root systems as per §5.1.2, we can pick a basis
$\\{e_{1},e_{2},...,e_{\lceil\frac{n_{a}-1}{2}\rceil}\\}$ of
$X^{*}(T_{\operatorname{SO}(M_{a,0})})$ and
$e_{\lceil\frac{n_{a}+1}{2}\rceil}\in X^{*}(\operatorname{im}\nu)$, such that
$\Phi_{\operatorname{SO}(M_{a})}$ and $\Phi_{\operatorname{SO}(M_{a})}^{+}$
are given by (5.1.1) (with $n$ replaced by $n_{a}$). Similarly, there is a
choice of linearly independent elements
$\\{e_{1}^{\prime},e^{\prime}_{2},...,e^{\prime}_{\frac{n_{b}}{2}}\\}$ of
$X^{*}(T_{\operatorname{U}(\varphi_{b})})$ and
$e^{\prime}_{\frac{n_{b}}{2}+1}\in X^{*}(\operatorname{im}\nu)$, such that
$\Phi_{\operatorname{U}(\varphi_{b})}$ and
$\Phi_{\operatorname{U}(\varphi_{b})}^{+}$ are given by (5.1.2) (with $n$
replaced by $n_{b}$). We must have
$e_{\lceil\frac{n_{a}+1}{2}\rceil}=e^{\prime}_{\frac{n_{b}}{2}+1}$, so we use
$e^{\prime\prime}$ to denote this element.

Now we show that either $n_{a}=0$ or $n_{b}=0$. Suppose $n_{b}\geq 2$, we must
show that $n_{a}=0$. We will use the fact that $\Phi_{G}$ is closed under
reflection along the plane that is perpendicular to a root. Note that
$e^{\prime\prime}-e_{1}^{\prime}\in\Phi^{+}_{G,\nu}$. Let $\Pi$ be the
hyperplane in $X^{*}(T_{H})_{\mathbb{R}}$ perpendicular to
$e^{\prime\prime}-e_{1}^{\prime}$. If $n_{a}$ is odd, then
$e^{\prime\prime}\in\Phi^{+}_{G,\nu}$. The reflection of $e^{\prime\prime}$
through $\Pi$, which is $e_{1}^{\prime}$, should also lie in $\Phi_{G}$. In
fact, it should lie in $\Phi_{G}\cap
X^{*}(T_{H_{0}})_{\mathbb{R}}=\Phi_{G_{0}}$. Unfortunately, $\Phi_{G_{0}}$
doesn’t contain $e_{1}^{\prime}$. This contradiction shows that $n_{a}$ must
be even. If $n_{a}\geq 2$, then $e^{\prime\prime}-e_{1}\in\Phi^{+}_{G,\nu}$.
Again, the reflection of $e^{\prime\prime}-e_{1}$ through $\Pi$, which is
$e_{1}+e_{1}^{\prime}$, should lie in $\Phi_{G_{0}}$. But $\Phi_{G_{0}}$
cannot contain such an element. Therefore we must have $n_{a}\leq 1$.
Combining these, we get $n_{a}=0$.

As we noted before, $G$ contains both $\operatorname{SO}(M_{a})$ and
$\operatorname{U}(\varphi_{b})$. If $n_{b}=0$, then $G=\operatorname{SO}(M)$.
If $n_{a}=0$, then from our assumption that
$G_{0}=\operatorname{SO}(M_{a,0})\times\operatorname{U}(\varphi_{b,0})$,
together with condition (2) and the Claim, we find that
$G=\operatorname{U}(\varphi_{b})$. $\hfill\square$

##  6\. Conjecture 4.5 and the Tate-linear conjecture

In this section we prove Conjecture 4.5 for GSpin Shimura varieties and
products of modular curves. By Proposition 4.9, this implies the Tate-linear
conjecture for these Shimura varieties. In §6.1 we introduce several important
lattices that arise from the formal torus $\mathscr{T}_{f,x}$. We then relate
these lattices to the monodromy of $F$-isocrystals in §6.2. After that, we use
the monodromy results and Lie theory lemmas (§5) to construct a finite subset
$\bm{\Delta}\subseteq\operatorname{End}(\mathscr{A}^{\operatorname{KS}}_{\mathbf{I},\overline{\eta}})\otimes\mathbb{Z}_{p}$
with certain special properties (such a subset is called sufficient, see
Definition 1). The existence of such a subset is sufficient for proving
Conjecture 4.5 in the case of products of modular curves (§6.3) and GSpin
Shimura varieties (§6.4).

###  6.1. The Tate-linear character and cocharacter lattices

We refer the readers to §4.1.3 and §4.2 for setups and notations. Let
$\Psi_{\mathbf{I}},\operatorname{Br}_{\mathbf{I}}$ be the products of
(extended) formal Brauer groups over indices running through $\mathbf{I}$.
These are $p$-divisible groups over
$\mathscr{S}_{\mathbf{I},\mathbb{F}}^{{\mathrm{ord}}}$. Let
${\mu}_{i}:\mathbb{G}_{m}\rightarrow\operatorname{GL}(H_{i,{\mathbb{Z}}_{p}})$
be the canonical Hodge cocharactes of
$\mathscr{A}^{\operatorname{KS}}_{i,x}[p^{\infty}]$ and
$\overline{\mu}_{i}:\mathbb{G}_{m}\rightarrow\operatorname{GL}(\overline{L}_{i,{\mathbb{Z}}_{p}})$
be the induced cocharacter, which is indeed the canonical Hodge cocharactes of
$\Psi_{i,x}$. We make the following convention

(6.1.1)
$\displaystyle\operatorname{GL}^{\prime}(H_{\mathbf{I}})=\prod_{i\in\mathbf{I}}\operatorname{GL}(H_{i}),$
$\displaystyle\operatorname{GL}^{\prime}(\overline{L}_{\mathbf{I},\mathbb{Z}_{p}})=\prod_{i\in\mathbf{I}}\operatorname{GL}(\overline{L}_{i,\mathbb{Z}_{p}}).$

Let
${\mu}_{\mathbf{I}}:\mathbb{G}_{m}\rightarrow\operatorname{GSpin}^{\prime}(L_{\mathbf{I},{\mathbb{Z}}_{p}})$
resp.
${\mu}_{\mathbf{I}}^{c}:\mathbb{G}_{m}\rightarrow\operatorname{SO}^{\prime}(L_{\mathbf{I},{\mathbb{Z}}_{p}})$
resp.
$\overline{\mu}_{\mathbf{I}}:\mathbb{G}_{m}\rightarrow\operatorname{GL}^{\prime}(\overline{L}_{\mathbf{I},{\mathbb{Z}}_{p}})$
be the product of $\mu_{i}$’s resp. $\mu_{i}^{c}$’s resp.
$\overline{\mu}_{i}$’s. As usual, let
$U_{\operatorname{GSpin}^{\prime},\mu_{\mathbf{I}}}$ resp.
$U_{\operatorname{SO}^{\prime},\mu^{c}_{\mathbf{I}}}$ resp.
$U_{\operatorname{GL}^{\prime},\overline{\mu}_{\mathbf{I}}}$ be the
corresponding unipotent and
$U_{\operatorname{GSpin}^{\prime},\mu_{\mathbf{I}}^{-1}}$ resp.
$U_{\operatorname{SO}^{\prime},\mu^{c,-1}_{\mathbf{I}}}$ resp.
$U_{\operatorname{GL}^{\prime},\overline{\mu}_{\mathbf{I}}^{-1}}$ be the
corresponding opposite unipotent. According to Remark 2.6, the arithmetic
deformation space $\mathscr{S}^{/x}_{\mathbf{I},W}$ admits three equivalent
formal torus structures:

(6.1.2) $\mathscr{S}^{/x}_{\mathbf{I},W}\simeq\left\\{\begin{aligned}
&U_{\operatorname{GSpin}^{\prime},{\mu}_{\mathbf{I}}}({\mathbb{Z}}_{p})\otimes_{\mathbb{Z}_{p}}\mathbb{G}_{m}^{\wedge},\\\
&U_{\operatorname{SO}^{\prime},{\mu}^{c}_{\mathbf{I}}}({\mathbb{Z}}_{p})\otimes_{\mathbb{Z}_{p}}\mathbb{G}_{m}^{\wedge},\\\
&U_{\operatorname{GL}^{\prime},\overline{\mu}_{\mathbf{I}}}({\mathbb{Z}}_{p})\otimes_{\mathbb{Z}_{p}}\mathbb{G}_{m}^{\wedge}.\end{aligned}\right.$

######  Definition 1.

  1. (1)

The Tate-linear cocharacter lattice is the saturated sublattice of
$X_{*}(\mathscr{S}_{\mathbf{I},\mathbb{F}}^{/x})$ defined as

$\mathcal{T}_{f,x}:=X_{*}(\mathscr{T}_{f,x})\subseteq
X_{*}(\mathscr{S}_{\mathbf{I},\mathbb{F}}^{/x}).$

We also denote its rational structure as
$T_{f,x}:=\mathcal{T}_{f,x}\otimes\mathbb{Q}_{p}\subseteq
X_{*}(\mathscr{S}_{\mathbf{I},\mathbb{F}}^{/x})\otimes{\mathbb{Q}_{p}}$. Under
identifications (6.1.2), ${T}_{f,x}$ gives rise to unipotent
$\mathbb{Q}_{p}$-subgroups of
$U_{\operatorname{GSpin}^{\prime},{\mu}_{\mathbf{I}}}$ ,
$U_{\operatorname{SO}^{\prime},{\mu}^{c}_{\mathbf{I}}}$ and
$U_{\operatorname{GL}^{\prime},\overline{\mu}_{\mathbf{I}}}$. When the context
is clear, these unipotent subgroups (or their $\mathbb{Q}_{p}$-points) will
again be denoted ${T}_{f,x}$.

  2. (2)

The Tate-linear character lattice is the saturated sublattice of
$X^{*}(\mathscr{S}_{\mathbf{I},\mathbb{F}}^{/x})$ defined as

$\mathcal{K}_{f,x}:=\ker(X^{*}(\mathscr{S}_{\mathbf{I},\mathbb{F}}^{/x})\rightarrow
X^{*}(\mathscr{T}_{f,x})).$

We also denote its rational structure as
$K_{f,x}:=\mathcal{K}_{f,x}\otimes\mathbb{Q}_{p}\subseteq
X^{*}(\mathscr{S}_{\mathbf{I},\mathbb{F}}^{/x})\otimes{\mathbb{Q}_{p}}$.
Again, ${K}_{f,x}$ gives rise to unipotent subgroups of
$U_{\operatorname{GSpin}^{\prime},{\mu}_{\mathbf{I}}^{-1}}$ ,
$U_{\operatorname{SO}^{\prime},{\mu}^{c,-1}_{\mathbf{I}}}$ and
$U_{\operatorname{GL}^{\prime},\overline{\mu}_{\mathbf{I}}^{-1}}$. These
subgroups (or their $\mathbb{Q}_{p}$-points) will again be denoted $K_{f,x}$.

  3. (3)

A subset
$\bm{\Delta}\subseteq\operatorname{End}(\mathscr{A}^{\operatorname{KS}}_{\mathbf{I},\overline{\eta}})\otimes\mathbb{Z}_{p}$
is called sufficient, if the corresponding deformation space

$\mathscr{D}_{\bm{\Delta}}=\operatorname{Def}(\bm{\Delta}/\mathscr{S}^{/x}_{\mathbf{I},W})\subseteq\mathscr{S}^{/x}_{\mathbf{I},W},$

(which is an obvious generalization of (2.3.2) to products of Shimura
varieties) satisfies $\ker(X^{*}({\mathscr{S}}_{\mathbf{I},W}^{/x})\rightarrow
X^{*}(\mathscr{D}_{\bm{\Delta}}))_{\mathbb{Q}_{p}}=K_{f,x}$ 888Note that
$X^{*}({\mathscr{S}}_{\mathbf{I},W}^{/x})$ can be canonically identified with
$X^{*}({\mathscr{S}}_{\mathbf{I},\mathbb{F}}^{/x})$..

######  Lemma 6.1.

If
$\operatorname{End}(\mathscr{A}^{\operatorname{KS}}_{\mathbf{I},\overline{\eta}})$
contains a sufficient subset, then Conjecture 4.5 holds.

###### Proof.

Let
$\Lambda=U_{\operatorname{GSpin}^{\prime},\mu^{-1}_{\mathbf{I}}}({\mathbb{Z}}_{p})$,
by Proposition 2.5, we can identify $\mathscr{S}^{/x}_{\mathbf{I},W}$ with
$\operatorname{Hom}_{\mathbb{Z}_{p}}(\Lambda,\mathbb{G}_{m}^{\wedge})$. Let
$\bm{\Delta}$ be the sufficient subset in question. By the theory of canonical
coordinates, there is a sublattice $\Lambda_{\bm{\Delta}}\subseteq\Lambda$
such that
$\mathscr{D}_{\bm{\Delta}}=\operatorname{Hom}_{\mathbb{Z}_{p}}\left(\Lambda/\Lambda_{\bm{\Delta}},\mathbb{G}^{\wedge}_{m}\right)$.
This $\Lambda_{\bm{\Delta}}$ is nothing other than
$\ker(X^{*}(\mathscr{S}^{/x}_{\mathbf{I},W})\rightarrow
X^{*}(\mathscr{D}_{\bm{\Delta}}))$. Since $\bm{\Delta}$ is sufficient,
$\mathcal{K}_{f,x}$ is the saturation of $\Lambda_{\bm{\Delta}}$. Therefore,
$\mathscr{D}_{\bm{\Delta},\mathbb{F}}$ admits $\mathscr{T}_{f,x}$ as the
induced reduced structure. On the other hand, $\bm{\Delta}$ deforms to
$\mathscr{X}_{f,W}^{/x,+}$ by Lemma 4.4(2), hence
$\mathscr{X}_{f,\mathbb{F},\operatorname{red}}^{/x}\subseteq\mathscr{D}_{\bm{\Delta},\mathbb{F},\operatorname{red}}=\mathscr{T}_{f,x}$.
Since $f$ factors through $\mathscr{X}_{f,\mathbb{F}}$ by Lemma 4.4(3), we
have $\mathscr{X}_{f,\mathbb{F},\operatorname{red}}^{/x}=\mathscr{T}_{f,x}$.
$\hfill\square$

###  6.2. Local and global monodromy of $\mathbb{L}^{-}_{\mathbf{I},p,X}$

Let $\mathbb{L}_{\mathbf{I},p,X}^{-}$ be the underlying $F$-isocrystal of
$\mathbb{L}_{\mathbf{I},p,X}$ and $\omega_{x}$ be the fiber functor of the
Tannakian category
$\langle\mathbb{L}^{-}_{\mathbf{I},p,X}\rangle^{\otimes}\subseteq\operatorname{\textbf{F-Isoc}}(X)$
defined in §3.2 999As we mentioned in §1.6, when talking about arithmetic
local systems and their monodromy groups, we always view $X$ as a variety over
a sufficiently large finite field.. We have two exact sequences

(6.2.1) $\displaystyle
0\rightarrow\mathbb{D}(\operatorname{Br}_{\mathbf{I},X})^{\vee}(-1)\rightarrow\mathbb{L}_{\mathbf{I},p,X}\rightarrow\mathbb{D}(\Psi_{\mathbf{I},X})\rightarrow
0.$ (6.2.2) $\displaystyle
0\rightarrow\mathbb{D}(\Psi_{\mathbf{I},X}^{{\text{\'{e}t}}})\rightarrow\mathbb{D}(\Psi_{\mathbf{I},X})\rightarrow\mathbb{D}(\operatorname{Br}_{\mathbf{I},X})\rightarrow
0.$

The existence of nondegenerate pairings $\mathbf{Q}_{i}$ over each $F$-crystal
$\mathbf{L}_{i,p}$ guarantees that the natural projection

(6.2.3) $G(\mathbb{L}^{-}_{\mathbf{I},p,X},x)\twoheadrightarrow
G(\mathbb{D}(\Psi_{\mathbf{I},X}),x)$

is an isomorphism. Therefore, to study the monodromy of
$\mathbb{L}^{-}_{\mathbf{I},p,X}$, it suffices to study the monodromy of
$\mathbb{D}(\Psi_{\mathbf{I},X})$. Since $\Psi_{\mathbf{I},X}$ is ordinary,
the techniques from §3.2.1 apply.

