathfrak{S}}^{tor,mod}_{K^{p}K_{p,n},\Sigma}$
where the inverse limit is taken in the category of $p$-adic formal schemes.
This inverse limit exists because the transition morphisms are affine.

######  Proposition 4.4.19.

The formal scheme ${\mathfrak{S}}^{tor,mod}_{K^{p},\Sigma}$ is integral
perfectoid and its generic fiber is $\mathcal{S}_{K^{p},\Sigma}^{tor}$.

###### Proof.

This follows from almost verbatim from [PS16], section A.12, taking into
account remark 4.4.18. ∎

We now let $\mathcal{U}\hookrightarrow\mathcal{FL}_{G,\mu}$ be a quasi-compact
open subset. Our goal is to define formal models for
$\pi_{HT}^{-1}(\mathcal{U})$ and $(\pi^{tor}_{HT})^{-1}(\mathcal{U})$.

We first need a formal model for $\mathcal{U}$. By [L9̈0], thm. 1.6, there
exists an ideal $\mathcal{I}$ of $\mathscr{O}_{\mathfrak{FL}_{G,\mu}}$ and an
open subscheme $\mathfrak{U}$ of
$\mathrm{NBL}_{\mathcal{I}}(\mathfrak{FL}_{G,\mu})$ such that the generic
fiber of $\mathfrak{U}$ is $\mathcal{U}$.

We can define
$\mathfrak{S}_{K^{p},\Sigma,\mathfrak{U}}^{tor,mod}\rightarrow\mathfrak{S}^{\star,mod}_{K^{p},\mathfrak{U}}$
which is a formal model for
$(\pi^{tor}_{HT})^{-1}(\mathcal{U})\rightarrow\pi_{HT}^{-1}(\mathcal{U})$ as
follows. The ideal $\mathcal{I}$ pulls back to ideals $\mathcal{I}_{1}$ and
$\mathcal{I}_{2}$ of $\mathfrak{S}_{K^{p},\Sigma}^{tor,mod}$ and
$\mathfrak{S}^{\star,mod}_{K^{p}}$ respectively. We now wish to consider the
normalized blow up of $\mathfrak{S}_{K^{p},\Sigma}^{tor,mod}$ and
$\mathfrak{S}^{\star,mod}_{K^{p}}$ at $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$
respectively. We actually show that the ideals come from finite level, perform
the normalized blow-up at a finite level, and then pass to the limit.

######  Lemma 4.4.20.

The ideal $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ are pull backs of ideals
$\mathcal{I}_{1,K_{p}}$ and $\mathcal{I}_{2,K_{p}}$ of
$\mathfrak{S}_{K^{p}K_{p},\Sigma}^{tor,mod}$ and
$\mathfrak{S}^{\star,mod}_{K^{p}K_{p,n}}$ for $K_{p}$ small enough.

###### Proof.

It suffices to prove the claim for $\mathcal{I}_{2}$. We need to prove that
$\mathcal{I}_{2}$ is locally generated by $K_{p,n}$-invariant sections for $n$
large enough. First observe that the ideal $\mathcal{I}$ contains $p^{s}$ for
some integer $s$. Over each standard affine $\mathfrak{U}_{i}$, we have
$\mathcal{I}(\mathfrak{U}_{i})=(s_{1,i},\cdots,s_{k,i})$. We can find sections
$s^{\prime}_{1,i},\cdots,s^{\prime}_{k,i}\in\mathrm{H}^{0}(\pi_{HT}^{-1}(\mathcal{U}_{i}),\mathscr{O}_{\mathcal{S}_{K^{p}}^{\star}}^{+})$
such that $s^{\prime}_{j,i}=s_{j,i}~{}\mod p^{s}$, and $s^{\prime}_{j,i}$
comes from some finite level $K_{p,n}$ by proposition 4.4.16, (3). Thus, the
$s^{\prime}_{j,i}$ and $p^{s}$ generate $\mathcal{I}_{2}$ over the image of
$\pi_{HT}^{-1}(\mathcal{U}_{i})$ in $\mathcal{S}^{\star}_{K^{p}K_{p,n}}$. ∎

We can therefore consider

$\mathrm{NBL}_{\mathcal{I}_{1,K_{p}}}(\mathfrak{S}_{K^{p}K_{p},\Sigma}^{tor,mod})$

and

$\mathrm{NBL}_{\mathcal{I}_{2,K_{p}}}(\mathfrak{S}_{K^{p}K_{p}}^{\star,mod})$

for $K_{p}$ small enough.

We let
$\mathrm{NBL}_{\mathcal{I}_{1}}(\mathfrak{S}_{K^{p},\Sigma}^{tor,mod})=\lim_{K_{p}}\mathrm{NBL}_{\mathcal{I}_{1,K_{p}}}(\mathfrak{S}_{K^{p}K_{p},\Sigma}^{tor,mod})$
and
$\mathrm{NBL}_{\mathcal{I}_{2}}(\mathfrak{S}_{K^{p},\Sigma}^{\star,mod})=\lim_{K_{p}}\mathrm{NBL}_{\mathcal{I}_{2,K_{p}}}(\mathfrak{S}_{K^{p}K_{p},\Sigma}^{\star,mod})$
(the limits are taken in the category of $p$-adic formal schemes).

We have maps

$\mathrm{NBL}_{\mathcal{I}_{1}}(\mathfrak{S}_{K^{p},\Sigma}^{tor,mod})\rightarrow\mathrm{NBL}_{\mathcal{I}_{2}}(\mathfrak{S}_{K^{p}}^{\star,mod})\rightarrow\mathrm{NBL}_{\mathcal{I}}(\mathfrak{FL}_{G,\mu}).$

We let $\mathfrak{S}_{K^{p},\Sigma,\mathfrak{U}}^{tor,mod}$ and
$\mathfrak{S}^{\star,mod}_{K^{p},\mathfrak{U}}$ be the preimages of
$\mathfrak{U}$. These open formal subschemes come from open formal subschemes
$\mathfrak{S}_{K^{p}K_{p},\Sigma,\mathfrak{U}}^{tor,mod}$ and
$\mathfrak{S}^{\star,mod}_{K^{p}K_{p},\mathfrak{U}}$ of
$\mathrm{NBL}_{\mathcal{I}_{1,K_{p}}}(\mathfrak{S}_{K^{p}K_{p},\Sigma}^{tor,mod})$
and
$\mathrm{NBL}_{\mathcal{I}_{2,K_{p}}}(\mathfrak{S}_{K^{p}K_{p}}^{\star,mod})$
for large enough $K_{p}$ (the equations defining $\mathfrak{U}$ are defined at
finite level).

The following theorem shows that the spaces
$\mathfrak{S}^{tor}_{K^{p}K_{p},\Sigma}$,
$\mathfrak{S}^{tor,mod}_{K^{p}K_{p},\Sigma}$ and
$\mathfrak{S}^{tor,mod}_{K^{p}K_{p},\Sigma,\mathfrak{U}}$ admit the usual
description at the boundary in terms of certain formal charts. The case of
$\mathfrak{S}^{tor}_{K^{p}K_{p},\Sigma}$ is available in the literature. The
other cases are deduced from the case of
$\mathfrak{S}^{tor}_{K^{p}K_{p},\Sigma}$, by tracing down what happens with
the various normalized blow ups at the level of formal charts. This is
possible because the Hodge-Tate period map behaves nicely for degenerations of
abelian varieties, since the Hodge-Tate period morphism for étale and
multiplicative $p$-divisible group is trivial.

######  Theorem 4.4.21.

  1. (1)

Let $K=K^{p}K_{p}$ with $K_{p}\subseteq\mathrm{GSp}_{2g}(\mathbb{Z}_{p})$. We
have a decomposition
$\mathfrak{S}^{tor}_{K^{p}K_{p},\Sigma}=\coprod_{\Phi}Z_{K}(\Phi)$ into
locally closed formal subschemes, indexed by certain cusp label
representatives $\Phi$.

  2. (2)

The formal completion
$\widehat{\mathfrak{S}^{tor}_{K^{p}K_{p},\Sigma}}^{Z_{K}(\Phi)}$ admits the
following canonical description:

     * •

There is a tower of $p$-adic formal schemes:

$\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi})\rightarrow\mathbf{S}_{K_{\Phi}}(\overline{Q}_{\Phi},\overline{D}_{\Phi})\rightarrow\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})$

where
$\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi})\rightarrow\mathbf{S}_{K_{\Phi}}(\overline{Q}_{\Phi},\overline{D}_{\Phi})$
is a torsor under a torus $\mathbf{E}_{K}(\Phi)$, and
$\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})$ is an integral model of a lower
dimensional Siegel variety.

In the situation that $K$ is the principal level $N$ congruence subgroup,
$\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})$ carries a full level $N$
structure, and
$\mathbf{S}_{K_{\Phi}}(\overline{Q}_{\Phi},\overline{D}_{\Phi})\rightarrow\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})$
is an abelian scheme torsor. It parametrizes semi-abelian schemes
$0\rightarrow T\rightarrow\mathcal{G}\rightarrow A\rightarrow 0$ (together
with certain level structure) where $A$ is the universal abelian scheme over
$\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})$ and $T$ is a split torus of
rank $g-\dim A$.

     * •

There is a twisted torus embedding
$\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi})\rightarrow\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$
depending on $\Sigma$.

     * •

There is an arithmetic group $\Delta_{K}(\Phi)$ acting on
$X^{\star}(\mathbf{E}_{K}(\Phi))$ and on
$\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi})\hookrightarrow\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$.

     * •

We have a $\Delta_{K}(\Phi)$-invariant closed subscheme
$\mathbf{Z}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))\hookrightarrow\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$.

     * •

There is a finite morphism
$\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})\rightarrow\mathfrak{S}^{\star}_{K}$.

     * •

There is a series of morphisms:

$\mathbf{Z}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))\rightarrow\mathbf{S}_{K_{\Phi}}(\overline{Q}_{\Phi},\overline{D}_{\Phi})\rightarrow\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h}).$

     * •

We have a canonical isomorphism
$Z_{K}(\Phi)=\Delta_{K}(\Phi)\backslash\mathbf{Z}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$.

     * •

We have a canonical isomorphism

$\widehat{\mathfrak{S}^{tor}_{K,\Sigma}}^{{Z}_{K}(\Phi)}\simeq\Delta_{K}(\Phi)\backslash\big{(}\widehat{\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))}^{\mathbf{Z}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))}\big{)}.$

  3. (3)

For small enough $K_{p}$:

     * •

The ideal $\mathcal{J}_{K_{p}}$ restricted to
$\widehat{\mathfrak{S}^{tor}_{K,\Sigma}}^{{Z}_{K}(\Phi)}$ is the pull back of
an ideal $\mathcal{J}_{K_{p},\Phi}$ of
$\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})$.

     * •

Let
$\mathbf{Z}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))=\mathrm{NBL}_{\mathcal{J}_{K_{p},\Phi}}\mathbf{Z}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$
and
$Z^{mod}_{K}(\Phi)=\Delta_{K}(\Phi)\backslash\mathbf{Z}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$.
We have a decomposition
$\mathfrak{S}^{tor,mod}_{K^{p}K_{p},\Sigma}=\coprod_{\Phi}Z^{mod}_{K}(\Phi)$.

     * •

Let
$\mathbf{S}^{mod}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})=\mathrm{NBL}_{\mathcal{J}_{K_{p},\Phi}}(\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h}))$.
We have a finite morphism
$\mathbf{S}^{mod}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})\rightarrow\mathfrak{S}^{\star,mod}_{K^{p}K_{p}}$.

     * •

Let
$\mathbf{S}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))=\mathrm{NBL}_{\mathcal{J}_{K_{p},\Phi}}\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$.
We have a canonical isomorphism

$\widehat{\mathfrak{S}^{tor,mod}_{K,\Sigma}}^{{Z}^{mod}_{K}(\Phi)}\simeq\Delta_{K}(\Phi)\backslash\big{(}\widehat{\mathbf{S}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))}^{\mathbf{Z}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))}\big{)}.$

  4. (4)

For all small enough $K_{p}$:

     * •

The ideal $\mathcal{I}_{1,K_{p}}$ restricted to
$\widehat{\mathfrak{S}^{tor,mod}_{K,\Sigma}}^{{Z}^{mod}_{K}(\Phi)}$ is the
pull back of an ideal $\mathcal{I}_{1,K_{p},\Phi}$ of
$\mathbf{S}^{mod}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})$.

     * •

We have a decomposition
$\mathfrak{S}^{tor,mod}_{K^{p}K_{p},\Sigma,\mathfrak{U}}=\coprod_{\Phi}Z^{mod}_{K}(\Phi)_{\mathfrak{U}}$
where $Z^{mod}_{K}(\Phi)_{\mathfrak{U}}$ is an open subset of
$\mathrm{NBL}_{\mathcal{I}_{1,K_{p}}}(Z^{mod}_{K}(\Phi))$.

     * •

There is an open subset
$\mathbf{S}^{mod}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})_{\mathfrak{U}}$ of
$\mathrm{NBL}_{\mathcal{I}_{1,K_{p},\Phi}}(\mathbf{S}^{mod}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h}))$
such that we have a finite morphism
$\mathbf{S}^{mod}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})_{\mathfrak{U}}\rightarrow\mathfrak{S}^{\star,mod}_{K^{p}K_{p},\mathfrak{U}}$.

     * •

There exists an open subset
$\mathbf{S}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))_{\mathfrak{U}}$
of
$\mathrm{NBL}_{\mathcal{I}_{1,K_{p},\Phi}}(\mathbf{S}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi)))$
and an open subset
$\mathbf{Z}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))_{\mathfrak{U}}$
of
$\mathrm{NBL}_{\mathcal{I}_{1,K_{p},\Phi}}(\mathbf{Z}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi)))$
such that we have a canonical isomorphism

$\widehat{\mathfrak{S}^{tor,mod}_{K,\Sigma,\mathfrak{U}}}^{{Z}^{mod}_{K}(\Phi)_{\mathfrak{U}}}\simeq\Delta_{K}(\Phi)\backslash\big{(}\widehat{\mathbf{S}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))_{\mathfrak{U}}}^{\mathbf{Z}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))_{\mathfrak{U}}}\big{)}.$

######  Remark 4.4.22.

The main observation needed to prove this theorem is the property that the
ideals $\mathcal{J}_{K_{p}}$ and $\mathcal{I}_{1,K_{p}}$ come from ideals
$\mathcal{J}_{K_{p},\Phi}$ and $\mathcal{I}_{1,K_{p},\Phi}$ on each formal
chart (first item in $(3)$ and $(4)$). This is reminiscent of the concept of
well positioned subset or subscheme of [LS18], def. 2.2.1.

###### Proof.

The first two items follow from [Lan17], [MP19] or [PS16] (for principal level
structures). The point $(3)$, follows from [PS16], section A.12. It remains to
check the point $(4)$. The key property is that the ideal
$\mathcal{I}_{1,K_{p}}$ restricted to
$\widehat{\mathfrak{S}^{tor,mod}_{K,\Sigma}}^{{Z}^{mod}_{K}(\Phi)}$ is the
pull back of an ideal $\mathcal{I}_{1,K_{p},\Phi}$ of
$\mathbf{S}^{mod}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})$. The rest follows easily.
We argue as follows. Let $\Phi$ be a cusp label at some finite level
$K^{p}K_{p}$. For all $n$ large enough such that $K_{p,n}\subseteq K_{p}$, we
let $\\{\Phi_{n}\\}_{n}$ be a compatible sequence of cusp label of level
$K^{p}K_{p,n}$ mapping to $\Phi$. We let
$Z^{mod}_{K^{p}}(\Phi_{\infty})=\lim_{n}Z^{mod}_{K^{p}K_{p,n}}(\Phi_{n})$. We
can consider the completion of $\mathfrak{S}^{tor,mod}_{K^{p},\Sigma}$ at
$Z^{mod}_{K^{p}}(\Phi_{\infty})$. Then we have a map
$\widehat{\mathfrak{S}^{tor,mod}_{K^{p},\Sigma}}^{Z^{mod}_{K^{p}}(\Phi_{\infty})}\rightarrow\mathbf{S}^{mod}_{K^{p}_{\Phi_{\infty}}}(G_{\Phi,h},D_{\Phi,h})$
where $\mathbf{S}^{mod}_{K^{p}_{\Phi_{\infty}}}(G_{\Phi,h},D_{\Phi,h})$ is a
perfectoid formal scheme attached to a Siegel Shimura datum of lower
dimension. Moreover, we have a factorization of the Hodge-Tate period map as
follows (compare with [CS19], coro. 4.2.2):

The sheaf of ideals $\mathcal{I}$ of $\mathfrak{FL}$ restricts to a sheaf of
ideals $\mathcal{I}_{\Phi}$ of $\mathfrak{FL}_{G_{\Phi,h},\mu_{G_{\Phi,h}}}$.
We deduce that at infinite level, the sheaf $\mathcal{I}_{1}$ restricted to
$\widehat{\mathfrak{S}^{tor,mod}_{K^{p},\Sigma}}^{Z^{mod}_{K^{p}}(\Phi_{\infty})}$
is the pull back of a sheaf of ideals $\mathcal{I}_{1,\Phi}$ on
$\mathbf{S}^{mod}_{K^{p}_{\Phi_{\infty}}}(G_{\Phi,h},D_{\Phi,h})$. We can then
prove that $\mathcal{I}_{1,\Phi}$ comes from finite level $K_{p}$ as in lemma
4.4.20. ∎

######  Theorem 4.4.23.

Consider the map
$\pi:\mathfrak{S}_{K^{p}K_{p},\Sigma,\mathfrak{U}}^{tor,mod}\rightarrow\mathfrak{S}^{\star,mod}_{K^{p}K_{p},\mathfrak{U}}$.
Then we have that
$\mathrm{R}^{i}\pi_{\star}\mathscr{O}_{\mathfrak{S}_{K^{p}K_{p},\Sigma,\mathfrak{U}}^{tor,mod}}(-nD)=0$
for all $i>0$ and $n\geq 1$.

###### Proof.

The proof of theorem 4.1.7 transposes verbatim, given theorem 4.4.21. ∎

####  4.4.24. The Hodge type case

We fix an embedding
$(G,X)\hookrightarrow(\mathrm{GSp}_{2g},\mathcal{H}^{\pm}_{g})$ of a Hodge
type Shimura datum into a Siegel Shimura datum. Let $V$ be a $2g$-dimensional
vector space over $\mathbb{Q}$, equipped with a symplectic pairing so that
$G\hookrightarrow\mathrm{GSp}_{2g}\hookrightarrow\mathrm{GL}(V)$. We can view
$G\hookrightarrow\mathrm{GL}(V)$ as the group stabilizing a finite number of
tensors $\\{s_{\alpha}\\}\in V^{\otimes}$ (where
$V^{\otimes}=\oplus_{m,n\in\mathbb{Z}_{\geq 0}}V^{\otimes
m}\otimes(V^{\vee})^{\otimes n}$). The group $\mathrm{GSp}_{2g}$ will be
denoted by $\tilde{G}$ in this section. All the objects corresponding to this
group will carry a $\sim$.