######  Remark 6.2.

In order to carry out concrete computations, it is convenient to have an
explicit description of the isomorphism (6.2.3). By projecting to each index,
we can assume that $\\#\mathbf{I}=1$. In the following, we drop all the
subscript $\mathbf{I}$ as usual. Arrange the basis $\\{e_{i}\\}$ of
$\omega_{x}(\mathbb{L}_{p,X})$ so that
$\omega_{x}(\mathbb{D}(\operatorname{Br}))=\operatorname{Span}_{\mathbb{Q}_{p}}\\{e_{b+2}\\}$,
$\omega_{x}(\mathbb{D}(\Psi))=\operatorname{Span}_{\mathbb{Q}_{p}}\\{e_{2},...,e_{b+2}\\}$
and the quadratic pairing over $\omega_{x}(\mathbb{L}_{p,x})$ is given by

$\mathbf{Q}=\begin{bmatrix}&&1\\\ \ &\mathbf{Q}_{0}&\\\ 1&&\end{bmatrix}.$

For a $\mathbb{Q}_{p}$-algebra $R$, an element $g\in
G(\mathbb{D}(\Psi_{X}),x)(R)$ is of the form

(6.2.4) $g=\begin{bmatrix}\mathbf{B}&\mathbf{v}\\\
&\lambda\end{bmatrix},\,\,\lambda\in\mathbb{G}_{m}(R),\,\,\mathbf{B}\in\operatorname{SO}(\mathbf{Q}_{0})(R),\,\,\mathbf{v}\in
U_{\operatorname{GL},\overline{\mu}}(\mathbb{Q}_{p})\otimes R.$

Then the preimage of $g$ under (6.2.3) is

(6.2.5)
$\begin{bmatrix}\lambda^{-1}&-\lambda^{-1}\mathbf{v}^{t}\mathbf{Q}_{0}\mathbf{B}&-\frac{1}{2}\lambda^{-1}\mathbf{v}^{t}\mathbf{Q}_{0}\mathbf{v}\\\
&\mathbf{B}&\mathbf{v}\\\ &&\lambda\end{bmatrix}.$

####  6.2.1. Local monodromy

Let $\Psi_{i}^{/x}$ be the pullback of $\Psi_{i}$ to
$\mathscr{S}^{/x}_{i,\mathbb{F}}$ and write
$\Psi_{\mathbf{I}}^{/x}=\prod_{i\in\mathbf{I}}\Psi_{i}^{/x}$. We have
$\mathbb{D}(\Psi_{\mathbf{I},X}^{/x})=\mathbb{D}(\Psi_{\mathbf{I},X})^{/x}$.
By Lemma 3.1, the local monodromy is

$G(\mathbb{D}(\Psi_{\mathbf{I},X})^{/x},x)=U(\mathbb{D}(\Psi_{\mathbf{I},X})^{/x},{x})\rtimes\operatorname{im}\overline{\mu}_{\mathbf{I}}.$

######  Proposition 6.3.

Notations being the same as §3.2. We have

  1. (1)

Regarding $T_{f,x}$ as a subgroup of
$U_{\operatorname{GL}^{\prime},\overline{\mu}_{\mathbf{I}}}$, we have

$U(\mathbb{D}(\Psi_{\mathbf{I},X}),{x})=U(\mathbb{D}(\Psi_{\mathbf{I},X})^{/x},{x})=T_{f,x}.$

  2. (2)

Regarding $T_{f,x}$ as a subgroup of
$U_{\operatorname{SO}^{\prime},{\mu}^{c}_{\mathbf{I}}}$, we have

$U(\mathbb{L}^{-}_{\mathbf{I},p,X},{x})=U(\mathbb{L}^{-,/x}_{\mathbf{I},p,X},{x})=T_{f,x}.$

###### Proof.

Providing the isomorphism (6.2.3), it suffices to prove (1). Let
$\Lambda=U_{\operatorname{GSpin}^{\prime},\mu^{-1}_{\mathbf{I}}}({\mathbb{Z}}_{p})$,
$\Lambda^{\vee}=U_{\operatorname{GSpin}^{\prime},\mu_{\mathbf{I}}}({\mathbb{Z}}_{p})$
and $X^{/x}=\operatorname{Spf}R$. Consider the pairing
$q\in\operatorname{Hom}(\Lambda,\mathbb{G}_{m}^{\wedge}(R))$ that arises from
$\Psi_{\mathbf{I},X}^{/x}$. Since $R$ is reduced, $\ker(q)$ is saturated in
$\Lambda$. Let $\ker(q)^{\perp}$ be the sub-lattice of $\Lambda^{\vee}$ that
pairs to 0 with $\ker(q)$. Then
$\ker(q)^{\perp}\otimes\mathbb{G}_{m}^{\wedge}$ is the smallest subtorus of
$\Lambda^{\vee}\otimes\mathbb{G}_{m}^{\wedge}=\mathscr{S}_{\mathbf{I},\mathbb{F}}^{/x}$
through which $f^{/x}$ factors. This shows that
$\mathcal{T}_{f,x}=\ker(q)^{\perp}$. We are done by Theorem 3.2.
$\hfill\square$

####  6.2.2. Global monodromy

By Theorem 3.2 and Proposition 6.3, the structure of the global monodromy can
be understood as

(6.2.6) $\displaystyle G(\mathbb{D}(\Psi_{\mathbf{I},X}),x)=T_{f,x}\rtimes
G(\operatorname{gr}\mathbb{D}(\Psi_{\mathbf{I},X}),x),$ (6.2.7) $\displaystyle
G(\mathbb{L}^{-}_{\mathbf{I},p,X},x)=T_{f,x}\rtimes
G(\operatorname{gr}\mathbb{L}^{-}_{\mathbf{I},p,X},x).$

The following recovers an independence result of Chai ([Ch03]) in the case of
products of GSpin Shimura varieties:

######  Corollary 6.4.

The rank of the formal subtorus $\mathscr{T}_{f,x}$ is independent of $x$
chosen. In particular, if $X$ is Tate-linear at one point, it is Tate-linear
at all points.

###### Proof.

For different $x$, the groups $G(\mathbb{D}(\Psi_{\mathbf{I},X}),{x})$ are
isomorphic. Similarly, for different $x$, the groups
$G(\operatorname{gr}\mathbb{D}(\Psi_{\mathbf{I},X}),{x})$ are also isomorphic.
Therefore $\dim U(\mathbb{D}(\Psi_{\mathbf{I},X})^{/x},{x})$ are the same for
all $x$. It now follows from Proposition 6.3 that
$\operatorname{rk}\mathscr{T}_{f,x}$ is independent of $x$ chosen.
$\hfill\square$

###  6.3. The case of products of modular curves

We prove the Tate-linear conjecture for products of modular curves. Let
$f_{i}$ be the composition of $f$ with the projection
$\mathscr{S}_{\mathbf{I},\mathbb{F}}\rightarrow\mathscr{S}_{i,\mathbb{F}}$.
More generally, for an index subset $\mathbf{J}\subset\mathbf{I}$, let
$f_{\mathbf{J}}$ be the composition of $f$ with the projection
$\mathscr{S}_{\mathbf{I},\mathbb{F}}\rightarrow\mathscr{S}_{\mathbf{J},\mathbb{F}}$.
In particular, $f=f_{\mathbf{I}}$.

Note that $\mathscr{T}_{f_{i},x}$ is both the smallest formal torus that
$f_{i}^{/x}$ factors through, and the projection of $\mathscr{T}_{f,x}$ to
$\mathscr{S}_{i,\mathbb{F}}^{/x}$. Without loss of generality, we can assume
that the projection of $\mathscr{T}_{f_{\mathbf{I}},x}$ to each
$\mathscr{S}_{i,\mathbb{F}}^{/x}$ is nontrivial. Since
$\mathscr{S}_{i,\mathbb{F}}^{/x}$ is one dimensional, this forces
$\mathscr{T}_{f_{i},x}=\mathscr{S}_{i,\mathbb{F}}^{/x}$.

####  6.3.1. The Tate-linear cocharacter ${T}_{f,x}$

To begin with, we show that only very special kind of subtori of
${\mathscr{S}}^{/x}_{\mathbf{I},\mathbb{F}}$ can arise as $\mathscr{T}_{f,x}$.
We first establish a special case of mod $p$ Mumford-Tate conjecture, which
plays an important role in revealing the structure of $\mathscr{T}_{f,x}$.

######  Lemma 6.5.

$G(\mathbb{L}_{i,p,X},x)^{\circ}=\operatorname{Hdg}(f_{i})_{\mathbb{Q}_{p}}=\operatorname{SO}(2,1)_{\mathbb{Q}_{p}}$.

###### Proof.

We know that
$G(\mathbb{L}_{i,p,X},x)^{\circ}\subseteq\operatorname{Hdg}(f_{i})_{\mathbb{Q}_{p}}=\operatorname{SO}(2,1)_{\mathbb{Q}_{p}}$
is a reductive subgroup. Since $\operatorname{rk}\mathscr{T}_{f_{i},x}=1$, we
see from Proposition 6.3 that $U(\mathbb{L}_{i,p,X}^{-},x)$ is one
dimensional. Since the only connected reductive subgroup of
$\operatorname{SO}(2,1)_{\mathbb{Q}_{p}}$ is $1$, $\mathbb{G}_{m}$ and
$\operatorname{SO}(2,1)_{\mathbb{Q}_{p}}$, we have
$G(\mathbb{L}_{i,p,X},x)^{\circ}=\operatorname{SO}(2,1)_{\mathbb{Q}_{p}}$.
$\hfill\square$

######  Corollary 6.6.

There is a partition $\mathbf{I}=\bigsqcup_{h\in\mathcal{H}}\mathbf{I}_{h}$
such that

  1. (1)

$G(\mathbb{L}_{\mathbf{I},p,X},x)^{\circ}=\prod_{h\in\mathcal{H}}G(\mathbb{L}_{\mathbf{I}_{h},p,X},x)^{\circ}$,
and for each $h\in\mathcal{H}$ and $i\in\mathbf{I}_{h}$, the natural
projection $G(\mathbb{L}_{\mathbf{I}_{h},p,X},x)^{\circ}\rightarrow
G(\mathbb{L}_{i,p,X},x)^{\circ}$ is an isomorphism.

  2. (2)

$\mathscr{T}_{f,x}=\prod_{h\in\mathcal{H}}\mathscr{T}_{f_{\mathbf{I}_{h}},x}$,
and for each $h\in\mathcal{H}$ and $i\in\mathbf{I}_{h}$, the projection
$\mathscr{T}_{f_{\mathbf{I}_{h}},x}\rightarrow\mathscr{T}_{f_{i},x}$ is an
isomorphism.

###### Proof.

By Tannakian formalism, there are natural projections
$G(\mathbb{L}_{\mathbf{I},p,X},x)^{\circ}\rightarrow
G(\mathbb{L}_{i,p,X},x)^{\circ}$ such that the induced morphism
$G(\mathbb{L}_{\mathbf{I},p,X},x)^{\circ}\rightarrow\prod_{i\in\mathbf{I}}G(\mathbb{L}_{i,p,X},x)^{\circ}$
is an embedding. It follows from Lemma 6.5 that each
$G(\mathbb{L}_{i,p,X},x)^{\circ}$ is isomorphic to
$\operatorname{SO}(2,1)_{\mathbb{Q}_{p}}$. Since
$\operatorname{SO}(2,1)_{\mathbb{Q}_{p}}$ is adjoint and simple, Goursat’s
lemma implies that there is a partition
$\mathbf{I}=\bigsqcup_{h\in\mathcal{H}}\mathbf{I}_{h}$ such that
$G(\mathbb{L}_{\mathbf{I},p,X},x)^{\circ}=\prod_{h\in\mathcal{H}}G(\mathbb{L}_{\mathbf{I}_{h},p,X},x)^{\circ}$
and, for each $h\in\mathcal{H}$ and $i\in\mathbf{I}_{h}$, the projection
$G(\mathbb{L}_{\mathbf{I}_{h},p,X},x)^{\circ}\rightarrow
G(\mathbb{L}_{i,p,X},x)^{\circ}$ is an isomorphism. This proves (1).

By Theorem 3.3, for each $h\in\mathcal{H}$ and $i\in\mathbf{I}_{h}$, the
projection $G(\mathbb{L}_{\mathbf{I}_{h},p,X}^{-},x)^{\circ}\rightarrow
G(\mathbb{L}_{i,p,X}^{-},x)^{\circ}$ is an isomorphism. Passing to unipotent
radical and using Proposition 6.3, we see that
${T}_{f,x}=\prod_{h\in\mathcal{H}}{T}_{f_{\mathbf{I}_{h}},x}$, and for each
$h\in\mathcal{H}$ and $i\in\mathbf{I}_{h}$, the projection
$T_{f_{\mathbf{I}_{h}},x}\rightarrow{T}_{f_{i},x}$ is an isomorphism. This
implies (2). $\hfill\square$

####  6.3.2. The crystalline endomorphisms
$\\{{\bm{\delta}}_{h,x}\\}_{h\in\mathcal{H}}$

Our goal is to construct a family of crystalline endomorphisms
$\\{{\bm{\delta}}_{h,x}\\}_{h\in\mathcal{H}}\in\operatorname{End}(\mathscr{A}_{\mathbf{I},\overline{\eta}}^{\operatorname{KS}})\otimes\mathbb{Z}_{p}$
indexed by the partition $\mathcal{H}$ as in Corollary 6.6, and show that
$\\{{\bm{\delta}}_{h,x}\\}_{h\in\mathcal{H}}$ meets the conditions of Lemma
6.1.

Possibly replacing $X$ by an étale cover, we can assume
$G(\mathbb{L}_{\mathbf{I},p,X},x)$ is connected. Therefore each
$G(\mathbb{L}_{\mathbf{I}_{h},p,X},x)$ and $G(\mathbb{L}_{i,p,X},x)$ are also
connected. Similar to Remark 6.2, for each $i\in\mathbf{I}$, we arrange the
basis $\\{e_{i,1},e_{i,2},e_{i,3}\\}$ of $\omega_{x}(\mathbb{L}_{i,p,X})$ so
that the quadratic pairing is

$\mathbf{Q}_{i}=\begin{bmatrix}&&1\\\ \ &1&\\\ 1&&\end{bmatrix},$

and the filtration is given by
$\omega_{x}(\mathbb{D}(\operatorname{Br}_{i}))=\operatorname{Span}_{\mathbb{Q}_{p}}\\{e_{i,3}\\}$
and
$\omega_{x}(\mathbb{D}(\Psi^{{\text{\'{e}t}}}_{i}))=\operatorname{Span}_{\mathbb{Q}_{p}}\\{e_{i,2},e_{i,3}\\}$.
For a $\mathbb{Q}_{p}$-algebra $R$, an element $g\in
G(\mathbb{L}_{i,p,X}^{-},x)(R)$ is of form

(6.3.1)
$g=\begin{bmatrix}\lambda^{-1}&-\lambda^{-1}\mathbf{v}^{t}&-\frac{1}{2}\lambda^{-1}\mathbf{v}^{t}\mathbf{v}\\\
&1&\mathbf{v}\\\
&&\lambda\end{bmatrix},\,\,\,\lambda\in\mathbb{G}_{m}(R),\mathbf{v}\in\operatorname{Span}_{\mathbb{Q}_{p}}\\{e_{i,3}^{\vee}\otimes
e_{i,2}\\}\otimes R.$

Fix an index $k\in\mathbf{I}_{h}$. Corollary 6.6(2) shows that there are
$a_{j}\in\mathbb{Z}_{p}^{*}$ for $j\in\mathbf{I}_{h}-\\{k\\}$, such that
$T_{f_{\mathbf{I}_{h}},x}$ is the graph of a linear morphism

$T_{f_{k},x}\xrightarrow{(a_{j})}\prod_{j\in\mathbf{I}_{h}-\\{k\\}}T_{f_{j},x}.$

We further set $a_{k}=1$ for convenience. Let
$A_{j}=\begin{bmatrix}a_{j}^{-1}&&\\\ &1&\\\ &&a_{j}\end{bmatrix}\text{ for
}j\in\mathbf{I}_{h}$. It follows from Corollary 6.6(1) and the explicit
expression (6.3.1) that $G(\mathbb{L}_{{\mathbf{I}_{h}},p,X},x)$ is the graph
of the following morphism

(6.3.2)
$G(\mathbb{L}_{{k},p,X},x)\xrightarrow{(\operatorname{ad}A_{j})}\prod_{j\in\mathbf{I}_{h}-\\{k\\}}G(\mathbb{L}_{j,p,X},x).$

One can then construct an endomorphism:

(6.3.3)
$d_{h,x}:\omega_{x}(\mathbb{L}_{\mathbf{I},p,X})\xrightarrow{\text{proj}_{k}}\omega_{x}(\mathbb{L}_{{k},p,X})\xrightarrow{(A_{j})}\bigoplus_{j\in\mathbf{I}_{h}}\omega_{x}(\mathbb{L}_{j,p,X})=\omega_{x}(\mathbb{L}_{\mathbf{I}_{h},p,X})\hookrightarrow\omega_{x}(\mathbb{L}_{\mathbf{I},p,X}).$

Using the explicit expression (6.3.2), we deduce that $d_{h,x}$ is an
endomorphism of $G(\mathbb{L}_{\mathbf{I},p,X},x)$-representations.
Furthermore, note that each $A_{j}$ is isometric, i.e.,
$\mathbf{Q}_{j}(A_{j}-,A_{j}-)=\mathbf{Q}_{k}(-,-)$, and since the
coefficients of $A_{j}$ all lie in $\mathbb{Z}_{p}$. Therefore, by the
functoriality of the Kuga–Satake construction, the equivalence between the
category of Dieudonné crystals and the category of $p$-divisible groups over
$X$ ([DJ95]), and crystalline isogeny theorem over finite generated fields
([DJ98]), $d_{h,x}$ gives rise to a crystalline endomorphism

(6.3.4)
$\bm{\delta}_{h,x}\in\operatorname{End}(\mathscr{A}_{\mathbf{I},{\eta}}^{\operatorname{KS}})\otimes\mathbb{Z}_{p}\subseteq\operatorname{End}(\mathscr{A}_{\mathbf{I},x}^{\operatorname{KS}})\otimes\mathbb{Z}_{p},$

where $\eta$ is the generic point of $X$.
Proof of Theorem 4.10 in the case of products of modular curves. We show that
the set $\bm{\Delta}=\\{\bm{\delta}_{h,x}\\}_{h\in\mathcal{H}}$ is sufficient
in the sense of Definition 1(3). Clearly, one can reduce to the case where
$\mathcal{H}=1$. In the following we assume $\mathcal{H}=1$ and drop the
subscript $h$. Let
$\Lambda=\operatorname{Span}_{\mathbb{Z}_{p}}\\{e_{i,2}^{\vee}\otimes
e_{i,3}\\}_{i\in\mathbf{I}}$. We can then identify

$X^{*}({\mathscr{S}}_{\mathbf{I},W}^{/x})\simeq
U_{\operatorname{GL}^{\prime},\overline{\mu}_{\mathbf{I}}^{-1}}(\mathbb{Z}_{p})\simeq\Lambda.$

Serre–Tate theory implies that
$q\in\operatorname{Hom}_{\mathbb{Z}_{p}}\left(\Lambda,\mathbb{G}^{\wedge}_{m}\right)$
lies in $\mathscr{D}_{\bm{\Delta}}$ if and only if for every $i\in\mathbf{I}$,
the identity $q(A_{i}e_{k,3}\otimes e^{\vee}_{i,2})=q(e_{k,3}\otimes
A_{i}^{t}e_{i,2}^{\vee})$ holds. If we write
$\mathscr{D}_{\bm{\Delta}}=\operatorname{Hom}_{\mathbb{Z}_{p}}\left(\Lambda/\Lambda_{\bm{\Delta}},\mathbb{G}^{\wedge}_{m}\right)$,
then

$\Lambda_{\bm{\Delta}}=\operatorname{Span}_{\mathbb{Z}_{p}}\\{e_{k,2}^{\vee}\otimes
e_{k,3}-a_{i}e_{i,2}^{\vee}\otimes e_{i,3}\\}_{i\in\mathbf{I}}.$

Clearly, $\Lambda_{\bm{\Delta}}=\mathcal{K}_{f,x}$, so $\bm{\Delta}$ is
sufficient. Lemma 6.1 implies that Conjecture 4.5 holds in the case of
products of modular curves. $\hfill\square$

###  6.4. The case of GSpin Shimura varieties

In this section we suppose that $\\#\mathbf{I}=1$. We drop the subscript
$\mathbf{I}$ as usual. Let $b=\dim\mathcal{S}$ and
$d=\operatorname{rk}\mathscr{T}_{f,x}$. Recall that $\mathbf{Q}$ and
$\mathbf{Q}_{0}$ are the quadratic pairings over $\omega_{x}(\mathbb{L}_{p})$
and $\omega_{x}(\mathbb{D}(\Psi^{{\text{\'{e}t}}}))$, respectively. The
following identifications are all canonical up to scalars

(6.4.1) $\displaystyle
U_{\operatorname{GL},\overline{\mu}}(\mathbb{Z}_{p})\simeq\omega_{x}(\mathbb{D}(\operatorname{Br}))^{\vee}\otimes\omega_{x}(\mathbb{D}(\Psi^{{\text{\'{e}t}}}))\simeq\omega_{x}(\mathbb{D}(\Psi^{{\text{\'{e}t}}})),$
(6.4.2) $\displaystyle
U_{\operatorname{GL},\overline{\mu}^{-1}}(\mathbb{Z}_{p})\simeq\omega_{x}(\mathbb{D}(\operatorname{Br}))\otimes\omega_{x}(\mathbb{D}(\Psi^{{\text{\'{e}t}}}))^{\vee}\simeq\omega_{x}(\mathbb{D}(\Psi^{{\text{\'{e}t}}}))^{\vee}.$

From (6.4.1), ${T}_{f,x}$ can be canonically considered as a quadratic
sublattice of $\omega_{x}(\mathbb{D}(\Psi^{{\text{\'{e}t}}}))$.

####  6.4.1. The Tate-linear cocharacter ${T}_{f,x}$

Similar to the case of products of modular curves, only very special kind of
subtori of $\mathscr{S}^{/x}_{\mathbb{F}}$ can arise as $\mathscr{T}_{f,x}$.
In fact, we show that, as a quadratic subspace of
$\omega_{x}(\mathbb{D}(\Psi^{{\text{\'{e}t}}}))$, $T_{f,x}$ can only be
nondegenerate or totally isotropic (Proposition 6.8).

Let $T^{\perp}_{f,x}$ be the orthogonal complement of $T_{f,x}$ in
$\omega_{x}(\mathbb{D}(\Psi^{{\text{\'{e}t}}}))$ and let $U_{f,x}:=T_{f,x}\cap
T^{\perp}_{f,x}$. Clearly, $T_{f,x},T^{\perp}_{f,x},U_{f,x}$ are
subrepresentations of $G(\mathbb{D}(\Psi^{{\text{\'{e}t}}}),x)$. Since
$G(\mathbb{D}(\Psi^{{\text{\'{e}t}}}),x)$ is reductive by Theorem 3.3, there
exist $G(\mathbb{D}(\Psi^{{\text{\'{e}t}}}),x)$-subrepresentations $V_{f,x}$,
$V^{\prime}_{f,x}$ and $U_{f,x}^{\prime}$ such that

(6.4.3)
$\displaystyle\omega_{x}(\mathbb{D}(\Psi^{{\text{\'{e}t}}}))=U_{f,x}\oplus
U_{f,x}^{\prime}\oplus V_{f,x}\oplus V_{f,x}^{\prime},$ $\displaystyle
T_{f,x}=V_{f,x}\oplus U_{f,x},$ $\displaystyle T_{f,x}^{\perp}=U_{f,x}\oplus
V_{f,x}^{\prime}.$

We note that $U_{f,x}$ is totally isotropic, while $V_{f,x},V_{f,x}^{\prime}$
and $U_{f,x}\oplus U_{f,x}^{\prime}$ are nondegenerate. Furthermore, $\dim
U_{f,x}=\dim U_{f,x}^{\prime}$, and $U_{f,x}$ is a maximal totally isotropic
subspace of $U_{f,x}\oplus U_{f,x}^{\prime}$. Therefore the map
$G(\mathbb{D}(\Psi^{{\text{\'{e}t}}}),x)\rightarrow\operatorname{SO}(U_{f,x}\oplus
U_{f,x}^{\prime})$ factors through a unitary subgroup associated to an
Hermitian form $\varphi_{f,x,0}$ with
$\operatorname{Tr}\varphi_{f,x,0}=\mathbf{Q}_{0}|_{U_{f,x}\oplus
U_{f,x}^{\prime}}$. Consequently, we might and do assume that
$U_{f,x}^{\prime}$ is totally isotropic and dual to $U_{f,x}$ with respect to
$\mathbf{Q}_{0}$. In a similar manner, we consider certain distinguished
subspaces of $\omega_{x}(\mathbb{L}_{p,X})$. Let
$B_{x}=\omega_{x}(\mathbb{D}(\operatorname{Br}))\oplus\omega_{x}(\mathbb{D}(\operatorname{Br})^{\vee}(-1))$.
We define

(6.4.4) $\displaystyle M^{{\text{\'{e}t}}}_{f,x}:=U_{f,x}\oplus
U_{f,x}^{\prime}\oplus V_{f,x},$ $\displaystyle M_{f,x}:=B_{x}\oplus
M^{{\text{\'{e}t}}}_{f,x},$ $\displaystyle
W_{f,x}:=U_{f,x}\oplus\omega_{x}(\mathbb{D}(\operatorname{Br})),$
$\displaystyle
W_{f,x}^{\prime}:=U_{f,x}^{\prime}\oplus\omega_{x}(\mathbb{D}(\operatorname{Br})^{\vee}(-1)).$

Note that $\omega_{x}(\mathbb{D}(\operatorname{Br}))$ and
$\omega_{x}(\mathbb{D}(\operatorname{Br})^{\vee}(-1))$ are one dimensional
subspaces of $\omega_{x}(\mathbb{L}_{p,X})$, which are totally isotropic and
dual to each other. So $W_{f,x}$ and $W_{f,x}^{\prime}$ are totally isotropic
and dual to each other. Furthermore, $W_{f,x}\oplus W_{f,x}^{\prime}$ is
equipped with an Hermitian form $\varphi_{f,x}$ such that
$\operatorname{Tr}\varphi_{f,x}=\mathbf{Q}|_{W_{f,x}\oplus W_{f,x}^{\prime}}$
and $\varphi_{f,x,0}=\varphi_{f,x}|_{U_{f,x}\oplus U^{\prime}_{f,x}}$.