Over $S_{K^{p}K_{p}}$ we have a pro-étale $K_{p}$-torsor represented by the
tower of Shimura varieties $\lim_{K^{\prime}_{p}\subseteq
K_{p}}S_{K^{p}K_{p}^{\prime}}$. By pushout along the map $K_{p}\rightarrow
G(\mathbb{Q}_{p})$ we get a $G(\mathbb{Q}_{p})$-torsor.

One can give a more “modular” description of this torsor using the closed
embedding ${S}_{K}\hookrightarrow\tilde{{S}}_{\tilde{K}}$, where
$\tilde{{S}}_{\tilde{K}}$ is a Siegel Shimura variety (over $E$) and the
embedding is the one induced by
$(G,X)\hookrightarrow(\mathrm{GSp}_{2g},\mathcal{H}^{\pm}_{g})$ for a suitable
compact open subgroup $\tilde{K}$. The tensors $\\{s_{\alpha}\\}$ can be used
to produce sections
$\\{s_{\alpha,p}\in\mathrm{H}^{0}({S}_{K},\mathrm{H}_{1}(A,\mathbb{Q}_{p})^{\otimes})\\}$.
More precisely, one first produces tensors
$\\{s_{\alpha,B}\in\mathrm{H}^{0}(S_{K}(\mathbb{C}),\mathrm{H}_{1}(A,\mathbb{Q})^{\otimes})\\}$
using the complex uniformization of $S_{K}(\mathbb{C})$. They give tensors
$\\{s_{\alpha,p}\in\mathrm{H}^{0}(S_{K}(\mathbb{C}),\mathrm{H}_{1}(A,\mathbb{Q}_{p})^{\otimes})\\}$.
Lemma 2.3.2 of [CS17] proves that these tensors are defined over $E$.
Therefore, we can consider the torsor of isomorphisms
$V\otimes_{\mathbb{Q}}\mathbb{Q}_{p}\rightarrow\mathrm{H}_{1}(A,\mathbb{Q}_{p})$,
preserving all the tensors $s_{\alpha,p}$.

There is also the $M_{\mu}$-torsor $M_{dR}$ over $S_{K}$. We recall its
description. The tensors $\\{s_{\alpha}\\}$ can be used to produce sections
$\\{s_{\alpha,dR}\in\mathrm{H}^{0}({S}_{K},\mathrm{H}_{1,dR}(A)^{\otimes})\\}$.
One first define the $P_{\mu}^{std}$-torsor $P_{dR}$, to be the the torsor of
isomorphisms
$V\otimes_{\mathbb{Q}}\mathscr{O}_{S_{K}}\rightarrow\mathrm{H}_{1,dR}(A)$
matching the filtration on $V$ corresponding to $P_{\mu}^{std}$ with the Hodge
filtration, and preserving the tensors $s_{\alpha,dR}$. By pushout along
$P_{\mu}^{std}\rightarrow M_{\mu}$, we have
$M_{dR}=P_{dR}\times^{P_{\mu}^{std}}M_{\mu}$.

######  Remark 4.4.25.

The closed immersion ${S}_{K}\hookrightarrow\tilde{{S}}_{\tilde{K}}$ extends
to a closed immersion $M_{dR}\hookrightarrow\tilde{M}_{dR}$ where
$\tilde{M}_{dR}$ is the $M_{\tilde{\mu}}$\- “de Rham” torsor over
$\tilde{{S}}_{\tilde{K}}$.

By pull back to the analytic space $\mathcal{S}^{an}_{K}$ we get a pro-étale
$G(\mathbb{Q}_{p})$-torsor $\mathcal{G}^{an}_{pet,p}$, as well as a
$\mathcal{M}_{\mu}^{an}$-torsor $\mathcal{M}_{dR}^{an}$.

In section 2.3 of [CS17] the authors define two other pro-étale torsors over
$\mathcal{S}^{an}_{K}$: $\mathcal{P}^{an}_{HT}$ and $\mathcal{M}^{an}_{HT}$,
under the groups $\mathcal{P}^{an}_{\mu}$ and $\mathcal{M}_{\mu}^{an}$. These
definitions extend those given in the Siegel case (see section 4.4.8) and use
the tensors $s_{\alpha,p}$. Moreover,
$\mathcal{P}^{an}_{HT}\times^{\mathcal{P}^{an}_{\mu}}\mathcal{G}^{an}=\mathcal{G}^{an}_{et,p}\times^{G(\mathbb{Q}_{p})}\mathcal{G}^{an}$
and
$\mathcal{M}^{an}_{HT}=\mathcal{P}^{an}_{HT}\times^{\mathcal{P}^{an}_{\mu}}\mathcal{M}^{an}_{\mu}$.
By [CS17], prop. 2.3.9, there is a canonical identification of torsors
$\mathcal{M}^{an}_{dR}$ and $\mathcal{M}^{an}_{HT}$

####  4.4.26. Perfectoid Hodge type Shimura varieties

By [Sch15], there is a perfectoid space
$\mathcal{S}^{an}_{K^{p}}\sim\lim_{K_{p}}\mathcal{S}^{an}_{K^{p}K_{p}}$. Since
the torsor $\mathcal{G}^{an}_{pet,p}$ becomes trivial over
$\mathcal{S}^{an}_{K^{p}}$, we obtain a Hodge-Tate period map ([CS17], thm
2.1.3): $\pi_{HT}:\mathcal{S}^{an}_{{K^{p}}}\rightarrow\mathcal{FL}_{G,\mu}$
which is $G(\mathbb{Q}_{p})$-equivariant.

A key property is that the pull back of the $\mathcal{M}_{\mu}^{an}$\- torsor
$\mathcal{G}^{an}/U_{\mathcal{P}_{\mu}^{an}}\rightarrow\mathcal{FL}_{G,\mu}$
via $\pi_{HT}$ is $\mathcal{M}_{HT}^{an}$ and this is canonically identified
with the pull back via
$\mathcal{S}^{an}_{{K^{p}}}\rightarrow\mathcal{S}^{an}_{{K^{p}K_{p}}}$ of
$\mathcal{M}_{dR}^{an}$.

Moreover, the relation with the Siegel datum is expressed by the following
diagram, where the horizontal maps are closed immersions:

####  4.4.27. Compactifications in the Hodge case

For any compact open $K\subset G(\mathbb{A}_{f})$, we have the minimal
compactification $\mathcal{S}^{\star}_{K}$ and there is a finite surjective
map $\mathcal{S}^{\star}_{K}\rightarrow\mathcal{S}^{\star}_{\overline{K}}$
where $\mathcal{S}^{\star}_{\overline{K}}$ is defined before [Sch15], thm. IV.
1.1. This is the schematic image of the morphism
$\mathcal{S}^{\star}_{K}\rightarrow\tilde{\mathcal{S}}^{\star}_{\tilde{K}}$
where $\tilde{\mathcal{S}}^{\star}_{\tilde{K}}$ is the minimal
compactification of the Shimura variety for $\tilde{G}$, and $\tilde{K}$ is a
small enough compact open subgroup of $\tilde{G}(\mathbb{A}_{f})$ such that
$\tilde{K}\cap G(\mathbb{A}_{f})=K$. By [Sch15], thm. IV. 1.1 there is a
perfectoid space
$\mathcal{S}^{\star}_{\overline{K^{p}}}\sim\lim_{K_{p}}\mathcal{S}^{\star}_{\overline{K^{p}K_{p}}}$
and there is a map
$\pi_{HT}:\mathcal{S}^{\star}_{\overline{K^{p}}}\rightarrow\mathcal{FL}_{{G},{\mu}}$.
Strictly speaking the map constructed in [Sch15], took values in
$\mathcal{FL}_{\tilde{G},\tilde{\mu}}$. Since
$\mathcal{FL}_{G,\mu}\hookrightarrow\mathcal{FL}_{\tilde{G},\tilde{\mu}}$ is
Zariski closed and the map is known to factors through $\mathcal{FL}_{G,\mu}$
over the Zariski dense open subset $\mathcal{S}^{an}_{{K^{p}}}$ by [CS17], thm
2.1.3, we deduce that we have a map
$\pi_{HT}:\mathcal{S}^{\star}_{\overline{K^{p}}}\rightarrow\mathcal{FL}_{G,\mu}$.

Since the tower $\\{\mathcal{S}^{\star}_{\overline{K^{p}K_{p}}}\\}_{K_{p}}$
carries a $G(\mathbb{Q}_{p})$-action, the space
$\mathcal{S}^{\star}_{\overline{K^{p}}}$ inherits a
$G(\mathbb{Q}_{p})$-action, and the map $\pi_{HT}$ is
$G(\mathbb{Q}_{p})$-equivariant. Moreover, the map $\pi_{HT}$ is affinoid in
the sense that there exists an affinoid covering
$\mathcal{FL}_{G,\mu}=\cup_{i}V_{i}$ such each $\pi_{HT}^{-1}(V_{i})$ is a
good affinoid perfectoid open subset of
$\mathcal{S}^{\star}_{\overline{K^{p}}}$ (see definition 4.4.3).

We also know that there is a perfectoid space
$\mathcal{S}^{\star}_{K^{p}}=\lim_{K_{p}}\mathcal{S}^{\star,\Diamond}_{K^{p}K_{p}}$
(this last inverse limit is taken in the category of diamonds) by [BS19], thm.
1.16. We therefore have a $G(\mathbb{Q}_{p})$-equivariant map
$\mathcal{S}^{\star}_{K^{p}}\rightarrow\mathcal{S}^{\star}_{\overline{K^{p}}}$.
It is not known whether
$\mathcal{S}^{\star}_{K^{p}}\sim\lim\mathcal{S}^{\star}_{K^{p}K_{p}}$ in
general.

By [Lan20], for a cofinal subset of cone decompositions $\Sigma$, we have a
perfectoid space
$\mathcal{S}^{tor}_{K^{p},\Sigma}\sim\lim\mathcal{S}^{tor}_{K^{p}K_{p},\Sigma}$.
More precisely, for each such cone decomposition $\Sigma$, there exists a cone
decomposition $\tilde{\Sigma}$ and a closed immersion of perfectoid spaces
$\mathcal{S}^{tor}_{K^{p},\Sigma}\hookrightarrow\mathcal{S}^{tor}_{\tilde{K}^{p},\tilde{\Sigma}}$.
Let us call these cone decompositions _perfect_ cone decompositions because
they give rise to perfectoid toroidal compactifications.

######  Remark 4.4.28.

We did not prove that there is a perfectoid space
$\mathcal{S}^{tor}_{K^{p},\Sigma}\sim\lim\mathcal{S}^{tor}_{K^{p}K_{p},\Sigma}$
for any cone decomposition $\Sigma$. It seems likely that one could reproduce
the argument of [PS16] in the Hodge case, using the explicit description of
the boundary of the integral toroidal compactifications given in [MP19].

For a perfect cone decomposition $\Sigma$, we have a series of maps
$\mathcal{S}^{tor}_{K^{p},\Sigma}\rightarrow\mathcal{S}^{\star}_{K^{p}}\rightarrow\mathcal{S}^{\star}_{\overline{K^{p}}}$
and we therefore get a map
$\pi_{HT}^{tor}:\mathcal{S}^{tor}_{K^{p},\Sigma}\rightarrow\mathcal{FL}_{G,\mu}$.

The relation to the Siegel Shimura varieties is given by the following diagram
where all horizontal maps are closed immersions:

Using the map $\pi_{HT}^{tor}$ we can define a canonical extension of the
torsors $\mathcal{P}_{HT}^{an}$ and $\mathcal{M}^{an}_{HT}$ to
$\mathcal{S}^{tor}_{K^{p},\Sigma}$, by simply by pulling back the universal
$\mathcal{P}_{\mu}^{an}$-torsor over $\mathcal{FL}_{G,\mu}$ and pushing out
along the map $\mathcal{P}_{\mu}^{an}\rightarrow\mathcal{M}_{\mu}^{an}$. There
is also the torsor $\mathcal{M}^{an}_{dR}$ over $\mathcal{S}_{K,\Sigma}^{tor}$
which is pulled back from the map
$\mathcal{S}_{K^{p},\Sigma}^{tor}\rightarrow\mathcal{S}_{K^{p}K_{p},\Sigma}^{tor}$.
The following proposition is corollary 5.2 of [EH19].

######  Proposition 4.4.29 ([EH19]).

The torsors $\mathcal{M}^{an}_{HT}$ and
$\mathcal{M}^{an}_{dR}\times_{\mathcal{S}_{K^{p}K_{p},\Sigma}^{tor}}\mathcal{S}^{tor}_{K^{p},\Sigma}$
are canonically isomorphic.

###### Proof.

Let us denote by $\tilde{M}$ the restriction to
$\mathcal{S}^{tor}_{K^{p},\Sigma}$ of the torsor
$\tilde{\mathcal{M}}^{an}_{dR}=\tilde{\mathcal{M}}^{an}_{HT}$ defined over the
Siegel perfectoid Shimura variety
$\tilde{\mathcal{S}}_{\tilde{K}^{p},\tilde{\Sigma}}$. By construction,
$\tilde{M}$ is a $\mathcal{M}^{an}_{\tilde{\mu}}$-torsor. The two torsors
$\mathcal{M}^{an}_{HT}$ and $\mathcal{M}^{an}_{dR}$ are
$\mathcal{M}_{\mu}^{an}$-reductions of this torsor. They coincide over
$\mathcal{S}^{an}_{K^{p}}$ by [CS17], prop. 2.3.9. But $\mathcal{M}^{an}_{HT}$
(resp. $\mathcal{M}^{an}_{dR}$) is equal to the Zariski closure of
$\mathcal{M}^{an}_{HT}|_{\mathcal{S}^{an}_{K^{p}}}$ (resp.
$\mathcal{M}^{an}_{dR}|_{\mathcal{S}^{an}_{K^{p}}}$) in $\tilde{M}$.
Therefore, these torsors have to coincide everywhere. ∎

######  Remark 4.4.30.

The isomorphism of torsors is Hecke equivariant.

####  4.4.31. Integral models in the Hodge case

We need to consider formal models. This section is entirely parallel to
section 4.4.12. Let $K=K^{p}K_{p}$ with $K=\tilde{K}\cap G(\mathbb{A}_{f})$.
Suppose that $\tilde{K}_{p}\subseteq\tilde{G}(\mathbb{Z}_{p})$. We can define
$\mathfrak{S}_{K^{p}K_{p}}$, $\mathfrak{S}^{\star}_{K^{p}K_{p}}$, and
$\mathfrak{S}^{tor}_{K^{p}K_{p},\Sigma}$ as the normalizations of
$\tilde{\mathfrak{S}}_{\tilde{K}}$,
$\tilde{\mathfrak{S}}^{\star}_{\tilde{K}}$, and
$\tilde{\mathfrak{S}}^{tor}_{\tilde{K},\tilde{\Sigma}}$ in
$\mathcal{S}_{K^{p}K_{p}}$, $\mathcal{S}^{\star}_{K^{p}K_{p}}$, and
$\mathcal{S}^{tor}_{K^{p}K_{p},\Sigma}$ respectively.

Similarly, for $K_{p}$ small enough, we define
$\mathfrak{S}^{\star,mod}_{K^{p}K_{p}}$,
$\mathfrak{S}^{tor,mod}_{K^{p}K_{p},\Sigma}$ as the normalizations of
$\tilde{\mathfrak{S}}^{\star,mod}_{\tilde{K}}$ and
$\tilde{\mathfrak{S}}^{tor,mod}_{\tilde{K},\Sigma}$ in
$\mathcal{S}^{\star}_{K^{p}K_{p}}$ and $\mathcal{S}^{tor}_{K^{p}K_{p},\Sigma}$
respectively. Alternatively, these can also be constructed as normalized blow
ups of $\mathfrak{S}^{\star}_{K^{p}K_{p}}$ and
$\mathfrak{S}^{tor}_{K^{p}K_{p},\Sigma}$ for the ideals $\mathcal{I}_{K_{p}}$
and $\mathcal{J}_{K_{p}}$ which are the pull back of the ideals
$\mathcal{I}_{\tilde{K}_{p}}$ and $\mathcal{J}_{\tilde{K}_{p}}$ from
$\tilde{\mathfrak{S}}^{\star}_{\tilde{K}}$ and
$\tilde{\mathfrak{S}}^{tor}_{\tilde{K},\Sigma}$.

We let
${\mathfrak{S}}^{\star,mod}_{K^{p}}=\lim_{K_{p}}{\mathfrak{S}}^{\star,mod}_{K^{p}K_{p}}$,
where the inverse limit is taken in the category of $p$-adic formal schemes.
This inverse limit exists because the transition morphisms are affine. In the
limit we have a map
${\mathfrak{S}}^{\star,mod}_{K^{p}}\rightarrow\tilde{\mathfrak{S}}^{\star,mod}_{\tilde{K}^{p}}$
and therefore a map
$\pi_{HT}:\mathfrak{S}^{\star,mod}_{K^{p}}\rightarrow\mathfrak{FL}_{\tilde{G},\tilde{\mu}}$.

######  Proposition 4.4.32.

The Hodge-Tate map factors through a map
$\pi_{HT}:\mathfrak{S}^{\star,mod}_{K^{p}}\rightarrow\mathfrak{FL}_{G,\mu}$.

###### Proof.

One first checks that the space $\mathfrak{S}^{mod}_{K^{p}}$ (the complement
of the boundary in $\mathfrak{S}^{\star,mod}_{K^{p}}$) is integral perfectoid
and its generic fiber is the quasi-compact open perfectoid Shimura variety
$\mathcal{S}_{K^{p}}$. This follows from the Siegel case, using that
$\mathcal{S}_{K^{p}}\hookrightarrow\tilde{S}_{\tilde{K}^{p}}$ is a Zariski
closed immersion. Therefore the factorization of the period morphism through
$\mathfrak{FL}_{G,\mu}$ holds over the Zariski dense subspace
$\mathfrak{S}^{mod}_{K^{p}}$, and thus everywhere. ∎

######  Remark 4.4.33.

We do not know if $\mathfrak{S}^{\star,mod}_{K^{p}}$ is integral perfectoid.