######  Lemma 6.7.

We have a splitting $\omega_{x}(\mathbb{L}_{p,X})=V_{f,x}^{\prime}\oplus
M_{f,x}$ as $G(\mathbb{L}_{p,X},x)$-representations. Equivalently,
$\mathbb{L}_{p,X}=\mathbb{V}^{\prime}_{p,f}\oplus\mathbb{M}_{p,f}$ in the
category of overconvergent $F$-isocrystals. Furthermore, if $V_{f,x}=0$, then
we have a further splitting $M_{f,x}=W_{f,x}\oplus W_{f,x}^{\prime}$ in the
category of $G(\mathbb{L}_{p,X},x)$-representations. Equivalently,
$\mathbb{M}_{p,f}=\mathbb{W}_{p,f}\oplus\mathbb{W}_{p,f}^{\prime}$ in the
category of overconvergent $F$-isocrystals, and $\mathbb{W}_{p,f}$ is dual to
$\mathbb{W}_{p,f}^{\prime}$. (Here we denote by $\mathbb{V}^{\prime}_{p,f}$,
$\mathbb{M}_{p,f}$, $\mathbb{W}_{p,f}$ and $\mathbb{W}_{p,f}^{\prime}$ the
overconvergent $F$-isocrystals arising from $V_{f,x}^{\prime}$, $M_{f,x}$,
$W_{f,x}$ and $W_{f,x}^{\prime}$ via Tannakian formalism. The subscript $p$,
as usual, stands for “$\mathrm{cris}$”).

###### Proof.

Easy computation involving explicit formulas in Remark 6.2 shows that
$V_{f,x}^{\prime}$ and $M_{f,x}$ are
$G(\mathbb{L}^{-}_{p,X},x)$-subrepresentations of
$\omega_{x}(\mathbb{L}_{p,X})$. So at least, we have
$\omega_{x}(\mathbb{L}_{p,X})=V_{f,x}^{\prime}\oplus M_{f,x}$ in the category
of $G(\mathbb{L}^{-}_{p,X},x)$-subrepresentations. Let
$\mathbb{V}^{\prime-}_{p,f}$ and $\mathbb{M}_{p,f}^{-}$ be the $F$-isocrystals
corresponding to $V^{\prime}_{f,x}$ and $M_{f,x}$. We have
$\mathbb{L}_{p,X}^{-}=\mathbb{V}^{\prime-}_{p,f}\oplus\mathbb{M}_{p,f}^{-}$ in
the category of $F$-isocrystals. Note that, there is an idempotent
$\alpha\in\operatorname{End}(\mathbb{L}_{p,X}^{-})$ such that
$\ker\alpha=\mathbb{V}^{\prime-}_{p,f}$ and
$\operatorname{im}\alpha=\mathbb{M}_{p,f}^{-}$. Since $\mathbb{L}_{p,X}$ is
overconvergent, Kedlaya’s result [Ked01] implies that
$\alpha\in\operatorname{End}(\mathbb{L}_{p,X})$. Therefore $\ker\alpha$ and
$\operatorname{im}\alpha$ are also overconvergent and $V_{f,x}^{\prime}$ and
$M_{f,x}$ are $G(\mathbb{L}_{p,X},x)$-subrepresentations of
$\omega_{x}(\mathbb{L}_{p,X})$.

Now suppose that $V_{f,x}=0$, easy computation again shows that
$M_{f,x}=W_{f,x}\oplus W_{f,x}^{\prime}$ in the category of
$G(\mathbb{L}_{p,X}^{-},x)$-representations. Let $\mathbb{W}^{-}_{p,f}$ and
$\mathbb{W}^{\prime-}_{p,f}$ be the $F$-isocrystals corresponding to $W_{f,x}$
and $W^{\prime}_{f,x}$. So
$\mathbb{M}_{p,f}=\mathbb{W}_{p,f}^{-}\oplus\mathbb{W}^{\prime-}_{p,f}$ at
least in the category of $F$-isocrystals. Similar to the previous case,
[Ked01] again implies that $\mathbb{W}_{p,f}^{-}$ and
$\mathbb{W}^{\prime-}_{p,f}$ are overconvergent. The assertion that
$\mathbb{W}_{p,f}$ and $\mathbb{W}^{\prime}_{p,f}$ are mutually dual follows
easily. $\hfill\square$

######  Proposition 6.8.

Either $V_{f,x}=0$ or $U_{f,x}=0$. In other words, $T_{f,x}$ is either
nondegenrate or totally isotropic. Furthermore, we have

$G(\mathbb{M}_{p,f},x)^{\circ}=\left\\{\begin{aligned}
&\operatorname{SO}(M_{f,x}),\,\,&U_{f,x}=0,\\\
&\operatorname{U}(\varphi_{f,x}),\,\,&V_{f,x}=0.\end{aligned}\right.$

###### Proof.

By Lemma 6.7, there is an overconvergent sub-$F$-isocrystal
$\mathbb{M}_{p,f}\subseteq\mathbb{L}_{p,X}$. Let
$\mathbb{M}_{p,f}^{{\text{\'{e}t}}}\subseteq\mathbb{D}(\Psi^{{\text{\'{e}t}}}_{X})$
be the sub-$F$-isocrystal corresponding to $M^{{\text{\'{e}t}}}_{f,x}$.
Theorem 3.3 implies that $G(\mathbb{M}_{p,f},x)^{\circ}$ is a reductive
subgroup of $\operatorname{SO}(M_{f,x})$ which admits
$G(\mathbb{M}_{p,f}^{-},x)^{\circ}$ as the parabolic subgroup corresponding to
$\mu^{c}$. The Levi of $G(\mathbb{M}_{p,f}^{-},x)^{\circ}$ is
$G(\mathbb{M}_{p,f}^{{\text{\'{e}t}}},x)^{\circ}\times\operatorname{im}\mu^{c}$.
The projection of $G(\mathbb{M}_{p,f}^{{\text{\'{e}t}}},x)^{\circ}$ to
$\operatorname{GL}(V_{f,x})$ resp. $\operatorname{GL}(U_{f,x}\oplus
U_{f,x}^{\prime})$ lies in $\operatorname{SO}(V_{f,x})$ resp.
$\operatorname{U}(\varphi_{f,x,0})$, so we have

(6.4.5)
$G(\mathbb{M}_{p,f}^{{\text{\'{e}t}}},x)^{\circ}\subseteq\operatorname{SO}(V_{f,x})\times\operatorname{U}(\varphi_{f,x,0}).$

The unipotent of $G(\mathbb{M}_{p,f}^{-},x)^{\circ}$ corresponding to
$\mu^{c}$ is $U(\mathbb{L}_{p,X}^{-},x)$, which by Proposition 6.3 is
$T_{f,x}$ (considered as a subgroup of $U_{\operatorname{SO},\mu^{c}}$).
Therefore

(6.4.6) $U(\mathbb{L}_{p,X}^{-},x)=U_{\operatorname{SO}(V_{f,x}\oplus
B_{x}),\mu^{c}}\times U_{\operatorname{U}(\varphi_{f,x}),\mu^{c}}.$

Upon base changing to $\overline{\mathbb{Q}}_{p}$, we are in the situation of
§5. In fact, let $K,M,M_{0}$ and $B$ in §5 be
$\overline{\mathbb{Q}}_{p},M_{f,x},M_{f,x}^{{\text{\'{e}t}}}$ and $B_{x}$,
respectively. Let $Q^{\prime},\nu,G$ and $G_{0}$ in §5 be
$\mathbf{Q}|_{M_{f,x}},\mu^{c}|_{\operatorname{SO}(M_{f,x})}$,
$G(\mathbb{M}_{p,f},x)^{\circ}$ and
$G(\mathbb{M}_{p,f}^{{\text{\'{e}t}}},x)^{\circ}$, respectively.

We wish to apply Lemma 5.3 to $G=G(\mathbb{M}_{p,f},x)^{\circ}$. In fact, it
suffices to take $M_{a,0}=V_{f,x}$, $M_{b,0}=U_{f,x}\oplus U_{f,x}^{\prime}$
and $(M_{b},\varphi_{b})=(U_{f,x}\oplus U_{f,x}^{\prime}\oplus
B_{x},\varphi_{f,x})$ (which is an Hermitian space respecting
$\mathbf{Q}|_{M_{b}}$ in the sense of §5.1.1). Now (6.4.5) and (6.4.6)
translates to the fact that $G$ meets conditions (1) and (2) of Lemma 5.3. It
follows from the lemma that either $U_{f,x}=0$ and
$G(\mathbb{M}_{p,f},x)^{\circ}=\operatorname{SO}(M_{f,x})$, or $V_{f,x}=0$ and
$G(\mathbb{M}_{p,f},x)^{\circ}=\operatorname{U}(\varphi_{f,x})$.
$\hfill\square$

######  Remark 6.9.

The identity (5.2.1) from Lemma 5.3 also translates to the following:

$G(\mathbb{M}_{p,f}^{{\text{\'{e}t}}},x)^{\circ}=\left\\{\begin{aligned}
&\operatorname{SO}(T_{f,x}),\,\,&T_{f,x}\text{ is nondegenerate},\\\
&\operatorname{U}(\varphi_{f,x,0}),\,\,&T_{f,x}\text{ is totally
isotropic}.\end{aligned}\right.$

####  6.4.2. The crystalline endomorphisms ${\bm{\delta}}_{\mathbf{v},x}$ and
${\bm{\delta}}_{\mathbf{w},x}$

We construct fundamental crystalline endomorphisms
$\bm{\Delta}\subseteq\operatorname{End}(\mathscr{A}_{\overline{\eta}}^{\operatorname{KS}})\otimes\mathbb{Z}_{p}$
that meet the conditions of Lemma 6.1. Possibly replacing $X$ by a finite
étale cover, we can assume that $G(\mathbb{L}_{p,X},x)$ is connected. Let
$\eta$ be the generic point of $X$. Let $\dim V_{f,x}^{\prime}=r$. By Lemma
6.7, we have an overconvergent $F$-isocrystal
$\mathbb{V}^{\prime}_{p,f}\subseteq\mathbb{L}_{p,X}$. From Remark 2.2, we know
that there is a chain of embeddings

(6.4.7)
$\det{\mathbb{V}}^{\prime}_{p,f}\subseteq\wedge^{r}\mathbb{L}_{p,X}\subseteq\mathcal{E}nd(\mathbb{H}_{p,X}).$

Since
$G(\mathbb{V}^{\prime}_{p,f},x)\subseteq\operatorname{SO}(V_{f,x}^{\prime})$,
the uniroot $F$-isocrystal $\det\mathbb{V}^{\prime}_{p,f}$ is constant. Pick a
$\mathbb{Q}_{p}$-generator $d_{\mathbf{v},x}$ of $\det V_{f,x}^{\prime}$. By
the equivalence between the category of Dieudonné crystals and the category of
$p$-divisible groups over $X$ ([DJ95]) and crystalline isogeny theorem over
finite generated fields ([DJ98]), we have

$d_{\mathbf{v},x}\in\operatorname{End}(\mathbb{H}_{p,\eta})=\operatorname{End}(\mathscr{A}^{\operatorname{KS}}_{\eta})\otimes\mathbb{Q}_{p}.$

Let ${\bm{\delta}}_{\mathbf{v},x}$ be a suitable $p$-power multiple of
$d_{\mathbf{v},x}$ that lies in
$\operatorname{End}(\mathscr{A}^{\operatorname{KS}}_{\eta})\otimes\mathbb{Z}_{p}$.

Now suppose $V_{f,x}=0$, i.e., $T_{f,x}$ is totally isotropic and
$\omega_{x}(\mathbb{L}_{p,X})=W_{f,x}\oplus W^{\prime}_{f,x}\oplus
V^{\prime}_{f,x}$. Let $1\neq\gamma\in\mathbb{Z}_{p}^{*}$. We define an
isometry $d_{\mathbf{w},x}\in\operatorname{End}(\omega_{x}(\mathbb{L}_{p,X}))$
by

(6.4.8) $d_{\mathbf{w},x}(v)=\left\\{\begin{aligned} &\gamma v,\,\,\,v\in
W_{f,x},\\\ &\gamma^{-1}v,\,\,\,v\in W_{f,x}^{\prime},\\\ &v,\,\,\,v\in
V^{\prime}_{f,x}.\end{aligned}\right.$

Since $G(\mathbb{L}_{p,X},x)\subseteq\operatorname{U}(W_{f,x}\oplus
W_{f,x}^{\prime})\times\operatorname{SO}(V_{f,x}^{\prime})$,
$d_{\mathbf{w},x}$ is also an isometric isomorphism of
$G(\mathbb{L}_{p,X},x)$-representations. By funtoriality of the Kuga–Satake
construction, the equivalence between the category of Dieudonné crystals and
the category of $p$-divisible groups, and the crystalline isogeny theorem over
finite generated fields, we again have
$d_{\mathbf{w},x}\in\operatorname{End}(\mathscr{A}^{\operatorname{KS}}_{\eta})\otimes\mathbb{Q}_{p}$.
Let ${\bm{\delta}}_{\mathbf{w},x}$ be a suitable $p$-power multiple of
$d_{\mathbf{w},x}$ that lies in
$\operatorname{End}(\mathscr{A}^{\operatorname{KS}}_{\eta})\otimes\mathbb{Z}_{p}$.
Proof of Theorem 4.10 for GSpin Shimura varieties. We show that, when
$T_{f,x}$ is nondegenerate resp. totally isotropic, then
$\bm{\Delta}=\\{\bm{\delta}_{\mathbf{v},x}\\}$ resp.
$\\{\bm{\delta}_{\mathbf{v},x},\bm{\delta}_{\mathbf{w},x}\\}$ is sufficient in
the sense of Definition 1(3). The conjecture will then follow from Lemma 6.1.
Let $\mathscr{D}_{\bm{\Delta},1}$ resp. $\mathscr{D}_{\bm{\Delta},2}$ be
$\operatorname{Def}(\\{\bm{\delta}_{\mathbf{v},x}\\}/\mathscr{S}_{W}^{/x})$
resp.
$\operatorname{Def}(\\{\bm{\delta}_{\mathbf{v},x},\bm{\delta}_{\mathbf{w},x}\\}/\mathscr{S}_{W}^{/x})$.