We let
${\mathfrak{S}}^{tor,mod}_{K^{p},\Sigma}=\lim_{K_{p}}{\mathfrak{S}}^{tor,mod}_{K^{p}K_{p},\Sigma}$
where the inverse limit is taken in the category of $p$-adic formal schemes.
This inverse limit exists because the transition morphisms are affine.

######  Proposition 4.4.34.

Assume that $\Sigma$ is perfect. The formal scheme
${\mathfrak{S}}^{tor,mod}_{K^{p},\Sigma}$ is integral perfectoid and its
generic fiber is $\mathcal{S}_{K^{p},\Sigma}^{tor}$.

###### Proof.

This follows from the Siegel case. ∎

######  Remark 4.4.35.

We do not know if $\mathfrak{S}^{tor,mod}_{K^{p},\Sigma}$ is integral
perfectoid for all $\Sigma$.

We now let $\mathcal{U}\hookrightarrow\mathcal{FL}_{G,\mu}$ be a quasi-compact
open subset. It is induced by a quasi-compact open subset
$\tilde{\mathcal{U}}$ of $\mathcal{FL}_{\tilde{G},\tilde{\mu}}$. Our goal is
to define formal models for $\pi_{HT}^{-1}(\mathcal{U})$ and
$(\pi^{tor}_{HT})^{-1}(\mathcal{U})$.

There exists an ideal $\tilde{\mathcal{I}}$ of
$\mathscr{O}_{\mathfrak{FL}_{\tilde{G},\tilde{\mu}}}$ and an open subscheme
$\tilde{\mathfrak{U}}$ of
$\mathrm{NBL}_{\tilde{\mathcal{I}}}(\mathfrak{FL}_{\tilde{G},\mu_{\tilde{G}}})$
such that the generic fiber of $\tilde{\mathfrak{U}}$ is
$\tilde{\mathcal{U}}$.

For $K_{p}$ small enough, we define
$\mathfrak{S}^{\star,mod}_{K^{p}K_{p},\mathfrak{U}}$,
$\mathfrak{S}^{tor,mod}_{K^{p}K_{p},\Sigma,\mathfrak{U}}$ as the
normalizations of
$\tilde{\mathfrak{S}}^{\star,mod}_{\tilde{K},\tilde{\mathfrak{U}}}$ and
$\tilde{\mathfrak{S}}^{tor,mod}_{\tilde{K},\tilde{\Sigma},\tilde{\mathfrak{U}}}$
in $\pi_{HT}^{-1}(\mathcal{U})$ and $(\pi^{tor}_{HT})^{-1}(\mathcal{U})$
respectively. Alternatively, these can also be constructed as suitable opens
in normalized blow ups of $\mathfrak{S}^{\star,mod}_{K^{p}K_{p}}$ and
$\mathfrak{S}^{tor,mod}_{K^{p}K_{p},\Sigma}$ at the ideals
$\mathcal{I}_{1,{K}_{p}}$ and $\mathcal{I}_{2,{K}_{p}}$ which are the pull
back of the ideals $\mathcal{I}_{1,\tilde{K}_{p}}$ and
$\mathcal{I}_{2,\tilde{K}_{p}}$ from
$\tilde{\mathfrak{S}}^{\star,mod}_{\tilde{K}}$ and
$\tilde{\mathfrak{S}}^{tor,mod}_{\tilde{K},\tilde{\Sigma}}$.

######  Theorem 4.4.36.

  1. (1)

Let $K=K^{p}K_{p}$ with $K_{p}\subseteq\tilde{G}(\mathbb{Z}_{p})$. We have a
decomposition
$\mathfrak{S}^{tor}_{K^{p}K_{p},\Sigma}=\coprod_{\Phi}Z_{K}(\Phi)$ into
locally closed formal subschemes, indexed by certain cusp label
representatives $\Phi$.

  2. (2)

The formal completion
$\widehat{\mathfrak{S}^{tor}_{K^{p}K_{p},\Sigma}}^{Z_{K}(\Phi)}$ admits the
following canonical description:

     * •

There is a tower of $p$-adic formal schemes:

$\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi})\rightarrow\mathbf{S}_{K_{\Phi}}(\overline{Q}_{\Phi},\overline{D}_{\Phi})\rightarrow\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})$

where
$\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi})\rightarrow\mathbf{S}_{K_{\Phi}}(\overline{Q}_{\Phi},\overline{D}_{\Phi})$
is a torsor under a torus $\mathbf{E}_{K}(\Phi)$, and
$\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})$ is an integral model of a lower
dimensional Shimura variety.

     * •

There is a twisted torus embedding
$\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi})\rightarrow\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$
depending on $\Sigma$.

     * •

There is an arithmetic group $\Delta_{K}(\Phi)$ acting on
$X^{\star}(\mathbf{E}_{K}(\Phi))$ and on
$\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi})\hookrightarrow\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$.

     * •

We have a $\Delta_{K}(\Phi)$-invariant closed subscheme
$\mathbf{Z}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))\hookrightarrow\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$.

     * •

There is a finite morphism
$\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})\rightarrow\mathfrak{S}^{\star}_{K}$.

     * •

There is a series of morphisms:

$\mathbf{Z}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))\rightarrow\mathbf{S}_{K_{\Phi}}(\overline{Q}_{\Phi},\overline{D}_{\Phi})\rightarrow\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h}).$

     * •

We have a canonical isomorphism
$Z_{K}(\Phi)=\Delta_{K}(\Phi)\backslash\mathbf{Z}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$.

     * •

We have a canonical isomorphism

$\widehat{\mathfrak{S}^{tor}_{K,\Sigma}}^{{Z}_{K}(\Phi)}\simeq\Delta_{K}(\Phi)\backslash\big{(}\widehat{\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))}^{\mathbf{Z}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))}\big{)}.$

  3. (3)

For small enough $K_{p}$:

     * •

The ideal $\mathcal{J}_{K_{p}}$ restricted to
$\widehat{\mathfrak{S}^{tor}_{K,\Sigma}}^{{Z}_{K}(\Phi)}$ is the pull back of
an ideal $\mathcal{J}_{K_{p},\Phi}$ of
$\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})$.

     * •

Let
$\mathbf{Z}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))=\mathrm{NBL}_{\mathcal{J}_{K_{p},\Phi}}\mathbf{Z}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$
and
$Z^{mod}_{K}(\Phi)=\Delta_{K}(\Phi)\backslash\mathbf{Z}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$.
We have a decomposition
$\mathfrak{S}^{tor,mod}_{K^{p}K_{p},\Sigma}=\coprod_{\Phi}Z^{mod}_{K}(\Phi)$.

     * •

Let
$\mathbf{S}^{mod}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})=\mathrm{NBL}_{\mathcal{J}_{K_{p},\Phi}}(\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h}))$.
We have a finite morphism
$\mathbf{S}^{mod}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})\rightarrow\mathfrak{S}^{\star,mod}_{K^{p}K_{p}}$.

     * •

Let
$\mathbf{S}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))=\mathrm{NBL}_{\mathcal{J}_{K_{p},\Phi}}\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$.
We have a canonical isomorphism

$\widehat{\mathfrak{S}^{tor,mod}_{K,\Sigma}}^{{Z}^{mod}_{K}(\Phi)}\simeq\Delta_{K}(\Phi)\backslash\big{(}\widehat{\mathbf{S}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))}^{\mathbf{Z}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))}\big{)}.$

  4. (4)

For all small enough $K_{p}$,

     * •

The ideal $\mathcal{I}_{1,K_{p}}$ restricted to
$\widehat{\mathfrak{S}^{tor,mod}_{K,\Sigma}}^{{Z}^{mod}_{K}(\Phi)}$ is the
pull back of an ideal $\mathcal{I}_{1,K_{p},\Phi}$ of
$\mathbf{S}^{mod}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})$.

     * •

We have a decomposition
$\mathfrak{S}^{tor,mod}_{K^{p}K_{p},\Sigma,\mathfrak{U}}=\coprod_{\Phi}Z^{mod}_{K}(\Phi)_{\mathfrak{U}}$
where $Z^{mod}_{K}(\Phi)_{\mathfrak{U}}$ is an open subset of
$\mathrm{NBL}_{\mathcal{I}_{1,K_{p}}}(Z^{mod}_{K}(\Phi))$.

     * •

There is an open subset
$\mathbf{S}^{mod}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})_{\mathfrak{U}}$ of
$\mathrm{NBL}_{\mathcal{I}_{1,K_{p},\Phi}}(\mathbf{S}^{mod}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h}))$
such that we have a finite morphism
$\mathbf{S}^{mod}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})_{\mathfrak{U}}\rightarrow\mathfrak{S}^{\star,mod}_{K^{p}K_{p},\mathfrak{U}}$.

     * •

There exists an open subset
$\mathbf{S}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))_{\mathfrak{U}}$
of
$\mathrm{NBL}_{\mathcal{I}_{1,K_{p},\Phi}}(\mathbf{S}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi)))$
and an open subset
$\mathbf{Z}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))_{\mathfrak{U}}$
of
$\mathrm{NBL}_{\mathcal{I}_{1,K_{p},\Phi}}(\mathbf{Z}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi)))$
such that we have a canonical isomorphism

$\widehat{\mathfrak{S}^{tor,mod}_{K,\Sigma,\mathfrak{U}}}^{{Z}^{mod}_{K}(\Phi)_{\mathfrak{U}}}\simeq\Delta_{K}(\Phi)\backslash\big{(}\widehat{\mathbf{S}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))_{\mathfrak{U}}}^{\mathbf{Z}^{mod}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))_{\mathfrak{U}}}\big{)}.$

###### Proof.

The first two items follow from [MP19]. The point $(3)$ and the point $(4)$
follow from the analogous statement in theorem 4.4.21.∎

######  Theorem 4.4.37.

Consider the map
$\pi:\mathfrak{S}_{K^{p}K_{p},\Sigma,\mathfrak{U}}^{tor,mod}\rightarrow\mathfrak{S}^{\star,mod}_{K^{p}K_{p},\mathfrak{U}}$.
Then we have that
$\mathrm{R}^{i}\pi_{\star}\mathscr{O}_{\mathfrak{S}_{K^{p}K_{p},\Sigma,\mathfrak{U}}^{tor,mod}}(-nD)=0$
for all $i>0$ and $n\geq 1$.

###### Proof.

The proof of theorem 4.1.7 transposes verbatim, given theorem 4.4.36. ∎

####  4.4.38. General Shimura varieties

We now extend part of the preceding to general Shimura varieties. For the
moment, let $(G,X)$ be an arbitrary Shimura datum and let
$K=K^{p}K_{p}\subseteq G(\mathbb{A}_{f})$ be a compact open subgroup. We
consider the compactified Shimura variety $\mathcal{S}_{K,\Sigma}^{tor}$. For
any algebraic representation $W$ of $G^{c}$, we have a local system on the
pro-kummer-étale site $\mathcal{W}_{p}$ as well as a filtered vector bundle
with integrable log-connection $\mathcal{W}_{dR}$. We need some period sheaves
relative to $\mathcal{S}_{K,\Sigma}^{tor}$ (see [Sch13a], sect. 6 and
[DLLZ19], sect. 2): $\mathbb{B}_{dR}^{+}$, $\mathbb{B}_{dR}$,
$\mathscr{O}\mathbb{B}^{+}_{dR,log}$ and $\mathscr{O}\mathbb{B}_{dR,log}$. The
following property is a consequence of [DLLZ19], thm. 5.3.1. The local system
$\mathcal{W}_{p}$ and the filtered vector bundle with integrable log-
connection $\mathcal{W}_{dR}$ are associated ([Sch13a], def 7.5) in the sense
that there is a canonical isomorphism compatible with filtrations, connection
and Hecke action:

$\mathcal{W}_{p}\otimes_{\mathbb{Q}_{p}}\mathscr{O}\mathbb{B}_{dR,log}=\mathcal{W}_{dR}\otimes_{\mathscr{O}_{\mathcal{S}_{K,\Sigma}^{tor}}}\mathscr{O}\mathbb{B}_{dR,log}$

######  Remark 4.4.39.

This identity is a consequence of the comparison theorems for (semi)-abelian
varieties in the Siegel case. In the Hodge case and outside of the boundary,
it is proved in [CS17] section 2.2 and 2.3, as a consequence of the comparison
theorems for abelian varieties, together with results on Hodge tensors under
comparison isomorphisms due to [Bla94].

Using this canonical isomorphism, we define the Hodge-Tate filtration on
$\mathcal{W}_{p}\otimes\hat{\mathscr{O}}_{\mathcal{S}_{K,\Sigma}^{tor}}$ as
follows. Let
$\mathbb{M}=\mathcal{W}_{p}\otimes_{\mathbb{Q}_{p}}\mathbb{B}^{+}_{dR,log}$
and
$\mathbb{M}_{0}=(\mathcal{W}_{dR}\otimes_{\mathscr{O}_{\mathcal{S}_{K,\Sigma}^{tor}}}\mathscr{O}\mathbb{B}^{+}_{dR,log})^{\nabla=0}$.

These are two $\mathbb{B}^{+}_{dR}$-local systems and are lattices inside
$\mathbb{M}\otimes_{\mathbb{B}^{+}_{dR}}\mathbb{B}_{dR}=\mathbb{M}_{0}\otimes_{\mathbb{B}^{+}_{dR}}\mathbb{B}_{dR}$.
There are therefore two filtrations on this $\mathbb{B}_{dR}$-local system
$\mathrm{Fil}^{i}\mathbb{M}$ and $\mathrm{Fil}^{i}\mathbb{M}_{0}$. One defines
an ascending filtration by:

$\mathrm{Fil}_{-j}\mathcal{W}_{p}\otimes_{\mathbb{Q}_{p}}\hat{\mathscr{O}}_{\mathcal{S}_{K,\Sigma}^{tor}}=(\mathbb{M}\cap\mathrm{Fil}^{j}\mathbb{M}^{0})/(\mathrm{Fil}^{1}\mathbb{M}\cap\mathrm{Fil}^{j}\mathbb{M}^{0}).$

and we have the relation

$\mathrm{Gr}_{j}\mathcal{W}_{p}\otimes_{\mathbb{Q}_{p}}\hat{\mathscr{O}}_{\mathcal{S}_{K,\Sigma}^{tor}}=\mathrm{Gr}^{j}\mathcal{W}_{dR}\otimes_{\mathscr{O}_{\mathcal{S}^{tor}_{K,\Sigma}}}\hat{\mathscr{O}}_{\mathcal{S}_{K,\Sigma}^{tor}}.$

This construction is functorial with respect to the Hecke action and
compatible with the Tannakian formalism. One can therefore derive consequences
at the level of torsors. Over $\mathcal{S}_{K,\Sigma}^{tor}$, we have a pro-
kummer-étale $G^{c}(\mathbb{Q}_{p})$-torsor $\mathcal{G}_{pet,p}^{an}$ as well
as a principal $\mathcal{G}^{c,an}$-torsor $\mathcal{G}^{an}_{dR}$ (defined
over the étale site). The torsor $\mathcal{G}^{an}_{dR}$ has a reduction of
structure group to a $\mathcal{P}_{\mu}^{std,c,an}$-torsor (corresponding to
the filtration on $W_{dR}$) that we denote by $\mathcal{P}^{an}_{dR}$, and we
have

$\mathcal{M}^{an}_{dR}=\mathcal{P}^{an}_{dR}\times^{\mathcal{P}_{\mu}^{std,c,an}}\mathcal{M}_{\mu}^{c,an}.$

On the other hand, the torsor
$\mathcal{G}_{pet,p}^{an}\times^{G^{c}(\mathbb{Q}_{p})}\mathcal{G}^{c,an}$ has
a reduction of structure group to a $\mathcal{P}_{\mu}^{c,an}$-torsor
(corresponding to the filtration on
$W_{p}\otimes_{\mathbb{Q}_{p}}\hat{\mathscr{O}}_{\mathcal{S}_{K,\Sigma}^{tor}}$)
that we denote by $\mathcal{P}^{an}_{HT}$, and we define

$\mathcal{M}^{an}_{HT}=\mathcal{P}^{an}_{HT}\times^{\mathcal{P}_{\mu}^{c,an}}\mathcal{M}_{\mu}^{c,an}.$

We have the identification of torsors on the pro-étale site
$\mathcal{M}^{an}_{HT}=\mathcal{M}^{an}_{dR}$, compatible with the Hecke
action. This implies that $\mathcal{M}^{an}_{HT}$ is already defined on the
étale site.

We can consider the inverse limit in the category of diamonds
$\mathcal{S}_{K^{p},\Sigma}^{tor,\Diamond}=\lim_{K_{p}}\mathcal{S}_{K^{p}K_{p},\Sigma}^{tor,\Diamond}$.

To summarize the situation, we have the following theorem, which is really a
consequence of [DLLZ19], thm. 5.3.1.

######  Theorem 4.4.40.

The torsor $\mathcal{G}_{pet,p}^{an}$ is trivial over
$\mathcal{S}_{K^{p},\Sigma}^{tor,\Diamond}$, and we therefore obtain a
morphism:

$\pi_{HT}^{tor}:\mathcal{S}_{K^{p},\Sigma}^{tor,\Diamond}\rightarrow\mathcal{FL}_{G,\mu}.$

Moreover, the pull back of the torsor
$\mathcal{G}^{c,an}/U_{\mathcal{P}_{\mu}^{an}}\rightarrow\mathcal{FL}_{G,\mu}$
via
$\pi_{HT}^{tor}:\mathcal{S}_{K^{p},\Sigma}^{tor,\Diamond}\rightarrow\mathcal{FL}_{G,\mu}$
is $\mathcal{M}^{an}_{HT}$, and this is also the pull back of
$\mathcal{M}_{dR}^{an}$ via the map
$\mathcal{S}_{K^{p},\Sigma}^{tor,\Diamond}\rightarrow\mathcal{S}_{K^{p}K_{p},\Sigma}^{tor}$.

We have thus extended the picture to the case of general Shimura varieties at
the expense of working with diamonds. This is unfortunately not enough
information for our purposes. In particular, we need to address the question
of affineness of the Hodge-Tate period map. For this reason, we restrict to
abelian type Shimura data and work out the connection between abelian and
Hodge type Shimura varieties in the next section.

####  4.4.41. Abelian type Shimura varieties

We now consider an abelian type Shimura datum $(G,X)$. This means that there
is a Hodge type Shimura datum $(G_{1},X_{1})$ such that there is a central
isogeny $G_{1}^{der}\rightarrow G^{der}$ inducing an isomorphism of the
associated connected Shimura data $(G^{ad},X^{+})$ and
$(G_{1}^{ad},X_{1}^{+})$ (see [Mil05], definition 4.4).