Let $\mathscr{V}_{x}^{\prime}$ be an étale $p$-divisible subgroup of
$\Psi_{x}$ with $\mathbb{D}(\mathscr{V}_{x}^{\prime})=V_{f,x}^{\prime}$, then
$\mathscr{V}_{x}^{\prime}$ splits from $\Psi_{x}$ up to isogeny. Deforming the
isogeny class of
$\bm{\delta}_{\mathbf{v},x}\in\operatorname{End}(\mathscr{A}_{x}^{\operatorname{KS}})\otimes\mathbb{Q}_{p}$
inside ${\mathscr{S}}_{W}^{/x}$ is the same as deforming the corresponding
global section of $\wedge\mathbb{L}_{p,x}^{r}$, which is equivalent to
deforming the splitting of $\mathscr{V}_{x}^{\prime}$ from $\Psi_{x}$ up to
isogeny.

In the following we make the identification (6.4.1) and (6.4.2). Recall that
the pairing $\mathbf{Q}_{0}$ induces a canonical splitting
$U_{\operatorname{GL},\overline{\mu}}(\mathbb{Q}_{p})=M^{{\text{\'{e}t}}}_{f,x}\oplus
V_{f,x}^{\prime}$. As a result, $(V_{f,x}^{\prime})^{\vee}$ canonically sits
inside $U_{\operatorname{GL},\overline{\mu}^{-1}}(\mathbb{Q}_{p})$ and is the
kernel of the map
$U_{\operatorname{GL},\overline{\mu}^{-1}}(\mathbb{Q}_{p})\rightarrow(M^{{\text{\'{e}t}}}_{f,x})^{\vee}$.
By the theory of canonical coordinates, there is some lattice
$\Lambda_{\bm{\Delta},1}$ such that
$\mathscr{D}_{\bm{\Delta},1}=\operatorname{Hom}(U_{\operatorname{GL},\overline{\mu}^{-1}}(\mathbb{Z}_{p})/\Lambda_{\bm{\Delta},1},\mathbb{G}^{\wedge}_{m})$.
As noted before, $\Lambda_{\bm{\Delta},1,\mathbb{Q}_{p}}$ corresponds to the
condition of deforming the splitting of $\mathscr{V}_{x}^{\prime}$ from
$\Psi_{x}$ up to isogeny. Therefore
$\Lambda_{\bm{\Delta},1,\mathbb{Q}_{p}}=(V_{f,x}^{\prime})^{\vee}$.

When $T_{f,x}$ is nondegenerate, we have $T_{f,x}=M^{{\text{\'{e}t}}}_{f,x}$,
hence $\Lambda_{\bm{\Delta},1,\mathbb{Q}_{p}}=K_{f,x}$. When $T_{f,x}$ is
totally isotropic, there is some lattice $\Lambda_{\bm{\Delta},2}$ such that
$\mathscr{D}_{\bm{\Delta},2}=\operatorname{Hom}(U_{\operatorname{GL},\overline{\mu}^{-1}}(\mathbb{Z}_{p})/\Lambda_{\bm{\Delta},2},\mathbb{G}^{\wedge}_{m})$.
We must have
$\Lambda_{\bm{\Delta},2,\mathbb{Q}_{p}}\supseteq\Lambda_{\bm{\Delta},1,\mathbb{Q}_{p}}=(V_{f,x}^{\prime})^{\vee}$.
Since any deformation should also preserve the class
$\bm{\delta}_{\mathbf{w},x}$, the theory of canonical coordinates implies that
$\Lambda_{\bm{\Delta},2,\mathbb{Q}_{p}}=(V_{f,x}^{\prime})^{\vee}\oplus(U^{\prime}_{f,x})^{\vee}$.
Since $T_{f,x}=U_{f,x}$, we have
$\Lambda_{\bm{\Delta},2,\mathbb{Q}_{p}}=K_{f,x}$. $\hfill\square$

##  7\. Characteristic $p$ analogue of the Mumford–Tate conjecture

In this section we prove Conjecture 4.7 for GSpin Shimura varieties and
products of modular curves. As note in Proposition 4.9, this proves Conjecture
1.1 for GSpin Shimura varieties and products of modular curves. The main input
is Theorem 4.10 established in §6 and the independence of monodromy groups in
a compatible system of coefficient objects ([DA20, Theorem 1.2.1]). For
$u\in\mathrm{fpl}(\mathbb{Q})$, we will adopt the identification
$\alpha^{\prime}_{u}:L_{\mathbf{I},\mathbb{Q}_{u}}\simeq\omega_{x}(\mathbb{L}_{\mathbf{I},u})$,
as explained in §4.1.2. Therefore ${\operatorname{Hdg}}(f)$ is a subgroup of
$\operatorname{GL}(L_{\mathbf{I},\mathbb{Q}})$, and
$G(\mathbb{L}_{\mathbf{I},u,X},x)$ is a subgroup of
$\operatorname{GL}(L_{\mathbf{I},\mathbb{Q}_{u}})$. We will be using the fact
that
$G(\mathbb{L}_{\mathbf{I},u,X},x)^{\circ}\subseteq{\operatorname{Hdg}}(f)_{\mathbb{Q}_{u}}$
(cf. Lemma 4.3) without mentioning.

###  7.1. The case of products of modular curves

The case of products of modular curves should already be known in the
literature. The result can be proved without the Tate-linear conjecture.
Nevertheless, to exhibit the close relation between the Tate-linear and the
Mumford–Tate conjectures, as well as to pave the way for future
generalizations, we present a proof using these somewhat heavier machineries.
In the following, we suppose, without loss of generality, that the projection
of $\mathscr{T}_{f,x}$ to each $\mathscr{S}_{i,\mathbb{F}}^{/x}$ is
nontrivial.

####  7.1.1. The structure of $\operatorname{Hdg}(f)$

From Lemma 6.5, we already know that for each $i$,
$\operatorname{Hdg}(f_{i})\simeq\operatorname{SO}(2,1)$. To understand
$\operatorname{Hdg}(f_{\mathbf{I}})$, let
$\mathbf{I}=\bigsqcup_{h\in\mathcal{H}}\mathbf{I}_{h}$ be the partition of
$\mathbf{I}$ as in Corollary 6.6.

######  Proposition 7.1.

$\operatorname{Hdg}(f)=\prod_{h\in\mathcal{H}}\operatorname{Hdg}(f_{\mathbf{I}_{h}})$
and, for each $h\in\mathcal{H}$ and $i\in\mathbf{I}_{h}$, the projection
$\operatorname{Hdg}(f_{\mathbf{I}_{h}})\rightarrow\operatorname{Hdg}(f_{i})$
is an isomorphism.

###### Proof.

By definition, we have
$\operatorname{Hdg}(f)\subseteq\prod_{i\in\mathbf{I}}\operatorname{Hdg}(f_{i})$.
Since the projection of $G(\mathbb{L}_{\mathbf{I},p,X},x)^{\circ}$ to each
$G(\mathbb{L}_{i,p,X},x)^{\circ}$ is surjective, Lemma 6.5 and Lemma 4.3 imply
that the projection of $\operatorname{Hdg}(f)$ to each
$\operatorname{Hdg}(f_{i})$ is surjective.

Since $\operatorname{SO}(2,1)$ is adjoint and simple, Goursat’s lemma implies
that there is a partition
$\mathbf{I}=\bigsqcup_{h^{\prime}\in\mathcal{H}^{\prime}}\mathbf{I}_{h^{\prime}}$
such that
$\operatorname{Hdg}(f)=\prod_{h^{\prime}\in\mathcal{H}^{\prime}}\operatorname{Hdg}(f_{\mathbf{I}_{h^{\prime}}})$
and, for each $h^{\prime}\in\mathcal{H}^{\prime}$ and
$i\in\mathbf{I}_{h^{\prime}}$, the projection
$\operatorname{Hdg}(f_{\mathbf{I}_{h^{\prime}}})\rightarrow\operatorname{Hdg}(f_{i})$
is an isomorphism. By Corollary 6.6(2), we have
${T}_{f,x}=\prod_{h\in\mathcal{H}}{T}_{f_{\mathbf{I}_{h}},x}$, and for each
$h\in\mathcal{H}$ and $i\in\mathbf{I}_{h}$, the projection
${T}_{f_{\mathbf{I}_{h}},x}\rightarrow{T}_{f_{i},x}$ is an isomorphism. Taking
the unipotent of $\operatorname{Hdg}(f)_{\mathbb{Q}_{p}}$ corresponding to
$\mu^{c}_{\mathbf{I}}$ and using Theorem 6.3, we find that the partition
$\mathbf{I}=\bigsqcup_{h^{\prime}\in\mathcal{H}^{\prime}}\mathbf{I}_{h^{\prime}}$
is identical to the partition
$\mathbf{I}=\bigsqcup_{h\in\mathcal{H}}\mathbf{I}_{h}$. So
$\operatorname{Hdg}(f)$ has the desired structure. $\hfill\square$
Proof of Theorem 4.12 for products of modular curves. Lemma 4.3, Proposition
7.1 and Corollary 6.6(1) imply that
$\operatorname{Hdg}(f)_{\mathbb{Q}_{p}}=G(\mathbb{L}_{\mathbf{I},p,X},x)^{\circ}$.
The independence of monodromy groups in a compatible system ([DA20, Theorem
1.2.1]) implies that
$\operatorname{Hdg}(f)_{\mathbb{Q}_{u}}=G(\mathbb{L}_{\mathbf{I},u,X},x)^{\circ}$
for all finite place $u$. From the explicit descriptions, it is an easy
exercise to show that $\operatorname{Hdg}(f)$ coincides with the generic Hodge
group of $\mathbb{L}_{\mathbf{I},\mathrm{B},\mathcal{X}_{f}^{+}}$.
$\hfill\square$

###  7.2. The case of GSpin Shimura varieties

As usual, let $\operatorname{rk}L=b+2$ and
$d=\operatorname{rk}\mathcal{T}_{f,x}$. Recall from §6.4 that we have
subspaces $V_{f,x}^{\prime},M_{f,x},W_{f,x},W^{\prime}_{f,x}$ of
$\omega_{x}(\mathbb{L}_{p,X})$. By Lemma 6.7, these give rise to
overconvergent sub-$F$-isocrystals $\mathbb{V}^{\prime}_{p,f}$,
$\mathbb{M}_{p,f}$, $\mathbb{W}_{p,f}$ and $\mathbb{W}_{p,f}^{\prime}$ of
$\mathbb{L}_{p,X}$. We have

(7.2.1) $G(\mathbb{L}_{p,X},x)^{\circ}\subseteq
G(\mathbb{V}_{p,f}^{\prime},x)^{\circ}\times G(\mathbb{M}_{p,f},x)^{\circ}.$

By Proposition 6.8, the subspace $T_{f,x}\in\omega_{x}(\mathbb{L}_{p,x})$ is
either nondegenerate or totally isotropic. In the nondegenerate case, we have
$G(\mathbb{M}_{p,f},x)^{\circ}=\operatorname{SO}(M_{f,x})$, while in the
totally isotropic case, we have
$G(\mathbb{M}_{p,f},x)^{\circ}=\operatorname{U}(\varphi_{f,x})$.

######  Lemma 7.2.

(7.2.1) is an equality.

###### Proof.

Possibly base change to an étale cover, we can assume that
$G(\mathbb{L}_{p,X},x)$ is connected. Since any object in the Tannakian
subcategory $\langle\mathbb{V}_{p,f}^{\prime}\rangle^{\otimes}$ has zero
slope, it suffices to show the following:

######  Claim.

An object in $\langle\mathbb{M}_{p,f}\rangle^{\otimes}$ has zero slope only
when it is a direct sum of trivial objects.

Indeed, by Goursat’s lemma, there is a common quotient $H$ of
$G(\mathbb{V}_{p,f}^{\prime},x)$ and $G(\mathbb{M}_{p,f},x)$, such that
$G(\mathbb{L}_{p,X},x)=G(\mathbb{V}_{p,f}^{\prime},x)\times_{H}G(\mathbb{M}_{p,f},x)$.
A faithful representation of $H$ gives rise to an object that lies in both
$\langle\mathbb{V}_{p,f}^{\prime}\rangle^{\otimes}$ and
$\langle\mathbb{M}_{p,f}\rangle^{\otimes}$. If the Claim is true, then any
faithful representation of $H$ is trivial, so $H=\\{1\\}$. Therefore
$G(\mathbb{L}_{p,X},x)=G(\mathbb{V}_{p,f}^{\prime},x)\times
G(\mathbb{M}_{p,f},x)$.

We now prove the Claim. Let $G=G(\mathbb{M}_{p,f},x)$ and
$\nu=\mu^{c}|_{\operatorname{SO}(M_{f,x})}$. An object
$\mathbb{X}\subseteq\langle\mathbb{M}_{p,f}\rangle^{\otimes}$ with zero slope
gives rise to a representation
$G\rightarrow\operatorname{GL}(\omega_{x}(\mathbb{X}))$. Let $N$ be the
kernel. Then $N$ is a normal subgroup containing $\operatorname{im}\nu$ and
$U_{G,\nu}$. It suffices to show that $N=G$. By Proposition 6.8, $G$ is either
$\operatorname{SO}(M_{f,x})$ or $\operatorname{U}(\varphi_{f,x})$, depending
on whether $T_{f,x}$ is nondegenerate or totally isotropic. In the
nondegenerate case, $N=\operatorname{SO}(M_{f,x})$ is immediate. For the
totally isotropic case, we note that $\operatorname{SU}(\varphi_{f,x})$ is
simple. So at least $\operatorname{SU}(\varphi_{f,x})\subseteq N$. Since $N$
also contains $\operatorname{im}\nu$, we have
$N=\operatorname{U}(\varphi_{f,x})$. The proof of the Claim is complete.
$\hfill\square$

####  7.2.1. The structure of ${\operatorname{Hdg}}(f)$

######  Proposition 7.3.

The possible structures of ${\operatorname{Hdg}}(f)$ lie in the following two
cases:
($T_{f,x}$ is nondegenerate) There is a quadraple
$(F,\tau,\mathbf{M}_{f},{Q}_{f})$ where $F$ is totally real field,
$\tau:F\rightarrow\mathbb{C}$ is an embedding, and $(\mathbf{M}_{f},{Q}_{f})$
is a quadratic space over $F$, such that

  1. (1)

$\operatorname{Hdg}(f)=\operatorname{Res}_{F/\mathbb{Q}}\operatorname{SO}(Q_{f})$.

  2. (2)

$Q_{f}$ has signature $(2,d)$ at place $\tau$ and is negative definite at all
other real places.

  3. (3)

$Q=\operatorname{Tr}_{F/\mathbb{Q}}({Q}_{f})\oplus Q^{\rho}$, where $Q^{\rho}$
is a negative definite quadratic form.

  4. (4)

Let $\mathfrak{p}$ be the place of $F$ given by
$F\xrightarrow{\tau}\mathbb{C}\simeq\overline{\mathbb{Q}}_{p}$. Then
$F_{\mathfrak{p}}=\mathbb{Q}_{p}$ and
$(\mathbf{M}_{f},{Q}_{f})_{F_{\mathfrak{p}}}\simeq(M_{f,x},\mathbf{Q}|_{M_{f,x}})$.

($T_{f,x}$ is totally isotropic) There is a quintuple
$(E,F,\tau,\mathbf{M}_{f},{{\phi}}_{f})$ where $F$ is a totally real field,
$\tau:F\rightarrow\mathbb{C}$ is an embedding, $E/F$ is a quadratic imaginary
extension, and $(\mathbf{M}_{f},{\bm{\phi}}_{f})$ is an Hermitian space over
$E$, such that

  1. (1)

$\operatorname{Hdg}(f)=\operatorname{Res}_{F/\mathbb{Q}}\operatorname{U}({{\phi}}_{f})$.

  2. (2)

$\phi_{f}$ has signature $(1,d)$ at place $\tau$ and is negative definite at
all other real places.

  3. (3)

$Q=\operatorname{Tr}_{E/\mathbb{Q}}({\phi}_{f})\oplus Q^{\rho}$, where
$Q^{\rho}$ is a negative definite quadratic form.

  4. (4)

Let $\mathfrak{p}$ be the place of $F$ given by
$F\xrightarrow{\tau}\mathbb{C}\simeq\overline{\mathbb{Q}}_{p}$. Then
$F_{\mathfrak{p}}=\mathbb{Q}_{p}$ and
$(\mathbf{M}_{f},\phi_{f})_{F_{\mathfrak{p}}}\simeq(M_{f,x},\varphi_{f,x})$.

###### Proof.

The following argument uses multiple ideas from [F18]. Consider the standard
representation
$\rho:\operatorname{Hdg}(f)\rightarrow\operatorname{SO}(L_{\mathbb{Q}})$. Let
$L_{\mathbb{Q}}^{\rho}\subseteq L_{\mathbb{Q}}$ be the subspace fixed by
$\operatorname{Hdg}(f)$ and let $Q^{\rho}:=Q|_{L_{\mathbb{Q}}^{\rho}}$. The
quadratic space $(L_{\mathbb{Q}}^{\rho},Q^{\rho})$ is negative definite, and
$\rho$ factors through
$\operatorname{SO}(L_{\mathbb{Q}}^{\rho,\perp})\subseteq\operatorname{SO}(L_{\mathbb{Q}})$.
Let $\mathbf{E}$ be the center of the subalgebra of
$\operatorname{End}(L_{\mathbb{Q}})$ generated by
$\rho(\operatorname{Hdg}(f))$, and $\mathbf{F}$ be the subalgebra fixed by the
adjoint involution induced by $Q$. Then $\mathbf{F}$ is totally real, and
decomposes into a finite product of totally real fields $\mathbf{F}=\prod
F_{\alpha}$. The idempotents of $\mathbf{F}$ induce splittings
$\mathbf{E}=\prod E_{\alpha}$ and
$L_{\mathbb{Q}}^{\rho,\perp}=\bigoplus_{\alpha}\mathbf{M}_{\alpha}$. Each
$\mathbf{M}_{\alpha}$ has a structure of an $F_{\alpha}$-vector space, it is
further equipped with an $F_{\alpha}$-valued quadratic form $Q_{\alpha}$ such
that
$Q|_{\mathbf{M}_{\alpha}}=\operatorname{Tr}_{F_{\alpha}/\mathbb{Q}}(Q_{\alpha})$.
So $\rho$ factors through
$\prod_{\alpha}\operatorname{Res}_{F_{\alpha}/\mathbb{Q}}\operatorname{SO}(Q_{\alpha})$.
The same argument in §3 of loc.cit shows that $\mathbf{F}$ has exactly one
factor $F_{f}$ with the corresponding quadratic space $(\mathbf{M}_{f},Q_{f})$
such that the form $\operatorname{Tr}_{F_{f}/\mathbb{Q}}(Q_{f})$ is
indefinite. Furthermore, there exists exactly one real place $\tau$ of $F_{f}$
over which $Q_{f}$ is indefinite.

######  Claim.

$\mathbf{F}=F_{f}$ and $L_{\mathbb{Q}}^{\rho,\perp}=\mathbf{M}_{f}$.

Consider the Shimura subvariety $\mathcal{H}$ with structure group
$\operatorname{SO}(\mathbf{M}_{f})$ and $\mathscr{H}$ be its closure in
$\mathscr{S}$. It is easy to see that the cocharacter
$h_{\tilde{x}_{\mathbb{C}}}$ factors through $\mathcal{H}$, and the component
of $\mathcal{X}_{f}$ containing $\tilde{x}_{\mathbb{C}}$ factors through
$\mathcal{H}$. As a result, $f$ factors through $\mathscr{H}_{\mathbb{F}}$.
Let
$\Sigma\subseteq\operatorname{End}^{0}(\mathscr{A}^{\operatorname{KS}}_{\tilde{x}_{\mathbb{C}}})$
be the set of special endomorphisms that arise from $\mathbf{M}_{f}^{\perp}$.
Then $\Sigma$ extends to $\mathscr{A}^{\operatorname{KS}}_{\mathcal{H}}$.
Applying Lemma 4.2 to a irreducible component of $\mathscr{H}_{\mathbb{F}}$
containing the image of $f$, we see that
$\Sigma\subseteq\operatorname{End}^{0}(\mathscr{A}^{\operatorname{KS}}_{X^{\prime}})$
for some integral finite cover $X^{\prime}/X$. Therefore,
$\Sigma\subseteq\operatorname{End}^{0}(\mathscr{A}^{\operatorname{KS}}_{\overline{\eta}})$.
The group $\operatorname{Hdg}(f)$, by definition, must also commute with
$\Sigma$. It follows that
$\operatorname{Hdg}(f)\subseteq\operatorname{SO}(\mathbf{M}_{f})$, hence
$\mathbf{F}=F_{f}$ and $L_{\mathbb{Q}}^{\rho,\perp}=\mathbf{M}_{f}$. The field
$\mathbf{E}$ is either identical to $\mathbf{F}$ or a nontrivial CM extension
of $\mathbf{F}$. An argument similar to [F18, Construction 3.5] puts us in the
following two cases:
(Case 1). $\mathbf{E}=\mathbf{F}$. Write $F$ for $\mathbf{F}$, we have
$Q=\operatorname{Tr}_{F/\mathbb{Q}}(Q_{f})\oplus Q^{\rho}$ and
$\operatorname{Hdg}(f)=\operatorname{Res}_{F/\mathbb{Q}}\operatorname{SO}(Q_{f})$.
There is some integer $l$ such that $\tau Q_{f}$ has signature $(2,l)$. We see
that $\dim\mathcal{X}_{f}=l$. On the other hand, Proposition 4.4(1) and
Theorem 4.10 for GSpin Shimura varieties implies that
$\dim\mathcal{X}_{f}=\operatorname{rk}\mathscr{T}_{f,x}=d$. It follows that
$l=d$.

In the following we show that this is exactly the case when $T_{f,x}$ is
nondegenerate. Let $\mathcal{P}$ be a place of $F$ lying over $p$ and
$\mathfrak{p}$ be the place of $F$ given by
$F\xrightarrow{\tau}\mathbb{C}\simeq\overline{\mathbb{Q}}_{p}$. Let
$\mu^{c}:\mathbb{G}_{m}\rightarrow L_{\mathbb{Q}_{p}}$ be the canonical Hodge
cocharacter at point $x$. Since $\operatorname{im}\mu^{c}\subseteq
G(\mathbb{L}_{p,X},x)^{\circ}\subseteq\operatorname{Hdg}(f)_{\mathbb{Q}_{p}}$,
we see that $\mu^{c}$ factors through

(7.2.2)
$\operatorname{Hdg}(f)_{\mathbb{Q}_{p}}=\prod_{\mathcal{P}|p}\operatorname{Res}_{F_{\mathcal{P}}/\mathbb{Q}_{p}}\operatorname{SO}(Q_{f})_{F_{\mathcal{P}}}.$

Indeed, it furthermore factors through
$\operatorname{Res}_{F_{\mathfrak{p}}/\mathbb{Q}_{p}}\operatorname{SO}(Q_{f})_{F_{\mathfrak{p}}}$.
By base change to $\overline{\mathbb{Q}}_{p}$, we have

(7.2.3)
$\left(\operatorname{Res}_{F_{\mathfrak{p}}/\mathbb{Q}_{p}}\operatorname{SO}(Q_{f})_{F_{\mathfrak{p}}}\right)\times\overline{\mathbb{Q}}_{p}=\prod_{\sigma:F_{\mathfrak{p}}\hookrightarrow\overline{\mathbb{Q}}_{p}}\operatorname{SO}(Q_{f})_{\sigma,\overline{\mathbb{Q}}_{p}}.$

Then $\operatorname{SO}(Q_{f})_{\tau,\overline{\mathbb{Q}}_{p}}$ is the unique
factor that $\mu^{c}$ factors through. On the other hand,
$g\in\operatorname{Gal}(F_{\mathfrak{p}}/\mathbb{Q}_{p})$ acts transitively on
the right of (7.2.3), and $g\cdot\mu^{c}$ factors through
$\operatorname{SO}(Q_{f})_{g\cdot\tau,\overline{\mathbb{Q}}_{p}}$. Since
$\mu^{c}$ is defined over $\mathbb{Q}_{p}$, we have $g\cdot\mu^{c}=\mu^{c}$.
Consequently, there is only one embedding of $F_{\mathfrak{p}}$ into
$\overline{\mathbb{Q}}_{p}$. Hence $F_{\mathfrak{p}}=\mathbb{Q}_{p}$. It then
follows from Theorem 3.3, Proposition 4.10 and (7.2.2) that

$T_{f,x}=U(\mathbb{L}_{p,X}^{-},x)\subseteq
U_{\operatorname{Hdg}(f)_{\mathbb{Q}_{p}},\mu^{c}}=U_{\operatorname{SO}(Q_{f})_{F_{\mathfrak{p}}},\mu^{c}}.$

Since $\dim U_{\operatorname{SO}(Q_{f})_{F_{\mathfrak{p}}},\mu^{c}}=d$, we
have $T_{f,x}=U_{\operatorname{SO}(Q_{f})_{F_{\mathfrak{p}}},\mu^{c}}$. Thus
$T_{f,x}$ is nondegenerate,
$(\mathbf{M}_{f},Q_{f})_{F_{\mathfrak{p}}}\simeq(M_{f,x},\mathbf{Q}|_{M_{f,x}})$,
and
$\operatorname{SO}(Q_{f})_{F_{\mathfrak{p}}}\simeq\operatorname{SO}(M_{f,x})$.
(Case 2). $\mathbf{E}$ is a CM extension of $\mathbf{F}$. We write $E,F$ for
$\mathbf{E},\mathbf{F}$. Then $Q_{f}$ is the trace of an $E$-valued Hermitian
form $\phi_{f}$ over $\mathbf{M}_{f}$. We have
$Q=\operatorname{Tr}_{E/\mathbb{Q}}(\phi_{f})\oplus Q^{\rho}$ and
$\operatorname{Res}_{F/\mathbb{Q}}\operatorname{SU}(\phi_{f})\subseteq\operatorname{Hdg}(f)\subseteq\operatorname{Res}_{F/\mathbb{Q}}\operatorname{U}(\phi_{f})$.
By the definitions of $\mathcal{X}_{f}$ and $\operatorname{MT}(f)$, we have
$\operatorname{Hdg}(\mathcal{X}_{f})\subseteq\operatorname{Hdg}(f)$. Now
Zarhin’s result [Z83] claims that $\operatorname{Hdg}(\mathcal{X}_{f})$ must
be the scalar restriction of a unitary group or an orthogonal group. This
forces
$\operatorname{Hdg}(f)=\operatorname{Res}_{F/\mathbb{Q}}\operatorname{U}(\phi_{f})$.

Let $r$ be the integer such that $\tau\phi_{f}$ has signature $(1,r)$. We have
$\dim\mathcal{X}_{f}=r$. On the other hand, Proposition 4.4(1) and Theorem
4.10 for GSpin Shimura varieties implies that
$\dim\mathcal{X}_{f}=\operatorname{rk}\mathscr{T}_{f,x}=d$. It follows that
$r=d$.

Like Case 1, $\mu^{c}$ factors through
$\operatorname{Res}_{F_{\mathfrak{p}}/\mathbb{Q}_{p}}\operatorname{U}(\phi_{f})_{F_{\mathfrak{p}}}$.
A similar argument as in Case 1 shows that $F_{\mathfrak{p}}=\mathbb{Q}_{p}$
and

$T_{f,x}=U(\mathbb{L}_{p,X}^{-},x)\subseteq
U_{\operatorname{Hdg}(f)_{\mathbb{Q}_{p}},\mu^{c}}=U_{\operatorname{U}(\phi_{f})_{F_{\mathfrak{p}}},\mu^{c}}.$

Since $\dim U_{\operatorname{U}(\phi_{f})_{F_{\mathfrak{p}}},\mu^{c}}=d$, we
have $T_{f,x}=U_{\operatorname{U}(\phi_{f})_{F_{\mathfrak{p}}},\mu^{c}}$. It
then follows that $T_{f,x}$ is totally isotropic,
$(\mathbf{M}_{f},\phi_{f})_{F_{\mathfrak{p}}}\simeq(M_{f,x},\varphi_{f,x})$,
and
$\operatorname{U}(\phi_{f})_{F_{\mathfrak{p}}}\simeq\operatorname{U}(\varphi_{f,x})$.
$\hfill\square$

######  Remark 7.4.

In Proposition 7.3, the existence of the place $\mathfrak{p}$ such that
$F_{\mathfrak{p}}=\mathbb{Q}_{p}$ is related to the fact that the image of $f$
lies in the ordinary stratum. The readers shall compare this to [DU18,
Corollary 1.0.2], where it is claimed that a Shimura subvariety (satisfying
certain assumptions) with reflex field $F$ admits a mod $\mathfrak{p}$
reduction with nontrivial ordinary locus if and only if
$F_{\mathfrak{p}}=\mathbb{Q}_{p}$.