We recall a nice way to connect the Shimura data $(G,X)$ and $(G_{1},X_{1})$
from [Lov17], section 4.6. Let $E$ be the composite of the reflex fields of
both Shimura data. One can construct a diagram of Shimura data

where $B_{1}\rightarrow G_{1}$ and $B\rightarrow G$ induce isomorphisms
$B_{1}^{der}\simeq G_{1}^{der}$ and $B^{der}\simeq G^{der}$, and the map
$B_{1}\rightarrow B$ is a central isogeny.

Actually
$B_{1}=G_{1}\times_{G_{1}^{ab}}\mathrm{Res}_{E/\mathbb{Q}}\mathbb{G}_{m}$ for
a suitable map $\mathrm{Res}_{E/\mathbb{Q}}\mathbb{G}_{m}\rightarrow
G_{1}^{ab}$ induced by the cocharacter $\mu_{G_{1}}$. We let
$T=\mathrm{Res}_{E/\mathbb{Q}}\mathbb{G}_{m}$.

Since we are going to deal with several Shimura varieties at the same time in
this section, we change our notation, and denote by $S(G,X)_{K}$ the Shimura
variety attached to a datum $(G,X)$ and compact open $K$. We will similarly
write $M_{dR}(G,X)$ rather than $M_{dR}$, etc. We also adopt the standard
notation that for $H/\mathbb{Q}$ a reductive group, $H^{ad}(\mathbb{Q})^{+}$
is the intersection of $H^{ad}(\mathbb{Q})$ with the identity component of
$H^{ad}(\mathbb{R})$, while $H(\mathbb{Q})_{+}$ is the preimage of
$H^{ad}(\mathbb{Q})^{+}$ under $H(\mathbb{Q})\to H^{ad}(\mathbb{Q})$.

Let us first recall some classical results. We choose connected components
$X_{B_{1}}^{+}$, $X_{B}^{+}$, $X_{1}^{+}$ and $X^{+}$ compatibly with our
morphisms of Shimura data. This allows us to identify the set of geometric
connected component of a Shimura variety $S(H,X_{H})_{K}$ with
${H(\mathbb{Q})_{+}}\backslash H(\mathbb{A}_{f})/K$. We let
$S^{0}(H,X_{H})_{K}$ be the connected component corresponding to the class of
$1$ (we allow ourselves to extend the base field). We have
$S^{0}(H,X_{H})_{K}(\mathbb{C})=\Gamma_{K}\backslash X_{H}^{+}$ where
$\Gamma_{K}=H(\mathbb{Q})_{+}\cap K$. We adopt the same notation of adding a
supscript $0$ for the connected components of minimal and toroidal
compactifications.

We recall a functoriality between Shimura data and compactifications. Let
$g:(H,X_{H})\rightarrow(S,X_{S})$ be a morphism of Shimura data. We assume
that $g$ induces an isomorphism $H^{ad}=S^{ad}$. Let $K^{\prime}\subseteq
H(\mathbb{A}_{f})$ and $K\subset S(\mathbb{A}_{f})$ be compact open subgroups
such that $g(K^{\prime})\subseteq K$. Let $\Sigma$ be a cone decomposition for
$S$.

######  Proposition 4.4.42.

The following point are satisfied.

  1. (1)

The morphism $S(H,X_{H})_{K^{\prime}}\rightarrow S(S,X_{S})_{K}$ is finite
étale.

  2. (2)

The cone decomposition $\Sigma$ for $S$ induces a cone decomposition for $H$
and the morphism $S^{tor}(H,X_{H})_{K^{\prime},\Sigma}\rightarrow
S^{tor}(S,X_{S})_{K,\Sigma}$ is finite.

  3. (3)

If $g(K^{\prime})$ is normal inside $K$, the morphism
$S^{0}(H,X_{H})_{K^{\prime}}\rightarrow S^{0}(S,X_{S})_{K}$ is Galois with
finite group $\Delta(K,K^{\prime})$. The action of $\Delta(K,K^{\prime})$
extends to an action on $S^{tor,0}(H,X_{H})_{K^{\prime},\Sigma}$, and
$S^{tor,0}(S,X_{S})_{K,\Sigma}$ is the quotient of
$S^{tor,0}(H,X_{H})_{K^{\prime}}$ by the action of $\Delta(K,K^{\prime})$.

###### Proof.

For the first point we reduce to check this on connected components over the
complex numbers. We have $X_{H}^{+}=X_{S}^{+}=X^{+}$. We have
$S^{0}(H,X_{H})_{K^{\prime}}=X^{+}/\Gamma_{K^{\prime}}$ and
$S^{0}(S,X_{S})_{K}=X^{+}/\Gamma_{K}$. The images of $\Gamma_{K}$ and
$\Gamma_{K}^{\prime}$ in $H^{ad}(\mathbb{Q})^{+}$, denoted respectively
$\Gamma^{ad}_{K}$ and $\Gamma^{ad}_{K^{\prime}}$, are arithmetic subgroups by
[Mil05], prop. 3.2. We deduce that $\Gamma^{ad}_{K^{\prime}}$ has finite index
in $\Gamma^{ad}_{K}$. We next claim that $\Sigma$ also induces a cone
decomposition for $H$. By our assumption, the map $P\mapsto g^{-1}(P)$ induces
a bijection between the rational parabolics of $S$ and $H$ and this is
basically all we need. See [Har89], sect. 2.5. The finiteness of the map
follows from the description of the local charts. For the last point, the
group $\Delta(K,K^{\prime})$ is $\Gamma_{K}^{ad}/\Gamma_{K^{\prime}}^{ad}$. By
normality, the action of $\Delta(K,K^{\prime})$ extends and the last point
follows. ∎

We also recall a construction of fiber products of torsors. Let $S$ be a
scheme and let $H_{1},H_{2},H_{3}$ be three group schemes with morphisms
$H_{1}\rightarrow H_{2}$ and $H_{3}\rightarrow H_{2}$. We assume that we have
$H_{i}$-torsors $P_{i}$ over $S$ and isomorphisms
$\theta_{1}:P_{H_{1}}\times^{H_{1}}H_{2}\simeq P_{H_{2}}$ and
$\theta_{3}:P_{H_{3}}\times^{H_{3}}H_{2}\simeq P_{H_{2}}$. Then one can form
$P_{H_{1}\times_{H_{2}}H_{3}}=P_{H_{1}}\times_{P_{H_{2}}}P_{H_{1}}$ which is
an $H_{1}\times_{H_{2}}H_{3}$ torsor.

######  Theorem 4.4.43.

The following points are satisfied:

  1. (1)

The towers of connected components of the Shimura varieties $\lim_{K\subseteq
B_{1}(\mathbb{A}_{f})}S^{0}(B_{1},X_{B_{1}})_{K}$ and $\lim_{K\subseteq
G_{1}(\mathbb{A}_{f})}S^{0}(G_{1},X_{1})_{K}$ are canonically isomorphic. The
same holds for the minimal and toroidal compactifications.

  2. (2)

Let $K^{p}\subseteq B_{1}(\mathbb{A}_{f}^{p})$ be compact open. There exists a
compact open subgroup $(K^{p})^{\prime}\subseteq G_{1}(\mathbb{A}_{f}^{p})$
such that we have a finite étale Galois morphism:

$\lim_{K^{\prime}_{p}\subseteq
G_{1}(\mathbb{Q}_{p})}S^{0}(G_{1},X_{1})_{(K^{p})^{\prime}K_{p}^{\prime}}\rightarrow\lim_{K_{p}\subseteq
B_{1}(\mathbb{Q}_{p})}S^{0}(B_{1},X_{B_{1}})_{K^{p}K_{p}}.$

  3. (3)

Let $K\subseteq B_{1}(\mathbb{A}_{f})$ be a compact open subgroup and $\Sigma$
be a cone decomposition for $G_{1}$. There are compact open subgroups
$K_{1}\subseteq G_{1}(\mathbb{A}_{f})$, $K_{2}\subseteq T(\mathbb{A}_{f})$,
$K_{3}\subseteq G^{ab}_{1}(\mathbb{A}_{f})$,

Denote by
$\pi_{3}:{S}^{tor}(B_{1},X_{B_{1}})_{{K},\Sigma}\rightarrow{S}(G_{1}^{ab},X_{G_{1}^{ab}})_{K_{3}}$.
The torsor $M_{dR}(B_{1},X_{B_{1}})$ over
${S}^{tor}(B_{1},X_{B_{1}})_{{K^{p}K_{p}},\Sigma}$ is canonically isomorphic
to the fiber product

$\pi_{1}^{\star}M_{dR}(G_{1},X_{1})\times_{\pi_{3}^{\star}M_{dR}(G_{1}^{ab},X_{G_{1}^{ab}})}\pi_{2}^{\star}M_{dR}(T,X_{T}).$

  4. (4)

Let $g:B_{1}\rightarrow G$. Let $K\subseteq G(\mathbb{A}_{f})$ be neat open
compact. There exists a compact open subgroup $K^{\prime}\subseteq
B_{1}(\mathbb{A}_{f})$ such that $g(K^{\prime})\subseteq K$ and the morphism
$S^{0}(B_{1},X_{B_{1}})_{K^{\prime}}\rightarrow S^{0}(G,X)_{K}$ is finite
étale and Galois with group $\Delta(K,K^{\prime})$.

  5. (5)

Let $\Sigma$ be a cone decomposition for $G$. The torsor $M_{dR}(G,X)$ over
$S^{tor,0}(G,X)_{K,\Sigma}$ is the quotient by $\Delta(K,K^{\prime})$ of the
torsor

$M_{dR}(B_{1},X_{1})\times^{M_{\mu_{B_{1}}}^{c}}M_{\mu_{G}}^{c}$

over $S^{tor,0}(B_{1},X_{B_{1}})_{K^{\prime},\Sigma}$.

  6. (6)

Let $K^{p}\subseteq G(\mathbb{A}^{p}_{f})$. There exists a compact open
subgroup $(K^{\prime})^{p}\subseteq B_{1}(\mathbb{A}_{f}^{p})$ such that
$g((K^{\prime})^{p})\subseteq K^{p}$ and the morphism

$\lim_{K_{p}^{\prime}\subseteq
B_{1}(\mathbb{Q}_{p})}S^{0}(B_{1},X_{B_{1}})_{(K^{\prime})^{p}K^{\prime}_{p}}\rightarrow\lim_{K_{p}\subseteq
G(\mathbb{Q}_{p})}S^{0}(G,X)_{K^{p}K_{p}}$

is finite étale and Galois.

###### Proof.

We recall that for a reductive group $H$ defined over $\mathbb{Q}$, the
topologies on $H^{ad}(\mathbb{Q})^{+}$ defined by the images of congruence
subgroups of $H(\mathbb{Q})_{+}$ and $H^{der}(\mathbb{Q})_{+}$ are the same.
This implies that the connected components of the tower of Shimura varieties
is determined by the connected Shimura datum. By construction, the datum
$(G_{1}^{der},X_{1}^{+})$ and $(B^{der}_{1},X_{B_{1}}^{+})$ are the same.

We now check the second point. Let us fix a decreasing sequence of compact
open subgroups $K_{p,n}\subseteq B_{1}(\mathbb{Q}_{p})$ with
$\cap_{n}K_{p,n}=\\{1\\}$ and $K_{p,n}$ normal inside $K_{p,1}$. Let
$K_{p,n}^{ad}$ be the image of $K_{p,n}$ in $B_{1}^{ad}(\mathbb{Q}_{p})$. Let
$\Gamma_{n}=B_{1}(\mathbb{Q})_{+}\cap K^{p}K_{p,n}$ and let $\Gamma_{n}^{ad}$
the image of $\Gamma_{n}$ in $B_{1}^{ad}(\mathbb{Q})^{+}$.

By the first point, there exists a compact open subgroup
$(K^{p})^{\prime}K_{p,1}^{\prime}\subseteq G_{1}(\mathbb{A}_{f})$ such that
$\Gamma^{\prime}_{1}=G_{1}(\mathbb{Q})_{+}\cap(K^{p})^{\prime}K^{\prime}_{p,1}$
has the property that its image in $B_{1}^{ad}(\mathbb{Q})^{+}$ is a finite
index normal subgroup of $\Gamma_{1}^{ad}$. Actually, we may also choose
$\Gamma^{\prime}_{1}$ such that $\Gamma^{\prime}_{1}\subseteq
G_{1}^{der}(\mathbb{Q})_{+}$ (this follows from the fact that $G_{1}$ is of
Hodge type, hence $G_{1}^{ab}(\mathbb{Q})$ is discrete in
$G_{1}^{ab}(\mathbb{A}_{f})$).

We may also assume that the map $G_{1}^{der}(\mathbb{Q})_{+}\rightarrow
B_{1}^{ad}(\mathbb{Q})^{+}$ induces an isomorphism from $\Gamma^{\prime}_{1}$
to its image (indeed, the kernel of this map is finite). Let us denote by
$\Gamma^{\prime}_{n}=\Gamma^{ad}_{n}\cap\Gamma^{\prime}_{1}$. Let
$\Delta_{n}=\Gamma_{n}^{ad}/\Gamma_{n}^{\prime}$. Clearly
$\Delta_{n}\rightarrow\Delta_{n-1}$ is injective and therefore
$\Delta=\lim_{n}\Delta_{n}$ is a finite group.

Since the map $G_{1}^{der}(\mathbb{Q}_{p})\rightarrow
B_{1}^{ad}(\mathbb{Q}_{p})$ is a local homeomorphism, there is for $n$ large
enough a subgroup $K_{p,n}^{der}\subseteq G_{1}^{der}(\mathbb{Q}_{p})$ which
maps isomorphically to $K_{p,n}^{ad}$, the image of $K_{p,n}$ in
$B_{1}^{ad}(\mathbb{Q}_{p})$. We deduce that
$\Gamma^{\prime}_{n}=\Gamma^{\prime}_{1}\cap K^{der}_{p,n}$. Finally, we can
find a decreasing sequence of compact open subgroups
$\\{K^{\prime}_{p,n}\subseteq G_{1}(\mathbb{Q}_{p})\\}$ such that
$\cap_{n}K^{\prime}_{p,n}=\\{1\\}$ and $K^{\prime}_{p,n}\cap
G^{der}_{1}(\mathbb{Q}_{p})=K_{p,n}^{der}$. We deduce that
$\lim_{K^{\prime}_{p,n}}S^{0}(G_{1},X_{B_{1}})_{(K^{p})^{\prime}K^{\prime}_{p,n}}\rightarrow\lim_{K_{p,n}}S^{0}(B_{1},X_{B_{1}})_{K^{p}K_{p,n}}$
is finite étale with group $\Delta$. For the third point, using the various
functorialities of Shimura varieties (see for instance [Lov17], lemma 3.1.6),
we see that we have a map of torsors

$M_{dR}(B_{1},X_{B_{1}})\rightarrow\pi_{1}^{\star}M_{dR}(G_{1},X_{1})\times_{\pi_{3}^{\star}M_{dR}(G_{1}^{ab},X_{G_{1}^{ab}})}\pi_{2}^{\star}M_{dR}(T,X_{T}).$

and a map of torsors is an isomorphism. We check the fourth point. We can find
a normal compact open subgroup $K^{\prime}\subseteq g^{-1}(K)$ and we apply
proposition 4.4.42. The fifth point follows again from the functorialities
between Shimura varieties. We now check the last point. Let us fix a
decreasing sequence of compact open subrgoups $\\{K_{p,n}\subseteq
G(\mathbb{Q}_{p})\\}$ with $\cap_{n}K_{p,n}=\\{1\\}$ and $K_{p,n}$ is normal
inside $K_{p,1}$. Let us define $\Gamma_{n}=G(\mathbb{Q})_{+}\cap
K^{p}K_{p,n}$. Let us fix a compact open subgroup
$(K^{\prime})^{p}K_{p,1}^{\prime}\subseteq B_{1}(\mathbb{A}_{f})$ with
$g((K^{\prime})^{p}K_{p,1}^{\prime})\subseteq K^{p}K_{p,n}$. Let
$K_{p,n}^{\prime}=g^{-1}(K_{p,n})\cap K^{\prime}_{p,1}$. Let
$K_{p,n}^{der}=K^{\prime}_{p,n}\cap B^{der}_{1}(\mathbb{Q}_{p})$. Let
$\Gamma^{\prime}_{n}=B_{1}(\mathbb{Q})_{+}\cap(K^{p})^{\prime}K^{\prime}_{p,n}$.
Let $\Gamma^{der}_{n}=\Gamma_{n}\cap B_{1}(\mathbb{Q})_{+}^{der}$. For any of
$\Gamma_{n},\Gamma^{\prime}_{n},\Gamma_{n}$, we add a subscript $ad$ to mean
their image in $G^{ad}(\mathbb{Q})^{+}$. We see that
$(\Gamma_{n}^{der})^{ad}\subseteq(\Gamma^{\prime}_{n})^{ad}\subseteq\Gamma_{n}^{ad}$
are all arithmetic subgroups. Let
$\Delta_{n}=\Gamma_{n}^{ad}/(\Gamma_{n}^{der})^{ad}$. We see that
$\Delta_{n}\rightarrow\Delta_{n-1}$ is injective and therefore
$\Delta=\lim_{n}\Delta_{n}$ is a finite group. We observe that
$\cap_{n}K_{p,n}^{der}=\mathrm{Ker}(B_{1}^{der}(\mathbb{Q}_{p})\rightarrow
G(\mathbb{Q}_{p}))$ is a finite group. Let us fix a compact open subgroup
$L_{p}\subseteq B_{1}^{der}(\mathbb{Q}_{p})$ such that
$L_{p}\cap\mathrm{Ker}(B_{1}^{der}(\mathbb{Q}_{p})\rightarrow
G(\mathbb{Q}_{p}))=\\{1\\}$. We let
$\Gamma^{der,^{\prime}}_{n}=\Gamma^{der}_{n}\cap L_{p}$. The morphism
$\lim_{n}\Gamma^{der,^{\prime}}_{n}\backslash
X^{+}\rightarrow\lim_{n}\Gamma_{n}\backslash X^{+}$ is finite étale and
Galois. Since this map factors the map $\lim_{K_{p}^{\prime}\subseteq
B_{1}(\mathbb{Q}_{p})}S^{0}(B_{1},X_{B_{1}})_{(K^{\prime})^{p}K^{\prime}_{p}}(\mathbb{C})\rightarrow\lim_{K_{p}\subseteq
G(\mathbb{Q}_{p})}S^{0}(G,X)_{K^{p}K_{p}}(\mathbb{C})$, we conclude. ∎

In this paper, we will be able to reduce a number of statements (and in
particular vanishing theorems) on abelian type Shimura varieties to the Hodge
type case using the following principle:

######  Principle 4.4.44.

  1. (1)

We can extend a construction (eg. of a torsor) from the Shimura variety for
$(G_{1},X_{1})$ to the Shimura variety for $(B_{1},X_{B_{1}})$ by performing a
(trivial) construction on the Shimura variety attached to $T$ and using the
identity
$M_{\mu_{B_{1}}}^{c}=M^{c}_{\mu_{G_{1}}}\times_{M^{ab,c}_{\mu_{G_{1}}}}T^{c}$.