####  7.2.2. The proof

By [DA20, Theorem 4.1.1], one can replace $X$ by a finite étale cover, so that
$G(\mathbb{L}_{u,X},x)$ is connected for all $u$. We will be assuming this in
the rest of this section.

We begin by studying the splitting behavior of $\operatorname{Hdg}(f)$ and
various local systems. Let $F,\tau$ be the totally real field and its complex
embedding from Proposition 7.3. We fix a sufficiently large Galois extension
$K/F$, as well as an embedding $K\subseteq\mathbb{C}$. Let $\Sigma_{F}$ be the
set of embeddings of $F$ into $K$. We will use $v$ to denote a place of $K$
over a finite place $u$ of $\mathbb{Q}$. The base change local system
$\mathbb{L}_{u,X}\otimes K_{v}$ will be denoted $\mathbb{L}_{v,X}$. From
Proposition 7.3, there is a $F$-group $\mathcal{G}$, which is either
$\operatorname{SO}(\mathbf{M}_{f})$ or $\operatorname{U}(\phi_{f})$, such that
$\operatorname{Hdg}(f)=\operatorname{Res}_{F/\mathbb{Q}}\mathcal{G}$. The
group $\mathcal{G}$ is equipped with a standard representation
$r_{f}:\mathcal{G}\rightarrow\operatorname{GL}(\mathbf{M}_{f})$. For a
$\sigma\in\Sigma_{F}$, we let $\mathcal{G}_{\sigma,K}$,
$\mathbf{M}_{f,\sigma,K}$, $r_{f,\sigma,K}$ be the base changes of
$\mathcal{G}$, $\mathbf{M}_{f}$, $r_{f}$ to $K$ via the embedding $\sigma$. We
have following decompositions

(7.2.4)
$\displaystyle\operatorname{Hdg}(f)_{K}\subseteq\prod_{\sigma\in\Sigma_{F}}\mathcal{G}_{\sigma,K},$
(7.2.5) $\displaystyle
L_{K}=\bigoplus_{\sigma\in\Sigma_{F}}\mathbf{M}_{\sigma,K},$ (7.2.6)
$\displaystyle\varrho_{f,K}=\bigoplus_{\sigma\in\Sigma_{F}}{r}_{f,\sigma,K}|_{\operatorname{Hdg}(f)_{K}},$

where we recall that $\varrho_{f}$ is the representation of
$\operatorname{Hdg}(f)$ on $L_{\mathbb{Q}}$. Note that for each $v$, the
monodromy group $G(\mathbb{L}_{v,X},x)$ sits inside
$\operatorname{Hdg}(f)_{K_{v}}$. The representation ${r}_{f,\sigma,K_{v}}$
induces a sub-local system $\mathbb{M}_{\sigma,v,X}\subseteq\mathbb{L}_{v,X}$
such that
$\mathbb{L}_{v,X}\subseteq\bigoplus_{\sigma\in\Sigma_{F}}\mathbb{M}_{\sigma,v,X}$.
Let $\mathbb{V}^{\prime}_{\sigma,v,X}$ be the image of $\mathbb{L}_{v,X}$ in
$\bigoplus_{\sigma\neq\sigma^{\prime}\in\Sigma_{F}}\mathbb{M}_{\sigma^{\prime},v,X}$.
Again, $\mathbb{M}_{\sigma,v,X}$ and $\mathbb{V}^{\prime}_{\sigma,v,X}$ are
arithmetic local systems, and shall be thought of as over a suitable finite
field model $X_{0}$. It is immediate from the definition that, for every
$\sigma$:

(7.2.7) $\displaystyle
G(\mathbb{M}_{\sigma,v,X},x)\subseteq\mathcal{G}_{\sigma,{K}_{v}},$ (7.2.8)
$\displaystyle
G(\mathbb{V}^{\prime}_{\sigma,v,X},x)\subseteq\prod_{\sigma\neq\sigma^{\prime}\in\Sigma_{F}}G(\mathbb{M}_{\sigma^{\prime},v,X},x),$
(7.2.9) $\displaystyle G(\mathbb{L}_{v,X},x)\subseteq
G(\mathbb{M}_{\sigma,v,X},x)\times G(\mathbb{V}^{\prime}_{\sigma,v,X},x).$

We know that $\\{\mathbb{L}_{v,X}\\}_{v\in\mathrm{fpl}(K)}$ is a compatible
system of coefficient objects. The following lemma says that the sub-local
systems that we have constructed as above are also compatible:

######  Lemma 7.5.

For each $\sigma\in\Sigma_{F}$, the collections
$\\{\mathbb{M}_{\sigma,v,X}\\}_{v\in\mathrm{fpl}(K)}$ and
$\\{\mathbb{V}^{\prime}_{\sigma,v,X}\\}_{v\in\mathrm{fpl}(K)}$ are compatible
systems of coefficient objects.

###### Proof.

We only prove the assertion for
$\\{\mathbb{M}_{\sigma,v,X}\\}_{v\in\mathrm{fpl}(K)}$, the assertion for
$\\{\mathbb{V}^{\prime}_{\sigma,v,X}\\}_{v\in\mathrm{fpl}(K)}$ is similar. Let
$X_{0}$ be a suitable finite field model of $X$. It suffices to check that, at
each closed point $x_{0}\in X_{0}$, the characteristic polynomial
$P(\mathbb{M}_{\sigma,v,x_{0}},t)$ of the geometric Frobenius on the fiber
$\mathbb{M}_{\sigma,v,x_{0}}$ is independent of $v$. Since $x_{0}$ is
ordinary, we can canonical lift the geometric Frobenius of
$\mathscr{A}^{\operatorname{KS}}_{x_{0}}$ to characteristic 0, this gives rise
to a conjugacy class of elements in $\operatorname{GL}(L_{\mathbb{Q}})$. We
denote by $P(L_{\mathbb{Q},\tilde{x}_{0}},t)$ the corresponding characteristic
polynomial. Then
$P(L_{\mathbb{Q},\tilde{x}_{0}},t)_{K_{v}}=P(\mathbb{L}_{v,x_{0}},t)$. From
(7.2.5) and (7.2.6), we have
$P(L_{\mathbb{Q},\tilde{x}_{0}},t)_{K}=\prod_{\sigma\in\Sigma_{F}}P(\mathbf{M}_{\sigma,K,\tilde{x}_{0}},t)$
and
$P(\mathbb{L}_{v,x_{0}},t)=\prod_{\sigma\in\Sigma_{F}}P(\mathbb{M}_{\sigma,v,x_{0}},t)$.
It then follows the definition that
$P(\mathbb{M}_{\sigma,v,x_{0}},t)=P(\mathbf{M}_{\sigma,K,\tilde{x}_{0}},t)_{K_{v}}$.
Therefore $P(\mathbb{M}_{\sigma,v,x_{0}},t)$ is independent of $v$.

The lemma can also be proved without the theory of canonical liftings.
Instead, one considers the splitting of the motive $\bm{L}_{x_{0}}$ as per
[MP152, §4] upon base changing to $K$. The proof is left to the readers.
$\hfill\square$

Proof of Theorem 4.12 for GSpin Shimura varieties. Notation as above. It
follows from Proposition 7.3 and Zarhin’s result [Z83] that
$\operatorname{Hdg}(\mathcal{X}_{f})=\operatorname{Hdg}(f)$.

We now show that
$\operatorname{Hdg}(f)_{\mathbb{Q}_{u}}=G(\mathbb{L}_{u,X},x)^{\circ}$ for
every $u\in\mathrm{fpl}(\mathbb{Q})$. Recall that we have replaced $X$ by a
finite étale cover so that $G(\mathbb{L}_{u,X},x)$ is connected for all $u$.
In the following will we omit the superscript “$\circ$” from all groups. Let
$\mathfrak{p}$ be the place of $F$ corresponding to the embedding $\tau$ (cf.
Proposition 7.3). Take a place $\mathfrak{P}|p$ of $K$ lying over
$\tau\mathfrak{p}$. Then $\mathbb{M}_{\tau,\mathfrak{P},X}$ and
$\mathbb{V}^{\prime}_{\tau,\mathfrak{P},X}$ are nothing other than
$\mathbb{M}_{p,f}\otimes K_{\mathfrak{P}}$ and
$\mathbb{V}^{\prime}_{p,f}\otimes K_{\mathfrak{P}}$. So by Proposition 6.8,
Lemma 7.2 and Proposition 7.3, we have

$G(\mathbb{L}_{\mathfrak{P},X},x)=G(\mathbb{M}_{\tau,\mathfrak{P},X},x)\times
G(\mathbb{V}^{\prime}_{\tau,\mathfrak{P},X},x),\,\,G(\mathbb{M}_{\tau,\mathfrak{P},X},x)=\mathcal{G}_{\tau,K_{\mathfrak{P}}}.$

Lemma 7.5 and the independence of monodromy groups in a compatible system
([DA20, Theorem 1.2.1]), together with (7.2.7) and (7.2.9) imply that, for
each finite place $v$ of $K$,

(7.2.10) $G(\mathbb{L}_{v,X},x)=G(\mathbb{M}_{\tau,v,X},x)\times
G(\mathbb{V}^{\prime}_{\tau,v,X},x),\,\,G(\mathbb{M}_{\tau,v,X},x)=\mathcal{G}_{\tau,{K}_{v}}.$

We now show by Galois theory, that (7.2.10) holds if one replaces $\tau$ by
any other $\sigma\in\Sigma_{F}$. Note that there is a natural bijection
$\Sigma_{F}\simeq\operatorname{Gal}(K/\mathbb{Q})/\operatorname{Gal}(K/\tau
F)$. So $\operatorname{Gal}({K}/\mathbb{Q})$ acts on $\Sigma_{F}$. In
particular, the decomposition group of a place $v$, which is identified with
$\operatorname{Gal}(K_{v}/\mathbb{Q}_{u})$, acts on $\Sigma_{F}$. For any
$\sigma\in\Sigma_{F}$, Chebotarev’s density theorem guarantees the existence
of a place $v$ and an element $g$ in
$\operatorname{Gal}(K_{v}/\mathbb{Q}_{u})$ such that $g\tau=\sigma$. Consider
the base change (7.2.10)$\times_{g}K_{v}$. We have

$\displaystyle\mathcal{G}_{\tau,{K}_{v}}\times_{g}{K}_{v}=\mathcal{G}_{\sigma,{K}_{v}},$
$\displaystyle
G(\mathbb{M}_{\tau,v,X},x)\times_{g}{K}_{v}=G(\mathbb{M}_{\sigma,v,X},x),$
$\displaystyle
G(\mathbb{V}^{\prime}_{\tau,v,X},x)\times_{g}{K}_{v}=G(\mathbb{V}^{\prime}_{\sigma,v,X},x),$
$\displaystyle
G(\mathbb{L}_{v,X},x)\times_{g}{K}_{v}=G(\mathbb{L}_{u,X},x)_{K_{v}}\times_{g}{K}_{v}=G(\mathbb{L}_{v,X},x).$

As a result, for this particular place $v$, we have

(7.2.11) $G(\mathbb{L}_{v,X},x)=G(\mathbb{M}_{\sigma,v,X},x)\times
G(\mathbb{V}^{\prime}_{\sigma,v,X},x),\,\,G(\mathbb{M}_{\sigma,v,X},x)=\mathcal{G}_{\sigma,{K}_{v}}.$

The independence of monodromy groups in a compatible family then implies that
for any finite place $v$ of $K$, (7.2.11) still holds. Since $\sigma$ is
arbitrary, we find that, for any $v$,

$G(\mathbb{L}_{v,X},x)=\prod_{\sigma\in\Sigma_{F}}\mathcal{G}_{\sigma,{K}_{v}}.$

Together with (7.2.4), we obtain

(7.2.12)
$G(\mathbb{L}_{u,X},x)=\operatorname{Hdg}(f)_{\mathbb{Q}_{u}}=(\operatorname{Res}_{F/\mathbb{Q}}\mathcal{G})_{\mathbb{Q}_{u}}.$

. $\hfill\square$

##  8\. Characteristic $p$ analogue of the André–Oort conjecture

In this section we prove the characteristic $p$ analogue of the André–Oort
conjecture (4.8) for GSpin Shimura varieties and products of modular curves.
Suppose $X$ contains a Zariski dense collection of positive dimensional
special subvarieties. In §8.2 we will construct certain large arithmetic
$p$-adic lisse sheaves on $X$ that arise from these special subvarieties.
After that, in §8.3 and §8.4, we use the established cases of the Tate-linear
conjecture and the characteristic $p$ analogue of the Mumford–Tate conjecture
to show that $X$ is special.

###  8.1. Setups

Notation being the same as Conjecture 4.8. Recall that $\bm{A}$ is a
collection of special subvarieties on $X$ and
$\mathbf{I}_{\bm{A}}\subseteq\mathbf{I}$ is the set of indices $i$ such that
$\bm{A}$ contains a Zariski dense collection of special subvarieties whose
projections to $\mathscr{S}_{i,\mathbb{F}}$ are positive dimensional. We will
always assume that $\mathbf{I}_{\bm{A}}\neq\emptyset$. In the following, the
letter “$Z$” is reserved for denoting special subvareities. $\bm{A}$ is said
to be normalized, if

  1. (1)

Each $Z\in\bm{A}$ is a positive dimensional connected smooth locally closed
subvariety of $X$,

  2. (2)

The projection of any $Z\in\bm{A}$ to
$\mathscr{S}_{\mathbf{I}-\mathbf{I}_{\bm{A}},\mathbb{F}}$ is a single point.

We call $\bm{A}$ is simple, if it further satisfies

  1. (3)

Any special subvariety in $\bm{A}$ has positive dimensional projections to
$\mathscr{S}_{i,\mathbb{F}}$ for all $i\in\mathbf{I}_{\bm{A}}$.