  2. (2)

The torsors we construct on a connected component of a Shimura variety for
$(G,X)$ is obtained from a torsor on a connected component of the Shimura
variety for $(B_{1},X_{B_{1}})$ by pushing out along the map
$B_{1}^{c}\rightarrow G^{c}$ and taking the quotient by a finite group.

We work until the end of this paragraph over the algebraically closed field
$\mathbb{C}\simeq\mathbb{C}_{p}$ (this means that all Shimura varieties are
base changed to $\mathbb{C}_{p}$). This simplifies our treatment of connected
components of Shimura varieties. We also recall that the rationality questions
with respect to the reflex field are irrelevant for us.

The principle 4.4.44 is based on the following theorem:

######  Theorem 4.4.45.

There is a commutative diagram

where $\mathcal{S}^{\star}(B_{1},X_{B_{1}})$,
$\mathcal{S}^{\star}(G_{1},X_{1})$, and $\mathcal{S}^{\star}(G,X)$ are
perfectoid spaces representing (in the category of diamonds)
$\lim_{K}\mathcal{S}^{\star}(B_{1},X_{B_{1}})_{K}$,
$\lim_{K}\mathcal{S}^{\star}(G_{1},X_{G_{1}})_{K}$, and
$\lim_{K}\mathcal{S}^{\star}(G,X)_{K}$. The maps are equivariant for the
respective actions of $B_{1}(\mathbb{A}_{f})$, $G_{1}(\mathbb{A}_{f})$, and
$G(\mathbb{A}_{f})$. The three maps to the flag variety
$\mathcal{FL}_{G,\mu}=\mathcal{FL}_{G_{1},\mu_{G_{1}}}=\mathcal{FL}_{B_{1},\mu_{B_{1}}}$
are Hodge-Tate period maps, and the composite map
$\mathcal{S}^{tor,\Diamond}(G,X)_{\Sigma}=\lim_{K}\mathcal{S}^{tor,\Diamond}(G,X)_{K,\Sigma}\rightarrow\mathcal{S}^{\star}(G,X)\rightarrow\mathcal{FL}_{G,\mu}$
is the map $\pi_{HT}^{tor}$ of theorem 4.4.40.

######  Remark 4.4.46.

In the course of the proof, we also introduce spaces
$\overline{\mathcal{S}}^{\star}(G,X)=\lim_{K}{\mathcal{S}}^{\star}(G,X)_{\overline{K}}$
which depend on the embedding of $(G_{1},X_{1})$ in a Siegel datum. We have
finite maps
${\mathcal{S}}^{\star}(G,X)_{{K}}\rightarrow{\mathcal{S}}^{\star}(G,X)_{\overline{K}}$
which are isomorphisms away from the boundary. We prove that
$\overline{\mathcal{S}}^{\star}(G,X)\sim\lim_{K}{\mathcal{S}}^{\star}(G,X)_{\overline{K}}$
and that the map $\pi_{HT}$ factors through an affinoid map
$\pi_{HT}:\overline{\mathcal{S}}^{\star}(G,X)\rightarrow\mathcal{FL}_{G,\mu}$.
See proposition 4.4.53.

######  Remark 4.4.47.

Perfectoid (minimal compactifications of) abelian type Shimura varieties have
been constructed for the first time in [HJ20]. Since we also need to construct
the Hodge-Tate period map and prove several compatibilities, we give a
complete argument. The argument is very similar to [HJ20] in the sense that
this is a reduction to the Hodge type case.

####  4.4.48. Proof of theorem 4.4.45

We first briefly recall how to reconstruct a Shimura variety from its
connected Shimura variety following [Del79], section 2.1. Let $(G,X)$ be a
Shimura datum. The center $Z(G)$ of $G$ is simply denoted by $Z$ unless some
confusion may arise.

The inverse system $S(G,X)=\lim_{K}S(G,X)_{K}$ has a right action of the group
$\mathcal{A}(G)=G(\mathbb{A}_{f})/\overline{Z(\mathbb{Q})}\ast_{G(\mathbb{Q})_{+}}G^{ad}(\mathbb{Q})^{+}$.
Here $\overline{Z(\mathbb{Q})}$ is the closure of $Z(\mathbb{Q})$ in
$G(\mathbb{A}_{f})$.

######  Remark 4.4.49.

If $Z_{s}=\\{1\\}$, then $\overline{Z(\mathbb{Q})}=Z(\mathbb{Q})$. If the
morphism $G(\mathbb{Q})\rightarrow G^{ad}(\mathbb{Q})$ is surjective (for
example, if $Z(G)$ is a split torus), then
$\mathcal{A}(G)=G(\mathbb{A}_{f})/\overline{Z(\mathbb{Q})}$.

There is a canonical bijection
$\pi_{0}(S(G,X))\rightarrow\overline{G(\mathbb{Q})_{+}}\backslash
G(\mathbb{A}_{f})$ of right $\mathcal{A}(G)$-profinite sets where
$\overline{G(\mathbb{Q})_{+}}$ is the closure of $G(\mathbb{Q})_{+}$ in
$G(\mathbb{A}_{f})$. This is also the completion of $G(\mathbb{Q})_{+}$ for
the topology where a basis of open neighborhoods of the identity is given by
the congruence subgroups of $G(\mathbb{Q})_{+}$.

Let us pick $1\in\overline{G(\mathbb{Q})_{+}}\backslash G(\mathbb{A}_{f})$ and
let $S^{0}(G,X)\hookrightarrow S(G,X)$ be the connected component
corresponding to $1$. This is the tower of connected Shimura varieties. The
stabilizer of $S^{0}(G,X)$ for the action of $\mathcal{A}(G)$ is
$\mathcal{A}^{0}(G)=\overline{G(\mathbb{Q})_{+}}/\overline{Z(\mathbb{Q})}\ast_{G(\mathbb{Q})_{+}}G^{ad}(\mathbb{Q})^{+}$
and we have the formula:

$S(G,X)=[S^{0}(G,X)\times\mathcal{A}(G)]/\mathcal{A}^{0}(G)$

for the action $(s,g).h=(sh,h^{-1}g)$. One important observation is that
$S^{0}(G,X)$ depends only on the connected Shimura datum $(G^{der},X^{+})$ and
the group $\mathcal{A}^{0}(G)$ depends also only on $G^{der}$. More precisely,
we have that
$\mathcal{A}^{0}(G)=\overline{G^{der}(\mathbb{Q})_{+}}\ast_{G^{der}(\mathbb{Q})_{+}}G^{ad}(\mathbb{Q})^{+}$,
and $\mathcal{A}^{0}(G)$ is therefore the completion of
$G^{ad}(\mathbb{Q})^{+}$ with respect to the topology where a basis of open
neighborhoods of $1$ is the images of congruence subgroups of
$G^{der}(\mathbb{Q})_{+}$. We have that $S^{0}(G,X)=\lim_{K}S^{0}(G,X)/K$
where the limit runs over the compact open subgroups of
$\overline{G^{der}(\mathbb{Q})_{+}}$, and
$(S^{0}(G,X)/K)(\mathbb{C})=\Gamma_{K}\backslash X^{+}$ where
$\Gamma_{K}=G^{der}(\mathbb{Q})_{+}\cap K$.

Of course, there is also a simpler formula: $S(G,X)=[S^{0}(G,X)\times
G(\mathbb{A}_{f})]/\overline{G(\mathbb{Q})_{+}}$, but the disadvantage of this
formula is that $\overline{G(\mathbb{Q})_{+}}$ depends on $G$!

All of this extends to the minimal compactification since a Shimura variety
and its minimal compactification have the same connected components, and group
actions extend by normality of minimal compactifications.

We now go back to our Hodge type datum $(G_{1},X_{1})$, and we fix an
embedding in a Siegel datum $(\tilde{G},\tilde{X})$. Attached to this fixed
embedding, for any compact open $K$ we have the minimal compactification
$S^{\star}(G_{1},X_{1})_{K}$ together with a finite surjective map to
$S^{\star}(G_{1},X_{1})_{\overline{K}}$, where
$S^{\star}(G_{1},X_{1})_{\overline{K}}$ is the schematic image of
$S^{\star}(G_{1},X_{1})_{K}\rightarrow
S^{\star}(\tilde{G},\tilde{X})_{\tilde{K}}$ for all small enough $\tilde{K}$
with $\tilde{K}\cap G(\mathbb{A}_{f})=K$. Let us denote
$\overline{S}^{\star}(G_{1},X_{1})=\lim_{K}S^{\star}(G_{1},X_{1})_{\overline{K}}$.
For the Siegel datum $(\tilde{G},\tilde{X})$, we have that
$\mathcal{A}(\tilde{G})=\tilde{G}(\mathbb{A}_{f})/Z(\tilde{G})(\mathbb{Q})$.
We deduce that the closed subgroup $\mathcal{A}(G_{1})$ of
$\mathcal{A}(\tilde{G})$ acts on $\overline{S}^{\star}(G_{1},X_{1})$ and
${S}^{\star}(G_{1},X_{1})$ in a compatible way. We also deduce that the Hodge-
Tate period map
$\mathcal{S}^{\star}(G_{1},X_{1})\rightarrow\overline{\mathcal{S}}^{\star}(G_{1},X_{1})\rightarrow\mathcal{FL}_{G_{1},\mu_{G_{1}}}$
is $\mathcal{A}(G_{1})$-equivariant (by reduction to the Siegel case). We
remark that the $\mathcal{A}(G_{1})$-action on
$\mathcal{FL}_{G_{1},\mu_{G_{1}}}$ factors through the map
$\mathcal{A}(G_{1})\rightarrow G_{1}^{ad}(\mathbb{A}_{f})\rightarrow
G^{ad}_{1}(\mathbb{Q}_{p})$.

We are now ready to extend things to $(B_{1},X_{B_{1}})$. First, we have that
$S^{\star}(B_{1},X_{B_{1}})=[S^{\star,0}(G_{1},X_{1})\times\mathcal{A}(B_{1})]/\mathcal{A}^{0}(B_{1})$.
We may also define
$\overline{S}^{\star}(B_{1},X_{B_{1}})=[\overline{S}^{\star,0}(G_{1},X_{1})\times\mathcal{A}(B_{1})]/\mathcal{A}^{0}(B_{1})$.
By taking $K$-invariants, we define
$S^{\star}(B_{1},X_{B_{1}})_{\overline{K}}=\overline{S}^{\star}(B_{1},X_{B_{1}})/K$.

Our first lemma is:

######  Lemma 4.4.50.

There is a perfectoid space
$\mathcal{S}^{\star}(B_{1},X_{B_{1}})_{\overline{K^{p}}}\sim\lim_{K_{p}}\mathcal{S}^{\star}(B_{1},X_{B_{1}})_{\overline{K^{p}K_{p}}}$
and a perfectoid space
$\mathcal{S}^{\star}(B_{1},X_{B_{1}})_{{K^{p}}}=\lim_{K_{p}}\mathcal{S}^{\star,\Diamond}(B_{1},X_{B_{1}})_{{K^{p}K_{p}}}$.

###### Proof.

We prove the first statement. Fix a compact open subgroup $K_{p}\subseteq
B_{1}(\mathbb{Q}_{p})$. It acts on the scheme
${S}^{\star}(B_{1},X_{B_{1}})_{\overline{K^{p}}}$, and it acts on the
connected components with finitely many orbits. We choose representatives for
the orbits, which are all of the form
${S}^{\star,0}(B_{1},X_{B_{1}})_{\overline{k_{i}K^{p}k_{i}^{-1}}}$ for
suitable elements $k_{i}\in B_{1}(\mathbb{A}_{f}^{p})$. Let $S(k_{i})$ be the
stabilizer of the connected component
${S}^{\star,0}(B_{1},X_{B_{1}})_{\overline{k_{i}K^{p}k_{i}^{-1}}}$ in $K_{p}$.
This is a closed subgroup of $K_{p}$. We deduce that
${S}^{\star}(B_{1},X_{B_{1}})_{\overline{K^{p}}}=\coprod_{i}[{S}^{\star,0}(B_{1},X_{B_{1}})_{\overline{k_{i}K^{p}k_{i}^{-1}}}\times
K_{p}]/{S(k_{i})}$.

By theorem 4.4.43 (2), we can find compact open subgroups
$(K_{i}^{p})^{\prime}\subseteq G_{1}(\mathbb{A}^{p}_{f})$ such that we have
finite morphisms
$S^{\star,0}(G_{1},X_{1})_{\overline{(K_{i}^{p})^{\prime}}}\rightarrow
S^{\star,0}(B_{1},X_{B_{1}})_{\overline{k_{i}K^{p}k_{i}^{-1}}}$, identifying
$S^{\star,0}(B_{1},X_{B_{1}})_{\overline{k_{i}K^{p}k_{i}^{-1}}}$ as the
quotient of $S^{\star,0}(G_{1},X_{1})_{\overline{(K_{i}^{p})^{\prime}}}$ by a
finite group.

Now we pass to analytic geometry. It follows from lemma 4.4.6 and the results
of 4.4.27 that the
$\mathcal{S}^{\star,0}(G_{1},X_{1})_{\overline{(K_{i}^{p})^{\prime}}}$ are
perfectoid. We deduce from lemma 4.4.7 that the
$\mathcal{S}^{\star,0}(B_{1},X_{B_{1}})_{\overline{k_{i}K^{p}k_{i}^{-1}}}$ are
perfectoid spaces.

We now want to define
$\mathcal{S}^{\star}(B_{1},X_{B_{1}})_{\overline{K^{p}}}=\coprod_{i}[\mathcal{S}^{\star,0}(B_{1},X_{B_{1}})_{\overline{k_{i}K^{p}k_{i}^{-1}}}\times
K_{p}]/{S(k_{i})}$. The product
$\mathcal{S}^{\star,0}(B_{1},X_{B_{1}})_{\overline{k_{i}K^{p}k_{i}^{-1}}}\times(K_{p}/{S(k_{i})})$
is a perfectoid space, being the product of a perfectoid space with a
profinite set. Checking that
$\mathcal{S}^{\star}(B_{1},X_{B_{1}})_{\overline{K^{p}}}\sim\lim_{K_{p}}\mathcal{S}^{\star}(B_{1},X_{B_{1}})_{\overline{K^{p}K_{p}}}$
is now an easy exercise, again using 4.4.27. We deduce that
$\mathcal{S}^{\star}(B_{1},X_{B_{1}})_{{K^{p}}}=\lim_{K_{p}}\mathcal{S}^{\star,\Diamond}(B_{1},X_{B_{1}})_{{K^{p}K_{p}}}$
by [BS19], thm. 1.16. ∎

We can now construct the Hodge-Tate period map.

######  Lemma 4.4.51.

We have a Hodge-Tate period map

$\pi_{HT}:\mathcal{S}^{\star}(B_{1},X_{B_{1}})_{{K^{p}}}\rightarrow\mathcal{S}^{\star}(B_{1},X_{B_{1}})_{\overline{K^{p}}}\rightarrow\mathcal{FL}_{B_{1},\mu_{B_{1}}}$

which agrees on connected components with Hodge-Tate period map for $G_{1}$.
The Hodge-Tate period map is $B_{1}(\mathbb{Q}_{p})$-equivariant and Hecke
equivariant away from $p$. The map
$\mathcal{S}^{\star}(B_{1},X_{B_{1}})_{\overline{K^{p}}}\rightarrow\mathcal{FL}_{B_{1},\mu_{B_{1}}}$
is affinoid in the sense that $\mathcal{FL}_{B_{1},\mu_{B_{1}}}$ has a cover
by affinoid opens whose preimages in
$\mathcal{S}^{\star}(B_{1},X_{B_{1}})_{\overline{K^{p}}}$ are good affinoid
perfectoid opens.

###### Proof.

We may pass to the limit over $K^{p}$ and consider the map
$\overline{\mathcal{S}}^{\star}(B_{1},X_{B_{1}})=[\overline{\mathcal{S}}^{\star,0}(G_{1},X_{1})\times\mathcal{A}(B_{1})]/\mathcal{A}^{0}(B_{1})\rightarrow\mathcal{FL}_{B_{1},\mu_{B_{1}}}$
sending
$(x,a)\in\overline{\mathcal{S}}^{\star,0}(G_{1},X_{1})\times\mathcal{A}(B_{1})$
to $a.\pi_{HT}(x)$ where
$\pi_{HT}(x)\in\mathcal{FL}_{B_{1},\mu_{B_{1}}}=\mathcal{FL}_{G_{1},\mu_{G_{1}}}$
and $\mathcal{A}(B_{1})$ acts via its quotient $B^{ad}_{1}(\mathbb{Q}_{p})$ on
$\mathcal{FL}_{B_{1},\mu_{B_{1}}}$. Since the map
$\overline{\mathcal{S}}^{\star,0}(G_{1},X_{1})\rightarrow\mathcal{FL}_{B_{1},\mu_{B_{1}}}$
is $\mathcal{A}^{0}(G_{1})$-equivariant, we deduce that we have a well-defined
$\mathcal{A}(B_{1})$-equivariant map
$\overline{\mathcal{S}}^{\star}(B_{1},X_{B_{1}})\rightarrow\mathcal{FL}_{B_{1},\mu_{B_{1}}}$.
The affine property follows from the Hodge case. ∎

We now consider the descent from $(B_{1},X_{B_{1}})$ to $(G,X)$. We have a map
$f:B_{1}\rightarrow G$ inducing an isogeny $B_{1}^{der}\rightarrow G^{der}$.

######  Lemma 4.4.52.

  1. (1)

We have a continuous surjective map
$\mathcal{A}^{0}(B_{1})\rightarrow\mathcal{A}^{0}(G)$ with kernel a pro-finite
group $\Delta$.

  2. (2)

We have a pro-finite étale morphism $S^{0}(B_{1},X_{B_{1}})\rightarrow
S^{0}(G,X)$ which is Galois with group $\Delta$.

###### Proof.

Let $Cong(B_{1})$ be the set of congruence subgroups of
$B_{1}^{der}(\mathbb{Q})_{+}$, and let $Cong(G)$ be the set of congruence
subgroups of $G^{der}(\mathbb{Q})_{+}$. We claim that the map
$f:B_{1}^{der}(\mathbb{Q})_{+}\rightarrow G^{der}(\mathbb{Q})_{+}$ induces a
map $f^{-1}:Cong(G)\rightarrow Cong(B_{1})$. Indeed, let $K\subseteq
G^{der}(\mathbb{A}_{f})$ be a compact open subgroup, then $f^{-1}(K)$ is
compact open in $B_{1}^{der}(\mathbb{A}_{f})$ and
$f^{-1}(G^{der}(\mathbb{Q})_{+}\cap K)=B_{1}^{der}(\mathbb{Q})_{+}\cap
f^{-1}(K)$. If $\Gamma\in Cong(B_{1})$, then $f(\Gamma)$ is an arithmetic
subgroup of $G^{der}(\mathbb{Q})_{+}$ but not necessarily a congruence
subgroup!

We consider the index set $I$ of compact open subgroups $(K,K^{\prime})\in
B_{1}^{der}(\mathbb{A}_{f})\times G^{der}(\mathbb{A}_{f})$ such that
$f(K)\subseteq K^{\prime}$.

Let $\Gamma_{K}=B_{1}^{der}(\mathbb{Q})_{+}\cap K$ and
$\Gamma_{K^{\prime}}=G^{der}(\mathbb{Q})_{+}\cap K^{\prime}$. Let
$\Delta(K,K^{\prime})=\Gamma_{K}\backslash\Gamma_{K^{\prime}}$.

We observe that for any sufficiently small $\Gamma\in Cong(B_{1})$ or
$\Gamma\in Cong(G)$, the map $f\circ g:\Gamma\rightarrow
G^{ad}(\mathbb{Q})^{+}$ or $f:\Gamma\rightarrow G^{ad}(\mathbb{Q})^{+}$ is
injective. Indeed, remark that the kernel of
$G^{der}(\mathbb{Q})_{+}\rightarrow G^{ad}(\mathbb{Q})^{+}$ is a finite group.

The groups $g\circ f(Cong(B_{1}))$ and $f(Cong(G))$ are the basis of open
neighborhoods of $1$ for two topologies $\tau(B_{1})$ and $\tau(G)$ on
$G^{ad}(\mathbb{Q})^{+}$. We know that $\mathcal{A}^{0}(B_{1})$ and
$\mathcal{A}^{0}(G)$ are the completion of $G^{ad}(\mathbb{Q})^{+}$ for the
topologies $\tau(B_{1})$ and $\tau(G)$ respectively. Therefore,
$\mathcal{A}^{0}(B_{1})=\lim_{\Gamma\in
Cong(B_{1})}G^{ad}(\mathbb{Q})^{+}/\Gamma$ and
$\mathcal{A}^{0}(G)=\lim_{\Gamma\in Cong(G)}G^{ad}(\mathbb{Q})^{+}/\Gamma$. By
the Mittag-Leffler criterion the sequence

$1\rightarrow\lim_{(K,K^{\prime})\in
I}\Delta(K,K^{\prime})\rightarrow\lim_{I}G^{ad}(\mathbb{Q})^{+}/(\Gamma_{K})\rightarrow\lim_{I}G^{ad}(\mathbb{Q})^{+}/\Gamma_{K^{\prime}}\rightarrow
1$

is exact and equals
$1\rightarrow\Delta\rightarrow\mathcal{A}^{0}(B_{1})\rightarrow\mathcal{A}^{0}(G)\rightarrow
1$. ∎

We are now ready to descend everything from $(B_{1},X_{B_{1}})$ to $(G,X)$.
First, we have that $S^{\star,0}(G,X)=S^{\star,0}(B_{1},X_{B_{1}})/\Delta$ and
we may also define
$\overline{S}^{\star,0}(G,X)=\overline{S}^{\star,0}(B_{1},X_{B_{1}})/\Delta$.
We have
$S^{\star}(G,X)=[S^{\star,0}(G,X)\times\mathcal{A}(G)]/\mathcal{A}^{0}(G)$. We
may also define
$\overline{S}^{\star}(G,X)=[\overline{S}^{\star,0}(G,X)\times\mathcal{A}(G)]/\mathcal{A}^{0}(G)$.
By taking $K$-invariants, we define
$\overline{S}^{\star}(G,X)/K={S}^{\star}(G,X)_{\overline{K}}$.

######  Proposition 4.4.53.

Let $(G,X)$ be an abelian type Shimura datum as above.

  1. (1)

There is a perfectoid space
$\mathcal{S}^{\star}(G,X)_{\overline{K^{p}}}\sim\lim_{K_{p}}\mathcal{S}^{\star}(G,X)_{\overline{K^{p}K_{p}}}$
and a perfectoid space
$\mathcal{S}^{\star}(G,X)_{{K^{p}}}=\lim_{K_{p}}\mathcal{S}^{\star,\Diamond}(G,X)_{{K^{p}K_{p}}}$.

  2. (2)

We have a Hodge-Tate period map
$\pi_{HT}:\mathcal{S}^{\star}(G,X)_{{K^{p}}}\rightarrow\mathcal{S}^{\star}(G,X)_{\overline{K^{p}}}\rightarrow\mathcal{FL}_{G,\mu}$
which is $G(\mathbb{Q}_{p})$-equivariant and Hecke equivariant. The map
$\mathcal{S}^{\star}(G,X)_{\overline{K^{p}}}\rightarrow\mathcal{FL}_{G,\mu}$
is affinoid in the sense that $\mathcal{FL}_{G,\mu}$ has a cover by affinoid
opens whose preimages in $\mathcal{S}^{\star}(G,X)_{\overline{K^{p}}}$ are
good affinoid perfectoid opens.

###### Proof.

For the first point, we observe that the group $\Delta$ acts trivially on the
flag variety. Since $\pi_{HT}$ was affinoid for $(B_{1},X_{B_{1}})$, we can
use theorem 4.4.43 (6), and apply lemma 4.4.6 and lemma 4.4.7 to deduce that
$\mathcal{S}^{\star,0}(G,X)_{\overline{K^{p}}}$ is perfectoid. We may now
extend the result from $\mathcal{S}^{\star,0}(G,X)_{\overline{K^{p}}}$ to
$\mathcal{S}^{\star}(G,X)_{\overline{K^{p}}}$ as in the proof of lemma 4.4.50.

Passing to the limit over $K^{p}$, we have a Hodge-Tate period map
$\overline{\mathcal{S}}^{\star,0}(B_{1},X_{B_{1}})/\Delta=\overline{\mathcal{S}}^{\star,0}(G,X)\rightarrow\mathcal{FL}_{G,\mu}=\mathcal{FL}_{B_{1},\mu_{B_{1}}}$
which is $\mathcal{A}^{0}(G)$-equivariant. We deduce that there is a
$\mathcal{A}(G)$-equivariant Hodge-Tate period map
$\overline{\mathcal{S}}^{\star}(G,X)\rightarrow\mathcal{FL}_{G,\mu}$. The
representability of
$\lim_{K_{p}}\mathcal{S}^{\star,\Diamond}(G,X)_{{K^{p}K_{p}}}$ follows from
[BS19], thm. 1.16.

∎

###  4.5. The truncated Hodge-Tate period map

In most of this paper, we work on finite level Shimura varieties rather than
perfectoid Shimura varieties. For this reason we introduce some truncated
Hodge-Tate period map.

Let $(G,X)$ be an abelian type Shimura datum. Let $K=K^{p}K_{p}\subseteq
G(\mathbb{A}_{f})$ be a compact open subgroup. The group $K_{p}$ acts on
$\mathcal{FL}_{G,\mu}$. We form the quotient space
$\mathcal{FL}_{G,\mu}/K_{p}$, equipped with the quotient topology from the
surjective map
$\pi_{K_{p}}:\mathcal{FL}_{G,\mu}\rightarrow\mathcal{FL}_{G,\mu}/K_{p}$. We
merely view $\mathcal{FL}_{G,\mu}/K_{p}$ as a topological space. We adopt some
definitions. We say that an open $U\subseteq\mathcal{FL}_{G,\mu}/K_{p}$ is
affinoid if $\pi_{K_{p}}^{-1}(U)$ is affinoid. If $V\subseteq
U\subseteq\mathcal{FL}_{G,\mu}/K_{p}$ are open we say that $V$ is a rational
subset of $U$ if $\pi_{K_{p}}^{-1}(V)$ is a rational subset of
$\pi_{K_{p}}^{-1}(U)$. We let the residue field of a point
$x\in\mathcal{FL}_{G,\mu}/K_{p}$ be the residue field of any lift of this
point to $\mathcal{FL}_{G,\mu}$. To illustrate all these definitions, we have
the following lemma:

######  Lemma 4.5.1.

Any point $x\in\mathcal{FL}_{G,\mu}/K_{p}$ with finite residue field over
$\mathbb{Q}_{p}$ (i.e. a classical point in the sense of rigid analytic
geometry) has a basis of neighborhoods consisting of affinoids
$\\{U_{n}\\}_{n\geq 1}$ with the property that $U_{n+1}\subseteq U_{n}$ is a
rational subset.

###### Proof.

The group $\mathcal{G}$ admits a filtration by open affinoid normal subgroups
$\mathcal{G}_{n}$ which form a basis of neighborhoods of the identity (take
$\mathcal{G}_{n}$ the subgroup of elements which reduce to $1$ modulo
$p^{n}$). Let $y\in\mathcal{FL}_{G,\mu}$ be a lift of $x$. Then
$y\mathcal{G}_{n}\hookrightarrow\mathcal{FL}_{G,\mu}$ is an affinoid for any
$n\geq 1$ (this is a closed tube centered at the point $y$, this is also where
we use that $x$ has finite residue field over $\mathbb{Q}_{p}$). We now
consider the group $\mathcal{G}_{n}K_{p}\subset\mathcal{G}$. The connected
component of the identity of this group is $\mathcal{G}_{n}$, and the quotient
$\mathcal{G}_{n}K_{p}/\mathcal{G}_{n}$ is a finite group. Then we find that
$y\mathcal{G}_{n}K_{p}=\coprod_{i\in I}yk_{i}\mathcal{G}_{n}$ for a finite set
$I$ and elements $k_{i}\in K_{p}$, and is therefore an affinoid which is
$K_{p}$-invariant. The $\\{y\mathcal{G}_{n}K_{p}/K_{p}\\}_{n\geq 1}$ form a
basis of open neighborhoods of $x$. Moreover, $y\mathcal{G}_{n}K_{p}\subseteq
y\mathcal{G}_{n-1}K_{p}$ is a rational subset because
$yk_{i}\mathcal{G}_{n}\subseteq yk_{i}\mathcal{G}_{n-1}$ is a rational subset.
∎

The main result of this section is the following:

######  Theorem 4.5.2.

There is a continuous map:

$\pi_{HT,K_{p}}:\mathcal{S}^{\star}_{K}\rightarrow\mathcal{FL}_{G,\mu}/K_{p}$

which is equivariant for the action of the Hecke algebra
$\mathcal{C}^{\infty}_{c}(K\backslash G(\mathbb{A}_{f})/K,\mathbb{Z})$ by
correspondences.

Moreover, any point $x\in\mathcal{FL}_{G,\mu}/K_{p}$ with finite residue field
over $\mathbb{Q}_{p}$ (i.e. a classical point in the sense of rigid analytic
geometry) has an affinoid neighborhood $U$ such that for any rational subset
$V\subseteq U$, $(\pi_{HT,K_{p}})^{-1}(V)$ is affinoid.

By proposition 4.4.53 there is a map
$\pi_{HT}:\mathcal{S}^{\star}_{\overline{K^{p}}}\rightarrow\mathcal{FL}_{G,\mu}$
which is equivariant for the action of the $G(\mathbb{Q}_{p})$. For any point
$x\in\mathcal{FL}_{G,\mu}$, there is an affinoid neighborhood $U$ of $x$ such
that $\pi_{HT}^{-1}(U)$ is affinoid, and moreover,
$\pi_{HT}^{-1}(U)=\lim_{K_{p}}\pi_{HT}^{-1}(U)_{K_{p}}$ where for $K_{p}$
small enough,
$\pi_{HT}^{-1}(U)_{K_{p}}\hookrightarrow\mathcal{S}^{\star}_{\overline{K^{p}K_{p}}}$
is affinoid. We call an open affinoid in $\mathcal{FL}_{G,\mu}$ with these
properties a _very good affinoid_. Clearly, a rational subset of a very good
affinoid is a very good affinoid.

We can define truncated Hodge-Tate period maps:

$\pi_{HT,K_{p}}:\mathcal{S}^{\star}_{K^{p}K_{p}}\rightarrow\mathcal{S}^{\star}_{\overline{K^{p}K_{p}}}\rightarrow\mathcal{FL}_{G,\mu}/K_{p}.$

######  Lemma 4.5.3.

The map $\pi_{HT,K_{p}}$ is continuous.

###### Proof.

We have a continuous map (of topological spaces)
$\lim_{K^{\prime}_{p}}\mathcal{S}^{\star}_{K^{p}K_{p}^{\prime}}\rightarrow\mathcal{S}^{\star}_{\overline{K^{p}}}\rightarrow\mathcal{FL}_{G,\mu}$.
For all normal subgroup $K^{\prime}_{p}\subseteq K_{p}$, the map
$\mathcal{S}^{\star}_{K^{p}K_{p}^{\prime}}\rightarrow\mathcal{S}^{\star}_{K^{p}K_{p}}$
is surjective and the target carries the quotient topology ([Han19], theorem
1.1). The continuity of the map of the lemma follows. ∎

######  Lemma 4.5.4.

Let $V\subseteq\mathcal{FL}_{G,\mu}$ be a very good affinoid invariant under a
compact open subgroup $K_{p}$. Let $\overline{V}$ be its image in
$\mathcal{FL}_{G,\mu}/K_{p}$. Then
$\pi_{HT,K_{p}}^{-1}(\overline{V})\subseteq\mathcal{S}^{\star}_{{K^{p}K_{p}}}$
is affinoid.

###### Proof.

It is part of the definition that $\pi_{HT}^{-1}(V)$ is the pullback of an
affinoid
$\pi_{HT}^{-1}(V)_{K^{\prime}_{p}}\subset\mathcal{S}^{\star}_{\overline{K^{p}K_{p}^{\prime}}}$
for some $K^{\prime}_{p}\subseteq K_{p}$, a normal compact open subgroup. We
can further pull back $\pi_{HT}^{-1}(V)_{K^{\prime}_{p}}$ to an affinoid
$\tilde{U}\subseteq\mathcal{S}^{\star}_{{K^{p}K_{p}^{\prime}}}$. The space
$\mathcal{S}^{\star}_{K^{p}K_{p}}$ is the categorical quotient of
$\mathcal{S}^{\star}_{K^{p}K_{p}^{\prime}}$ by $K_{p}/K^{\prime}_{p}$, and the
image of $\tilde{U}$ in $\mathcal{S}^{\star}_{K^{p}K_{p}}$ is indeed affinoid.
∎

###### Proof of Theorem 4.5.2.

The continuity of the map is lemma 4.5.3. Let $x\in\mathcal{FL}_{G,\mu}/K_{p}$
with finite residue field over $\mathbb{Q}_{p}$. Let $y$ be a lift of $x$ in
$\mathcal{FL}_{G,\mu}$. For $n$ large enough,
$x\mathcal{G}_{n}\hookrightarrow\mathcal{FL}_{G,\mu}$ is a very good affinoid
(where $\mathcal{G}_{n}$ is the subgroup of elements which reduce to $1$
modulo $p^{n}$ as in lemma 4.5.1). We form the group $\mathcal{G}_{n}K_{p}$.
Then $y\mathcal{G}_{n}K_{p}=\coprod_{i\in I}xk_{i}\mathcal{G}_{n}$ for a
finite set $I$ and elements $k_{i}\in K_{p}$. Moreover,
$yk_{i}\mathcal{G}_{n}=y\mathcal{G}_{n}k_{i}$ is a very good affinoid (because
the property of being a very good affinoid is preserved under
$G(\mathbb{Q}_{p})$-action). A finite disjoint union of very good affinoids is
a very good affinoid. We can apply lemma 4.5.4, to $y\mathcal{G}_{n}K_{p}$.
The image of $y\mathcal{G}_{n}K_{p}$ in $\mathcal{FL}_{G,\mu}/K_{p}$ provides
the open neighborhood $U$ of $x$ as in the theorem. Finally, since any
rational subset of a very good affinoid is again a very good affinoid, a
second application of lemma 4.5.4 proves the last point. ∎

We also adopt the notation
$\pi_{HT,K_{p}}^{tor}:\mathcal{S}^{tor}_{K^{p}K_{p},\Sigma}\rightarrow\mathcal{S}^{\star}_{K^{p}K_{p}}\stackrel{{\scriptstyle\pi_{HT,K_{p}}}}{{\rightarrow}}\mathcal{FL}_{G,\mu}/K_{p}$.

###  4.6. Reductions of the torsor $\mathcal{M}^{an}_{dR}$

The group $G$ of the Shimura datum is defined over $\mathbb{Q}$. We recall
that we have fixed a finite extension $F$ of $\mathbb{Q}_{p}$ which splits
$G$. We have also fixed a representative of the cocharacter $\mu$ over $F$ and
let $P_{\mu}$ and $M_{\mu}$ be the corresponding parabolic and Levi subgroups
of $G$. We will soon assume that $G$ is quasi-split over $\mathbb{Q}_{p}$.
When this is the case we assume that $P_{\mu}$ contains a Borel $B$ defined
over $\mathbb{Q}_{p}$. We take a reductive model for $G_{F}$ defined over
$\mathrm{Spec}~{}\mathcal{O}_{F}$. By abuse of notation, we also denote this
model by $G$. We also then have models for $P_{\mu}$ and $M_{\mu}$ over
$\mathrm{Spec}~{}\mathcal{O}_{F}$. On the analytic side, we have the (non-
quasi-compact) groups $\mathcal{G}^{an}$, $\mathcal{P}_{\mu}^{an}$, and
$\mathcal{M}_{\mu}^{an}$, all considered over
$\mathrm{Spa}(F,\mathcal{O}_{F})$, and there is an embedding
$G(\mathbb{Q}_{p})\hookrightarrow\mathcal{G}^{an}$. Because we have fixed an
integral model for $G$, we also have the quasi-compact groups
$\mathcal{G}\hookrightarrow\mathcal{G}^{an}$,
$\mathcal{P}_{\mu}\hookrightarrow\mathcal{P}_{\mu}^{an}$, and
$\mathcal{M}_{\mu}\hookrightarrow\mathcal{M}_{\mu}^{an}$.

The goal of this section is to use the Hodge-Tate period morphism to produce
some finer structure on the torsor $\mathcal{M}^{an}_{dR}$. These result
generalize those obtained in [AIP15], prop. 4.3.1 for example. These refined
structure will allow us to $p$-adically interpolate automorphic vector
bundles.

####  4.6.1. Preparations

We start by giving a detailed description of the torsor pulled back from the
map
$\pi_{HT}^{tor}:\mathcal{S}^{tor}_{K^{p},\Sigma}\rightarrow\mathcal{FL}_{G,\mu}$.

If $(G,X)$ is Hodge type Shimura datum, and $\Sigma$ is a perfect cone
decomposition, then $\mathcal{S}^{tor}_{K^{p},\Sigma}$ is a perfectoid space.
In the general abelian case, it is only known to be a diamond.

Let $S\rightarrow\mathcal{S}^{tor}_{K^{p},\Sigma}$ be a map from a perfectoid
space $S$. By composing with $\pi_{HT}^{tor}$, we get a map
$S\rightarrow\mathcal{FL}_{G,\mu}$ which can be described as follows. Over
$S$, we have a map
$\mathcal{P}^{an}_{HT}\hookrightarrow\mathcal{G}^{c,an}\times S$, from the
$\mathcal{P}^{c,an}_{\mu}$-torsor $\mathcal{P}^{an}_{HT}$ to the trivial
$\mathcal{G}^{c,an}$-torsor, which is equivariant for the natural morphism of
groups $\mathcal{P}^{c,an}_{\mu}\rightarrow\mathcal{G}^{c,an}$.

There is an étale cover $\tilde{S}\rightarrow S$ such that the torsor
$\mathcal{P}^{an}_{HT}\times_{S}\tilde{S}$ becomes trivial and aquires a
section $g_{\tilde{S}}\in\mathcal{G}^{c,an}(\tilde{S})$, unique up to left
multiplication by elements of $\mathcal{P}^{c,an}_{\mu}(\tilde{S})$, so that
we get a commutative diagram of torsors:

Over $\tilde{S}\times_{S}\tilde{S}$, there is a section
$h_{\tilde{S}\times_{S}\tilde{S}}\in\mathcal{P}^{c,an}_{\mu}(\tilde{S}\times_{S}\tilde{S})$
such that $h_{\tilde{S}\times_{S}\tilde{S}}\cdot
p_{1}^{\star}g_{\tilde{S}}=p_{2}^{\star}g_{\tilde{S}}$. The section
$h_{\tilde{S}\times_{S}\tilde{S}}$ is a $1$-cocycle which describes the
original torsor $\mathcal{P}^{an}_{HT}$ . Changing $g_{\tilde{S}}$ by left
multiplication by an element of $\mathcal{P}^{c,an}_{\mu}(\tilde{S})$ will
change $h_{\tilde{S}\times_{S}\tilde{S}}$ by a coboundary.

The image of $g_{\tilde{S}}\in\mathcal{FL}_{G,\mu}(\tilde{S})$ descends to
give a point in $\mathcal{FL}_{G,\mu}({S})$: the morphism
$S\rightarrow\mathcal{FL}_{G,\mu}$ we started with.

The group $\mathcal{G}^{an}$ acts on the right on $\mathcal{FL}_{G,\mu}$.
Concretely this action sends $g_{\tilde{S}}$ to $g_{\tilde{S}}.g$. This action
does not affect the construction of the torsor $\mathcal{P}_{HT}^{an}$ which
is indeed $\mathcal{G}^{an}$-equivariant.

####  4.6.2. Integral structure

Recall that $\mathcal{M}_{\mu}^{c}$ is a (quasi-compact) open subgroup of
$\mathcal{M}_{\mu}^{c,an}$. The following proposition can be interpreted as
the existence of an integral structure on the torsors $\mathcal{M}_{dR}^{an}$
or $\mathcal{M}_{HT}^{an}$.

######  Proposition 4.6.3.

Let $K_{p}\subset G(\mathbb{Q}_{p})\cap G(\mathcal{O}_{F})$. The étale torsor
$\mathcal{M}_{dR}^{an}=\mathcal{M}_{HT}^{an}$ over
$\mathcal{S}_{K^{p}K_{p},\Sigma}^{tor}$ has a reduction of structure group to
an étale $\mathcal{M}^{c}_{\mu}$-torsor $\mathcal{M}_{dR}=\mathcal{M}_{HT}$.

######  Remark 4.6.4.

In the Siegel case, the torsor $\mathcal{M}_{dR}^{an}$ is (ignoring the
center) the torsor of trivializations of the vector bundle $\omega_{A}$, the
conormal sheaf of the universal semi-abelian scheme over
$\mathcal{S}^{tor}_{K^{p}K_{p},\Sigma}$ (well defined up to prime to
$p$-isogeny by our choice of level structure). A possible integral structure
is obtained by declaring that an invariant differential form is integral if it
extends to an invariant differential form on an integral model of the
universal semi-abelian scheme. However, the integral structure we consider
here is different. Namely, we declare that a differential form is integral if
it is in the span of the image of the integral Tate module for the Hodge-Tate
period map. By [Far11], Theorem 2, section 5.3.2. we can explicitly bound the
difference between both integral structures.

###### Proof.

We work over $\mathcal{S}_{K^{p},\Sigma}^{tor}$. Let
$S\rightarrow\mathcal{S}_{K^{p},\Sigma}^{tor}$ be a map from a perfectoid
space. We first explain how to define the torsor $\mathcal{M}_{HT}$ over $S$
(in a functorial way).

The map $S\rightarrow\mathcal{FL}_{G,\mu}$ is described by an element
$g_{\tilde{S}}\in\mathcal{G}^{c,an}(\tilde{S})$ for some étale cover
$\tilde{S}\rightarrow S$. We are free to change $g_{\tilde{S}}$ by left
multiplication by an element of $\mathcal{P}^{c,an}_{\mu}(\tilde{S})$. Thus,
up to passing to some further cover of $\tilde{S}$, we may actually assume
that $g_{\tilde{S}}\in\mathcal{G}^{c}(\tilde{S})$ (because
$\mathcal{FL}_{G,\mu}=\mathcal{P}^{c}_{\mu}\backslash\mathcal{G}^{c}=\mathcal{P}^{c,an}_{\mu}\backslash\mathcal{G}^{c,an}$),
and this new element is well defined up to multiplication by an element of
$\mathcal{P}^{c}_{\mu}(\tilde{S})$. The torsor $\mathcal{P}^{an}_{HT}$ is
defined by the cocycle
$h_{\tilde{S}\times_{S}\tilde{S}}=p_{2}^{\star}g_{\tilde{S}}\cdot(p_{1}^{\star}g_{\tilde{S}})^{-1}\in\mathcal{P}^{c}_{\mu}(\tilde{S}\times_{S}\tilde{S})\subset\mathcal{P}^{c,an}_{\mu}(\tilde{S}\times_{S}\tilde{S})$.
We therefore have produced a reduction of the torsor $\mathcal{P}^{an}_{HT}$
to a torsor $\mathcal{P}_{HT}$ under the group $\mathcal{P}_{\mu}^{c}$. We can
take the pushout under the map
$\mathcal{P}^{c}_{\mu}\rightarrow\mathcal{M}^{c}_{\mu}$ (which amounts to
projecting $h_{\tilde{S}\times_{S}\tilde{S}}$ in
$\mathcal{M}^{c}_{\mu}(\tilde{S}\times_{S}\tilde{S})$) to get the desired
torsor $\mathcal{M}_{HT}$.

We now proceed to descend from $\mathcal{S}^{tor}_{K^{p},\Sigma}$ to
$\mathcal{S}_{K^{p}K_{p},\Sigma}^{tor}$. We have an étale torsor
$\mathcal{M}_{HT}^{an}\rightarrow\mathcal{S}_{K^{p}K_{p},\Sigma}^{tor}$, and
we have defined an open subset
$\mathcal{M}_{HT}\subset\mathcal{M}_{HT}^{an}\times_{\mathcal{S}_{K^{p}K_{p},\Sigma}^{tor}}\mathcal{S}^{tor}_{K^{p},\Sigma}$.
We claim that this open descends to an open subset of $\mathcal{M}_{HT}^{an}$.
The map
$\mathcal{M}_{HT}^{an}\times_{\mathcal{S}_{K^{p}K_{p},\Sigma}^{tor}}\mathcal{S}_{K^{p},\Sigma}^{tor}\rightarrow\mathcal{M}_{HT}^{an}$
identifies the topological space $|\mathcal{M}^{an}_{HT}|$ with the quotient
of
$|\mathcal{M}_{HT}^{an}\times_{\mathcal{S}_{K^{p}K_{p},\Sigma}^{tor}}\mathcal{S}_{K^{p},\Sigma}^{tor}|$
by the action of $K_{p}$. We have an identification between $K_{p}$-invariant
open subsets of
$\mathcal{M}_{HT}^{an}\times_{\mathcal{S}_{K,\Sigma}^{tor}}\mathcal{S}_{K^{p},\Sigma}^{tor}$
and open subsets of $\mathcal{M}_{HT}^{an}$. The only thing to check is
therefore that $\mathcal{M}_{HT}$ is indeed invariant under the action of
$K_{p}$. We go back to considering a map
$S\rightarrow\mathcal{S}_{K^{p},\Sigma}^{tor}$, described by an element
$g_{\tilde{S}}\in\mathcal{G}^{c}(\tilde{S})$. Under right multiplication by
$k\in K_{p}$, we get a new element
$g_{\tilde{S}}k\in\mathcal{G}^{c}(\tilde{S})$ (it is crucial here that $k\in
K_{p}\subseteq G(\mathcal{O}_{F})$) and the corresponding reduction of the
torsor described by the element
$p_{2}^{\star}g_{\tilde{S}}k.(p_{1}^{\star}g_{\tilde{S}}.k)^{-1}=p_{2}^{\star}g_{\tilde{S}}.(p_{1}^{\star}g_{\tilde{S}})^{-1}$
doesn’t depend on $k$.

We need to prove that the action map
$\mathcal{M}^{an}_{\mu}\times\mathcal{M}^{an}_{HT}\rightarrow\mathcal{M}^{an}_{HT}\times\mathcal{M}^{an}_{HT}$
(which is an isomorphism), induces an isomorphism
$\mathcal{M}_{\mu}\times\mathcal{M}_{HT}\rightarrow\mathcal{M}_{HT}\times\mathcal{M}_{HT}$.
In other words, we need to prove that the two open subsets
$\mathcal{M}_{\mu}\times\mathcal{M}_{HT}$ and
$\mathcal{M}_{HT}\times\mathcal{M}_{HT}$ identify via the action map. This can
be checked after pull back to $S^{tor}_{K^{p},\Sigma}$ and this is true.
Finally, the morphism
$\mathcal{M}_{HT}\rightarrow\mathcal{S}_{K,\Sigma}^{tor}$ is smooth, and
surjective on geometric points. Therefore, there are sections étale-locally
and $\mathcal{M}_{HT}$ is an étale torsor. ∎

######  Remark 4.6.5.

The above argument uses crucially that we know that $\mathcal{M}_{HT}^{an}$
descends from $\mathcal{S}^{tor}_{K^{p},\Sigma}$ to
$\mathcal{S}^{tor}_{K^{p}K_{p},\Sigma}$, because pro-étale descent is not
effective in general.

######  Remark 4.6.6.

We briefly explain how the construction in the abelian case connects with the
construction in the Hodge case, in order to illustrate principle 4.4.44. We
consider a diagram of Shimura datum with $(G,X)$ of abelian type and
$(G_{1},X_{1})$ of Hodge type as in section 4.4.41:

We may assume that the various morphisms between the groups $G_{1},B_{1},B,G$
over $\mathbb{Q}$ extend to morphisms over $\mathcal{O}_{F}$. We assume that
we have a diagram for suitable compacts $K^{\prime},K_{1},K_{2},K_{3}$ with
$K^{\prime}_{p}\subseteq B_{1}(\mathcal{O}_{F})$ and $K_{1,p}\subseteq
G_{1}(\mathcal{O}_{F})$, and cone decomposition for $G_{1}$:

We have the first formula:

$\mathcal{M}_{HT}(B_{1},X_{B_{1}})=\pi_{1}^{\star}\mathcal{M}_{HT}(G_{1},X_{1})\times_{\pi^{\star}\mathcal{M}_{HT}(G^{ab}_{1},X_{G^{ab}_{1}})}\pi_{2}^{\star}\mathcal{M}_{HT}(T,X_{T})$

Secondly, we have a finite map:
${S}^{tor,0}(B_{1},X_{B_{1}})_{{K^{\prime}},\Sigma}\rightarrow{S}^{tor,0}(G,X)_{{K},\Sigma}$,
generically finite étale with group $\Delta(K,K^{\prime})$ (possibly after
refining $\Sigma$). and the second formula

$(\mathcal{M}_{HT}(B_{1},X_{B_{1}})\times^{\mathcal{M}^{c}_{\mu_{B_{1}}}}\mathcal{M}_{\mu_{G}}^{c})/\Delta(K,K^{\prime})=\mathcal{M}_{HT}(G,X_{G})$

(valid for the restrictions of the torsor to the connected component).

For $\kappa\in X^{\star}(T^{c})^{M_{\mu},+}$, the sheaf $\mathcal{V}_{\kappa}$
is modeled on the representation $V_{\kappa}$ of $\mathcal{M}_{\mu}^{an}$
defined over $F$. Recall that $V_{\kappa}$ is defined as the module of
sections
$f(m)\in\mathrm{H}^{0}(\mathcal{M}^{an}_{\mu},\mathscr{O}_{\mathcal{M}^{an}_{\mu}})$
such that $f(mb)=-w_{0,M}\kappa(b)f(m)$ for
$b\in\mathcal{B}^{an}\cap\mathcal{M}^{an}_{\mu}$. Using that

$\mathcal{M}^{an}_{\mu}/(\mathcal{B}^{an}\cap\mathcal{M}^{an}_{\mu})=\mathcal{M}_{\mu}/(\mathcal{B}\cap\mathcal{M}_{\mu})$

we find that this is also the module of sections
$f(m)\in\mathrm{H}^{0}(\mathcal{M}_{\mu},\mathscr{O}_{\mathcal{M}_{\mu}})$
such that $f(mb)=-w_{0,M}\kappa(b)f(m)$ for
$b\in\mathcal{B}\cap\mathcal{M}_{\mu}$.

We can define an $\mathcal{O}_{F}$-submodule $V_{\kappa}^{+}\subset
V_{\kappa}$ by considering sections
$f(m)\in\mathrm{H}^{0}(\mathcal{M}_{\mu},\mathscr{O}_{\mathcal{M}_{\mu}}^{+})$
such that $f(mb)=-w_{0,M_{\mu}}\kappa(b)f(m)$. Since
$\mathrm{H}^{0}(\mathcal{M}_{\mu},\mathscr{O}_{\mathcal{M}_{\mu}}^{+})$ is
open and bounded in
$\mathrm{H}^{0}(\mathcal{M}_{\mu},\mathscr{O}_{\mathcal{M}_{\mu}})$, we deduce
that $V_{\kappa}^{+}$ is a lattice in $V_{\kappa}$, stable under the action of
$\mathcal{M}_{\mu}$.

######  Corollary 4.6.7.

Assume that $K_{p}\subseteq G(\mathcal{O}_{F})$. For all $\kappa\in
X^{\star}(T^{c})^{M_{\mu},+}$, the locally free sheaf $\mathcal{V}_{\kappa}$
over $\mathcal{S}^{tor}_{K,\Sigma}$ has an integral structure
$\mathcal{V}_{\kappa}^{+}$ in the sense of definition 2.6.1.

###### Proof.

We consider the map
$g:\mathcal{M}_{dR}\rightarrow\mathcal{S}^{tor}_{K,\Sigma}$. We let
$\mathcal{V}_{\kappa}^{+}$ be the subsheaf of
$g_{\star}\mathscr{O}^{+}_{\mathcal{M}_{dR}}$ of sections $f(m)$ which satisfy
$f(mb)=-w_{0,M}\kappa(b)f(m)$. ∎

####  4.6.8. Further reductions of the group structure

We assume that $K_{p}=K_{p,m^{\prime},b^{\prime}}$ for $m^{\prime}\geq
b^{\prime}\in\mathbb{Z}_{\geq 0}$ and $m^{\prime}>0$ (see section 3.5.1). This
is a compact open subgroup of $G(\mathbb{Q}_{p})$ with an Iwahori
decomposition. For any $w\in{}^{M}W$, we let $K_{p,w,M_{\mu}}$ be the
projection of $wK_{p}w^{-1}\cap\mathcal{P}_{\mu}$ to $\mathcal{M}_{\mu}$.

We will describe this group. We can define
$N_{p,w,M_{\mu}}=K_{p,w,M_{\mu}}\cap U_{M_{\mu}}$ and
$\overline{N}_{p,w,M_{\mu}}=K_{p,w,M_{\mu}}\cap\overline{U}_{M_{\mu}}$, where
we let $B_{M_{\mu}}$ be the Borel subgroup of $M_{\mu}$ which is the image of
$B$ in $M_{\mu}$, $\overline{B}_{M_{\mu}}$ the opposite Borel, and
$U_{M_{\mu}}$ (resp. $\overline{U_{M_{\mu}}}$) be the unipotent radical of
$B_{M_{\mu}}$ (resp. $\overline{B}_{M_{\mu}}$). We also recall that
$T_{b^{\prime}}=\mathrm{Ker}(T(\mathcal{O}_{F})\rightarrow
T(\mathcal{O}_{F}/p^{b^{\prime}}))\cap T(\mathbb{Q}_{p})=T\cap K_{p}$.

######  Proposition 4.6.9.

For any $w\in{}^{M}W$, the group $K_{p,w,M_{\mu}}$ is a subgroup of the
Iwahori subgroup of $M_{\mu}(\mathcal{O}_{F})$. Moreover, it admits an Iwahori
decomposition. Namely, the product map:

$N_{p,w,M_{\mu}}\times
wT_{b^{\prime}}w^{-1}\times\overline{N}_{p,w,M_{\mu}}\rightarrow
K_{p,w,M_{\mu}}$

is an isomorphism.

When $G$ is unramified, $M_{\mu}$ is defined over $\mathbb{Q}_{p}$, $w$ is
$\mathrm{Gal}(F/\mathbb{Q}_{p})$-invariant, and $K_{p}=K_{p,1,0}$ is the
Iwahori subgroup of $G(\mathbb{Z}_{p})$, then $K_{p,w,M_{\mu}}$ is the Iwahori
subgroup of $M_{\mu}(\mathbb{Z}_{p})$.

###### Proof.

Let $U$ and $\overline{U}$ be respectively the unipotent radicals of $B$ and
$\overline{B}$. We have the Iwahori decomposition
$K_{p}=\overline{N}_{p}\times T_{b^{\prime}}(\mathbb{Z}_{p})\times N_{p}$
where $N_{p}=K_{p}\cap U$ is $U(\mathbb{Q}_{p})\cap U(\mathcal{O}_{F})$ and
$\overline{N}_{p}=K_{p}\cap\overline{U}\subseteq\overline{U}(\mathbb{Q}_{p})$
is the subgroup of elements reducing to $1$ modulo $p^{m^{\prime}}$ It is
useful to give a more precise version of this decomposition. We let $\Phi$ be
the set of roots (defined over $F$). We have
$\Phi=\Phi^{+}_{M}\coprod\Phi^{-}_{M}\coprod\Phi^{+,M}\coprod\Phi^{-,M}$,
where $\Phi_{M}=\Phi^{+}_{M}\coprod\Phi^{-}_{M}$ is the set of roots in $M$.
We also let $\Phi_{0}=\Phi/\mathrm{Gal}(F/\mathbb{Q}_{p})$ and have
$\Phi_{0}=\Phi_{0}^{+}\coprod\Phi_{0}^{-}$ (because $G$ is quasi-split).

For all $\alpha_{0}\in\Phi_{0}$, we let $U_{\alpha_{0}}\hookrightarrow G$ be
the corresponding unipotent group. For all $\alpha\in\Phi$ we also denote by
$U_{\alpha}\hookrightarrow G_{\mathcal{O}_{F}}$ the one parameter subgroup. We
have
$U_{\alpha_{0}}\times_{\mathrm{Spec}~{}\mathbb{Q}_{p}}\mathrm{Spec}~{}F=\prod_{\alpha\in\Phi,\alpha\mapsto\alpha_{0}}U_{\alpha}\times_{\mathrm{Spec}~{}\mathcal{O}_{F}}\mathrm{Spec}~{}F$.
We consider the product map (in any order of the factors):

$\prod_{\alpha_{0}}U_{\alpha_{0}}(\mathbb{Q}_{p})\times
T(\mathbb{Q}_{p})\rightarrow G(\mathbb{Q}_{p}).$

This maps induces a bijection between $K_{p}$ and the set of elements
$((n_{\alpha_{0}})_{\alpha_{0}\in\Phi_{0}},t)$ which satisfy:
$n_{\alpha_{0}}$, $t\in G(\mathcal{O}_{F})$, $n_{\alpha_{0}}=1~{}\mod
p^{m^{\prime}}$ if $\alpha_{0}\in\Phi_{0}^{-}$, and $t=1~{}\mod
p^{b^{\prime}}$. We get that $wK_{p}w^{-1}\cap P_{\mu}$ identifies (via the
product map) with the set of elements
$((wn_{\alpha_{0}}w^{-1})_{\alpha_{0}\in\Phi_{0}}\in
wU_{\alpha_{0}}(\mathbb{Q}_{p})w^{-1},wtw^{-1}\in wT(\mathbb{Q}_{p})w^{-1})$
such that:

  * •

$n_{\alpha_{0}},~{}t\in G(\mathcal{O}_{F})$,

  * •

$n_{\alpha_{0}}=1~{}\mod p^{m^{\prime}}$ if $\alpha_{0}\in\Phi_{0}^{-}$,

  * •

$t=1~{}\mod p^{b^{\prime}}$,

  * •

$n_{\alpha_{0}}\in
U^{\prime}_{\alpha_{0}}=\mathrm{Ker}\big{(}U_{\alpha_{0}}(\mathbb{Q}_{p})\cap
G(\mathcal{O}_{F})\rightarrow\prod_{\alpha\in
w^{-1}\Phi^{-,M},\alpha\mapsto\alpha_{0}}U_{\alpha}(\mathcal{O}_{F})\big{)}$.

We let
$U^{\prime\prime}_{\alpha_{0}}=\mathrm{Im}\big{(}U^{\prime}_{\alpha_{0}}\rightarrow\prod_{\alpha\in
w^{-1}\Phi_{M},\alpha\mapsto\alpha_{0}}U_{\alpha}(\mathcal{O}_{F})\big{)}$. We
deduce that $K_{p,w,M_{\mu}}$ is in bijection (via the product map) with the
elements $((wn_{\alpha_{0}}w^{-1})_{\alpha_{0}\in\Phi_{0}}\in
wU^{\prime\prime}_{\alpha_{0}}w^{-1},wtw^{-1}\in wT_{b^{\prime}}w^{-1})$ such
that $n_{\alpha_{0}}=1~{}\mod p^{m^{\prime}}$ if $\alpha_{0}\in\Phi_{0}^{-}$.
The fact that any element of $K_{p,w,M_{\mu}}$ can be written in this way
follows from the previous discussion. The injectivity of the product map is a
general fact.

We now observe that $w\in{}^{M}W$ and therefore $\Phi_{M}^{+}\subset
w(\Phi^{+})$ and $\Phi_{M}^{-}\subset w(\Phi^{-})$. We deduce that
$w^{-1}\Phi_{M}\cap\Phi^{-}=w^{-1}\Phi_{M}^{-}$ and
$w^{-1}\Phi_{M}\cap\Phi^{+}=w^{-1}\Phi_{M}^{+}$.

It follows that if $\alpha_{0}\in\Phi_{0}^{+}$,
$wU^{\prime\prime}_{\alpha_{0}}w^{-1}\subset U_{M_{\mu}}(\mathcal{O}_{F})$ and
if $\alpha_{0}\in\Phi_{0}^{-}$,
$wU^{\prime\prime}_{\alpha_{0}}w^{-1}\subset\overline{U}_{M_{\mu}}(\mathcal{O}_{F})$.
Therefore, we deduce that $K_{p,w,M_{\mu}}=N_{p,w,M_{\mu}}\times
wT_{b^{\prime}}w^{-1}\times\overline{N}_{p,w,M_{\mu}}$. By the condition that
$n_{\alpha_{0}}=1~{}\mod p^{m^{\prime}}$ if $\alpha_{0}\in\Phi_{0}^{-}$, we
deduce that $K_{p,w,M_{\mu}}$ is a subgroup of the Iwahori of
$M_{\mu}(\mathcal{O}_{F})$.

Finally, if $G$ is unramified, and $M_{\mu}$ is defined over $\mathbb{Q}_{p}$,
the partition
$\Phi=\Phi^{+}_{M}\coprod\Phi^{-}_{M}\coprod\Phi^{+,M}\coprod\Phi^{-,M}$
descends to a partition
$\Phi_{0}=\Phi^{+}_{0,M}\coprod\Phi^{-}_{0,M}\coprod\Phi_{0}^{+,M}\coprod\Phi_{0}^{-,M}$.
If $w$ is rational, then it acts on $\Phi_{0}$. Assume that $K_{p}=K_{p,1,0}$.
The description of $K_{p,w,M_{\mu}}$ simplifies and we find that
$U^{\prime\prime}_{\alpha_{0}}=\\{1\\}$ if $\alpha_{0}\notin
w^{-1}\Phi_{0,M}$,
$U^{\prime\prime}_{\alpha_{0}}=U_{\alpha_{0}}(\mathbb{Z}_{p})$ if
$\alpha_{0}\in w^{-1}\Phi_{0,M}$. It follows that $K_{p,w,M_{\mu}}$ is the
Iwahori of $M_{\mu}(\mathbb{Z}_{p})$.

∎

######  Example 4.6.10.

The group $K_{p,w,M_{\mu}}$ may be a little strange. Let us consider the
following example. We assume that $G_{\mathbb{Q}_{p}}$ is
$\mathrm{Res}_{\mathbb{Q}_{p^{2}}/\mathbb{Q}_{p}}\mathrm{GL}_{2}$, with
standard diagonal torus $T_{\mathbb{Q}_{p}}$ and upper triangular borel
$B_{\mathbb{Q}_{p}}$. We let $K_{p}=K_{p,1,0}$. We identify
$G_{\mathbb{Q}_{p^{2}}}=\mathrm{GL}_{2}\times\mathrm{GL}_{2}$. We assume that
$\mu$ is defined over $\mathbb{Q}_{p^{2}}$ and is given by the cocharacter
$t\mapsto\mathrm{diag}(t,1)\times\mathrm{diag}(1,1)$ of
$T_{\mathbb{Q}_{p^{2}}}$. We deduce that $P_{\mu}$ is
$B_{\mathbb{Q}_{p^{2}}}\times\mathrm{GL}_{2}$, and that $M_{\mu}$ is
$T_{\mathbb{Q}_{p^{2}}}\times GL_{2}$. We finally observe that $K_{p}\cap
P_{\mu}$ is $B(\mathbb{Z}_{p})$ and therefore $K_{p,1,\mu}$ is the image of
$B(\mathbb{Z}_{p})$ in $T(\mathbb{Q}_{p^{2}})\times
GL_{2}(\mathbb{Q}_{p^{2}})$.

For all $m,n\in\mathbb{Q}_{\geq 0}$, we let $\mathcal{G}^{1}_{m,n}$ be the
subgroup of $\mathcal{G}$ of elements which reduce to $\mathcal{U}$ modulo
$p^{m+\epsilon}$ for all $\epsilon>0$ and to $\overline{\mathcal{U}}$ modulo
$p^{n}$ (see section 3.3.10). We let
$\mathcal{M}^{1}_{\mu,m,n}\subseteq\mathcal{M}_{\mu}$ be the group of elements
which reduce to $\mathcal{U}_{\mathcal{M}_{\mu}}$ modulo $p^{m+\epsilon}$ for
all $\epsilon>0$ and to $\overline{\mathcal{U}}_{\mathcal{M}_{\mu}}$ modulo
$p^{n}$.

######  Lemma 4.6.11.

Let $w\in{}^{M}W$ and let $K_{p}=K_{p,m^{\prime},b^{\prime}}$ with
$m^{\prime}\in\mathbb{Z}_{>0}$ and $b^{\prime}\in\mathbb{Z}_{\geq 0}$. Let
$m,n\geq 0$ and assume that $0\leq m-n\leq m^{\prime}-1$. Then
$K_{p,w,M_{\mu}}$ normalizes $\mathcal{M}^{1}_{\mu,m,n}$.

###### Proof.

Observe that $K_{p}\subseteq\mathcal{G}_{m-n,0}$. By lemma 3.3.15, $K_{p}$
normalizes $\mathcal{G}^{1}_{m,n}$. Moreover,
$\mathrm{Im}(w\mathcal{G}^{1}_{m,n}w^{-1}\cap\mathcal{P}_{\mu}\rightarrow\mathcal{M}_{\mu})=\mathcal{M}^{1}_{\mu,m,n}$.

∎

It follows from this lemma that $K_{p,w,M_{\mu}}\mathcal{M}^{1}_{\mu,m,n}$ is
a subgroup of $\mathcal{M}_{\mu}$.

######  Proposition 4.6.12.

Let $w\in{}^{M}W$ and let $K_{p}=K_{p,m^{\prime},b^{\prime}}$ with
$m^{\prime}\in\mathbb{Z}_{>0}$ and $m^{\prime}\geq b^{\prime}$. Let $m,n\geq
0$ and assume that $0\leq m-n\leq m^{\prime}-1$. Over
$(\pi^{tor}_{HT,K_{p}})^{-1}(]C_{w,k}[_{m,n}K_{p})\subseteq\mathcal{S}_{K^{p}K_{p},\Sigma}^{tor}$,
the torsor $\mathcal{M}_{HT}$ has a reduction to an étale torsor
$\mathcal{M}_{HT,m,n,K_{p}}$ under the group
$K^{c}_{p,w,M_{\mu}}\mathcal{M}^{1,c}_{\mu,m,n}$.

###### Proof.

The proof is very similar to the proof of proposition 4.6.3. We observe that
$]C_{w,k}[_{m,n}K_{p}=\mathcal{P}_{\mu}^{an}\backslash\mathcal{P}_{\mu}^{an}w\mathcal{G}^{1}_{m,n}K_{p}$.
Since $K_{p}$ normalizes $\mathcal{G}^{1}_{m,n}$,
$\mathcal{G}^{1}_{m,n}K_{p}=K_{p}\mathcal{G}^{1}_{m,n}$ is a group. It follows
that

$]C_{w,k}[_{m,n}K_{p}=(\mathcal{P}_{\mu}^{an}\cap
wK_{p}\mathcal{G}^{1}_{m,n}w^{-1})\backslash
wK_{p}\mathcal{G}^{1}_{m,n}w^{-1}.w$

Observe that we have the following $K_{p}$-equivariant
$K^{c}_{p,w,M_{\mu}}\mathcal{M}^{1,c}_{\mu,m,n}$-torsor:

$(\mathcal{U}_{\mu}^{an,c}\cap
wK^{c}_{p}\mathcal{G}^{1,c}_{m,n}w^{-1})\backslash
wK^{c}_{p}\mathcal{G}^{1,c}_{m,n}w^{-1}.w\rightarrow(\mathcal{P}_{\mu}^{an}\cap
wK_{p}\mathcal{G}^{1}_{m,n}w^{-1})\backslash
wK_{p}\mathcal{G}^{1}_{m,n}w^{-1}.w$

and we proceed to pull back and descend this torsor.

We first construct the torsor $\mathcal{M}_{HT,m,n,K_{p}}$ as an open subset
of

$\mathcal{M}_{HT}\times_{S^{tor}_{K,\Sigma}}(\pi^{tor}_{HT})^{-1}(]C_{w,k}[_{m,n}K_{p}),$

and prove it is $K_{p}$-invariant to descend it to an open subset of

$\mathcal{M}_{HT}\times_{S^{tor}_{K,\Sigma}}(\pi^{tor}_{HT,K_{p}})^{-1}(]C_{w,k}[_{m,n}K_{p}).$

Let $S\rightarrow(\pi^{tor}_{HT})^{-1}(]C_{w,k}[_{m,n}K_{p})$ be a map from a
perfectoid space $S$. The torsor $\mathcal{P}_{HT}$ is described as follows:
there is a cover $\tilde{S}\rightarrow S$ and an element
$g_{\tilde{S}}\in\mathcal{G}(\tilde{S})$ such that
$h_{\tilde{S}\times_{S}\tilde{S}}=p_{2}^{\star}g_{\tilde{S}}.(p_{1}^{\star}g_{\tilde{S}})^{-1}\in\mathcal{P}_{\mu}(\tilde{S}\times_{S}\tilde{S})$
is a $1$-cocycle describing the torsor. By assumption, we may assume
$g_{\tilde{S}}\in w\mathcal{G}^{1}_{m,n}K_{p}$, so that
$h_{\tilde{S}\times_{S}\tilde{S}}\in\mathcal{P}_{\mu}(\tilde{S}\times_{S}\tilde{S})\cap
w\mathcal{G}^{1}_{m,n}K_{p}w^{-1}$ and its image in $\mathcal{M}_{\mu}$
describes a reduction $\mathcal{M}_{HT,m,n,K_{p}}$ of
$\mathcal{M}_{HT}\times_{S^{tor}_{K,\Sigma}}(\pi^{tor}_{HT})^{-1}(]C_{w,k}[_{m,n}K_{p})$
to a $K_{p,w,M_{\mu}}\mathcal{M}^{1}_{\mu,m,n}$-torsor. One checks easily that
this torsor is $K_{p}$-invariant and therefore descends to a torsor over
$(\pi^{tor}_{HT,K_{p}})^{-1}(]C_{w,k}[_{m,n}K_{p})$. ∎

######  Remark 4.6.13.

The connection between the abelian case and Hodge case can be described
similarly as in remark 4.6.6. We may use section 3.5.5 to make sure that the
subset $]C_{w,k}[_{m,n}K_{p}$ of the flag variety doesn’t change when we
consider the different groups $G_{1},B_{1},G$.

######  Proposition 4.6.14.

Let $w\in{}^{M}W$ and let $K_{p}=K_{p,m^{\prime},b^{\prime}}$ with
$m^{\prime}\in\mathbb{Z}_{>0}$ and $m^{\prime}\geq b^{\prime}$. Let
$K_{p}^{\prime}=K_{p,m^{\prime\prime},b^{\prime\prime}}$ with
$m^{\prime\prime}\geq m^{\prime}$ and $m^{\prime\prime}\geq
b^{\prime\prime}\geq b^{\prime}$. Let $m,n\geq 0$ and assume that $0\leq
m-n\leq m^{\prime}-1$. Let $r,s\geq 0$ with $r\geq m$, $s\geq m$ and $0\leq
r-s\leq m^{\prime\prime}-1$. There is a commutative diagram:

The top horizontal map is equivariant for the map
$K^{\prime}_{p,w,M_{\mu}}\mathcal{M}^{1}_{\mu,r,s}\rightarrow
K_{p,w,M_{\mu}}\mathcal{M}^{1}_{\mu,m,n}$.

###### Proof.

This follows from the construction of the torsors. ∎

We have a map $\mathcal{M}^{1}_{\mu,n,n}\rightarrow\mathcal{M}_{\mu,n}$ where
$\mathcal{M}_{\mu,n}$ is the sub-group of $\mathcal{M}_{\mu}$ of elements
reducing to $1$ modulo $p^{n}$. Let $K_{p}=K_{p,m^{\prime},b^{\prime}}$ with
$m^{\prime}>0$, $m^{\prime}\geq b^{\prime}$ and let $m,n\geq 0$ be such that
$0\leq m-n\leq m^{\prime}-1$. Over
$(\pi^{tor}_{HT,K_{p}})^{-1}(]C_{w,k}[_{n,n}K_{p})$, we define the pushout:

$\mathcal{M}_{HT,n,K_{p}}=\mathcal{M}_{HT,n,n,K_{p}}\times^{K^{c}_{p,w,M_{\mu}}\mathcal{M}^{1,c}_{\mu,n,n}}K^{c}_{p,w,M_{\mu}}\mathcal{M}^{c}_{\mu,n}.$

It is sometimes more convenient to work with this torsor because the group
$\mathcal{M}^{c}_{\mu,n}$ is affinoid.

######  Proposition 4.6.15.

Assume that $(G,X)$ is a Hodge-type Shimura datum and $\Sigma$ a perfect cone
decomposition. Let $K_{p}=K_{p,m^{\prime},b}$. For any affinoid open
$\mathrm{Spa}(R,R^{+})\rightarrow(\pi^{tor}_{HT,K_{p}})^{-1}(]C_{w,k}[_{n,n}K_{p})$
which we assume to be pregood (see definition 4.4.3), there exists
$K^{\prime}_{p}\subseteq K_{p}$ such that over
$\mathrm{Spa}(R,R^{+})\times_{\mathcal{S}_{K^{p}K_{p},\Sigma}^{tor}}\mathcal{S}_{K^{p}K_{p}^{\prime},\Sigma}^{tor}$
the torsor $\mathcal{M}_{HT,n,K_{p}}$ is trivial.

######  Remark 4.6.16.