######  Lemma 8.1.

Possibly shrinking $\bm{A}$ and replacing a special subvariety in $\bm{A}$ by
an open dense subset, there is a normalized collection $\bm{A}^{\prime}$ such
that $\mathbf{I}_{\bm{A}^{\prime}}=\mathbf{I}_{\bm{A}}$. Therefore, to prove
Conjecture 4.8, we can always assume $\bm{A}$ is normalized.

###### Proof.

First, for any $Z\in\bm{A}$, replace $Z$ by its irreducible components
$Z_{1},...,Z_{n}$. If $Z_{i}$ is zero dimensional, we throw it out. Otherwise,
replace $Z_{i}$ by an open dense subset which is smooth. As a result, we get a
collection $\bm{A}^{\prime\prime}$ satisfying (1) with
$\mathbf{I}_{\bm{A}}=\mathbf{I}_{\bm{A}^{\prime\prime}}$. Then, let $\bm{B}$
be the collection of special subvarieties in $\bm{A}^{\prime\prime}$ whose
projections to
$\mathscr{S}_{\mathbf{I}-\mathbf{I}_{\bm{A}^{\prime\prime}},\mathbb{F}}$ are
positive dimensional. $\bm{B}$ can not be a Zariski dense collection. For
otherwise there must be an index
$i\in\mathbf{I}-\mathbf{I}_{\bm{A}^{\prime\prime}}$ such that
$\bm{A}^{\prime\prime}$ contains a Zariski dense collection of special
subvarieties whose projections to $\mathscr{S}_{i,\mathbb{F}}$ are positive
dimensional, hence $i\in\mathbf{I}_{\bm{A}^{\prime\prime}}$, a contradiction.
Now let $\bm{A}^{\prime}=\bm{A}^{\prime\prime}-\bm{B}$. $\hfill\square$

As a result of Lemma 8.1, we will always assume that $\bm{A}$ is normalized.
We call a morphism $X^{\prime}\rightarrow X$ an étale open dense subset, if
$X^{\prime}$ is finite étale over an open dense subset of $X$. If $X^{\prime}$
is an étale open dense subset of $X$, we denote by $\bm{A}_{X^{\prime}}$ the
pullback of $\bm{A}$ to $X^{\prime}$. The definition of $\mathbf{I}_{\bm{A}}$,
and the notion of being normalized and simple, extends to
$\bm{A}_{X^{\prime}}$. We have
$\mathbf{I}_{\bm{A}}=\mathbf{I}_{\bm{A}_{X^{\prime}}}$. Furthermore,
$\bm{A}_{X^{\prime}}$ is normalized resp. simple, if $\bm{A}$ is normalized
resp. simple.

###  8.2. $p$-adic lisse sheaves arising from dense collections of special
subvarieties

####  8.2.1. Tautological deformation spaces

For any subvariety of $\mathscr{S}_{\mathbf{I},\mathbb{F}}$, we use $\Delta$
to denote its immersion into $\mathscr{S}_{\mathbf{I},\mathbb{F}}$. There are
formal schemes $(X\times X)^{/\Delta}$,
$(X\times\mathscr{S}_{\mathbf{I},\mathbb{F}})^{/\Delta}$ over $X$ and
$(Z\times Z)^{/\Delta}$ over $Z$ sitting inside following diagram

${(Z\times Z)^{/\Delta}}$${(X\times
X)^{/\Delta}}$${(X\times\mathscr{S}_{\mathbf{I},\mathbb{F}})^{/\Delta}}$${Z}$${X}$${X}$

These formal schemes shall be thought of as variations of deformation spaces
over the base scheme. More precisely, over each $x\in X(\mathbb{F})$, the
fiber of $(X\times\mathscr{S}_{\mathbf{I},\mathbb{F}})^{/\Delta}$ at $x$ is
just $\mathscr{S}_{\mathbf{I},\mathbb{F}}^{/x}$, while the fiber of $(X\times
X)^{/\Delta}$ at $x$ is $X^{/x}$.

By Chai’s theory of global Serre–Tate coordinates ([Ch03, §2]), there is an
arithmetic lisse sheaf of $\mathbb{Z}_{p}$-modules over $X$, namely,
$\mathcal{E}_{\mathbf{I}}=\bigoplus_{i\in\mathbf{I}}X_{*}(\operatorname{Br}_{i,X})\otimes_{\mathbb{Z}_{p}}T_{p}(\Psi^{{\text{\'{e}t}}}_{i,X})$,
such that

(8.2.1)
$(X\times\mathscr{S}_{\mathbf{I},\mathbb{F}})^{/\Delta}=\mathcal{E}_{\mathbf{I},X}\otimes_{\mathbb{Z}_{p}}\mathbb{G}_{m}^{\wedge}.$

Here $\mathbb{G}_{m}^{\wedge}$ stands for the formal torus over $\mathbb{F}$.
For later use, we also set
$\mathcal{E}_{\mathbf{J}}=\bigoplus_{i\in\mathbf{J}}X_{*}(\operatorname{Br}_{i,X})\otimes_{\mathbb{Z}_{p}}T_{p}(\Psi^{{\text{\'{e}t}}}_{i,X})$
for any subset $\mathbf{J}\subseteq\mathbf{I}$.

Since $Z$ is special, [N96, Theorem 3.7] implies that $(Z\times
Z)^{/\Delta}\subseteq(Z\times\mathscr{S}_{\mathbf{I},\mathbb{F}})^{/\Delta}$
is a subtorus of the formal torus (8.2.1) over $Z$. So there exists a
saturated arithmetic lisse subsheaf
$\mathcal{F}_{Z}\subseteq\mathcal{E}_{\mathbf{I},Z}$, such that

(8.2.2) $(Z\times
Z)^{/\Delta}=\mathcal{F}_{Z}\otimes_{\mathbb{Z}_{p}}\mathbb{G}_{m}^{\wedge}.$

####  8.2.2. The induced $p$-adic lisse sheaves

Let $\mathfrak{X}$ be a Noetherian formal scheme and
$\mathcal{O}_{\mathfrak{X}}$ be its structure sheaf. An open formal subscheme
of $\mathfrak{X}$ is a pair
$(\mathfrak{X}^{\prime},\mathcal{O}_{\mathfrak{X}}|_{\mathfrak{X}^{\prime}})$,
where $\mathfrak{X}^{\prime}$ is an open subset of $\mathfrak{X}$. On the
other hand, according to [EGA1, §10.14], a coherent ideal
$\mathscr{A}\subseteq\mathcal{O}_{\mathfrak{X}}$ defines a closed formal
subscheme
$(\mathfrak{Y},\mathcal{O}_{\mathfrak{X}}/\mathscr{A}|\mathfrak{Y})$, where
$\mathfrak{Y}$ is the support of $\mathcal{O}_{\mathfrak{X}}/\mathscr{A}$, and
every closed formal subscheme arise this way. Note that a closed formal
subscheme is again Noetherian, and the intersection of two closed formal
subschemes is again a closed formal subscheme. An open formal subscheme of a
closed formal subscheme of $\mathfrak{X}$ is called a locally closed formal
subscheme.

Let $\mathfrak{X}=(X\times\mathscr{S}_{\mathbf{I},\mathbb{F}})^{/\Delta}$.
Then for every $Z\in\bm{A}$, $(Z\times Z)^{/\Delta}$ is a locally closed
formal subscheme of $\mathfrak{X}$. Let $\mathfrak{Z}$ be the smallest closed
formal subscheme of $(X\times\mathscr{S}_{\mathbf{I},\mathbb{F}})^{/\Delta}$
containing all $(Z\times Z)^{/\Delta},Z\in\bm{A}$. From the discussion made in
last paragraph, such $\mathfrak{Z}$ always exists, and is a Noetherian
subscheme of $\mathfrak{X}$. Since $(X\times X)^{/\Delta}$ is a closed formal
subscheme containing all $(Z\times Z)^{/\Delta}$, we have
$\mathfrak{Z}\subseteq(X\times X)^{/\Delta}$.

######  Lemma 8.2.

There is an étale open dense subset $X^{\prime}$ of $X$, together with
geometric $p$-adic lisse sheaves $\\{\mathcal{H}_{k}\\}_{k=1}^{n}$ over
$X^{\prime}$, such that the irreducible components of
$\mathfrak{Z}_{X^{\prime}}:=\mathfrak{Z}\times_{X}X^{\prime}$ are exactly
$\\{\mathcal{H}_{1}\otimes\mathbb{G}_{m}^{\wedge}\\}_{k=1}^{n}$.

###### Proof.

For every $n\in\mathbb{Z}$, there is a scaling by $1+p^{n}$ morphism
$\mathfrak{s}_{n}$ over
$\mathcal{E}_{\mathbf{I},X}\otimes_{\mathbb{Z}_{p}}\mathbb{G}_{m}^{\wedge}$.
Clearly, every $\mathfrak{s}_{n}$ is an isomorphism and takes each $(Z\times
Z)^{/\Delta}$ to itself. Therefore, $\mathfrak{s}_{n}$ takes $\mathfrak{Z}$ to
itself. By a rigidity result of Chai ([Chai08]), any irreducible component of
$\mathfrak{Z}_{\overline{\eta}}$ is a formal subtorus of
$\mathcal{E}_{\mathbf{I},\overline{\eta}}\otimes_{\mathbb{Z}_{p}}\mathbb{G}_{m}^{\wedge}$,
where $\eta$ is the generic point of $X$. As a result, there is an étale open
dense subset $X^{\prime}$ of $X$, with generic point $\eta^{\prime}$, such
that the irreducible components of $\mathfrak{Z}_{X^{\prime}}$ are in
bijection with the irreducible components of $\mathfrak{Z}_{\overline{\eta}}$.
Let $\mathfrak{Y}_{1},,...,\mathfrak{Y}_{n}$ be the irreducible components of
$\mathfrak{Z}_{X^{\prime}}$.

It follows that every $\mathfrak{Y}_{k,\eta^{\prime}}$ is a formal subtorus of
$\mathcal{E}_{\mathbf{I},\eta^{\prime}}\otimes_{\mathbb{Z}_{p}}\mathbb{G}_{m}^{\wedge}$.
Taking cocharacter lattices, we see that $\mathfrak{Y}_{k,\eta^{\prime}}$
gives rise to a saturated lisse subsheaf
$\mathcal{H}_{k,\eta^{\prime}}\subseteq\mathcal{E}_{\mathbf{I},\eta^{\prime}}$.
Since $X^{\prime}$ is smooth, there is a surjection
$\pi_{1}(\eta^{\prime},\overline{\eta})\rightarrow\pi_{1}(X^{\prime},\overline{\eta})$,
so we can spread out $\mathcal{H}_{k,\eta^{\prime}}$ to a saturated lisse
subsheaf $\mathcal{H}_{k}\subseteq\mathcal{E}_{\mathbf{I},X^{\prime}}$. Since
$\mathcal{H}_{k,\eta^{\prime}}\otimes\mathbb{G}_{m}^{\wedge}=\mathfrak{Y}_{k,\eta^{\prime}}$,
by further shrinking $X^{\prime}$, we can assume that
$\mathcal{H}_{k}\otimes\mathbb{G}_{m}^{\wedge}=\mathfrak{Y}_{k}$. This
finishes the proof. $\hfill\square$

######  Proposition 8.3.

There is an étale open dense subset $X^{\prime}$ of $X$, such that for every
$Z\in\bm{A}_{X^{\prime}}$, there exists a saturated arithmetic lisse subsheaf
$\mathcal{F}[Z]\subseteq\mathcal{E}_{\mathbf{I},X^{\prime}}$, such that:

  1. (1)

$\mathcal{F}_{Z}$ is an arithmetic lisse subsheaf of the restriction of
$\mathcal{F}[Z]$ to $Z$,

  2. (2)

$\mathcal{F}[Z]\otimes\mathbb{G}_{m}^{\wedge}\subseteq(X^{\prime}\times
X)^{/\Delta}$ (here $\Delta:X^{\prime}\rightarrow X$ is the étale morphism),

  3. (3)

if the projection of $Z$ to $\mathscr{S}_{\mathbf{J},\mathbb{F}}$ is a single
point, then
$\mathcal{F}[Z]\subseteq\mathcal{E}_{\mathbf{I}-\mathbf{J},X^{\prime}}$.

###### Proof.

By Lemma 8.2, there is a finite étale cover $X^{\prime}$ of $X$, together with
geometric $p$-adic lisse sheaves $\\{\mathcal{H}_{k}\\}_{k=1}^{n}$ over
$X^{\prime}$, such that the irreducible components of
$\mathfrak{Z}_{X^{\prime}}:=\mathfrak{Z}\times_{X}X^{\prime}$ are exactly
$\\{\mathcal{H}_{1}\otimes\mathbb{G}_{m}^{\wedge}\\}_{k=1}^{n}$. Since
$\mathfrak{Z}\subseteq(X\times X)^{/\Delta}$, we have

(8.2.3)
$\mathcal{H}_{k}\otimes\mathbb{G}_{m}^{\wedge}\subseteq\mathfrak{Z}_{X^{\prime}}\subseteq(X^{\prime}\times
X)^{/\Delta},\,\,1\leq k\leq m.$

For a $Z\in\bm{A}_{X^{\prime}}$, there exists a finite field $\mathbb{F}_{q}$,
together with $\mathbb{F}_{q}$-models $Z_{0}$ and $X_{0}^{\prime}$ of $Z$ and
$X^{\prime}$, such that $Z_{0}\subseteq X_{0}^{\prime}$ is an immersion,
$Z_{0}(\mathbb{F}_{q})\neq\emptyset$, and $\mathcal{F}_{Z}$ resp.
$\mathcal{E}_{\mathbf{I}}$ is the base change to $Z$ resp. $X^{\prime}$ of an
arithmetic lisse sheaf over $Z_{0}$ resp. $X^{\prime}_{0}$. Pick a point
$x_{0}\in Z_{0}(\mathbb{F}_{q})$ and a point $x\in Z_{0}(\mathbb{F})$ mapping
to $x_{0}$. We see that $\mathcal{F}_{Z,x}$ is invariant under
$\operatorname{Gal}(x|x_{0})$. Let $\mathcal{F}[Z]_{x}$ be the saturated
submodule of $\mathcal{E}_{\mathbf{I},x}$ generated by
$\\{g\cdot\mathcal{F}_{Z,x}|g\in\pi_{1}(X^{\prime},x)\\}$. Since
$\pi_{1}(X_{0}^{\prime},x)=\pi_{1}(X^{\prime},x)\rtimes\operatorname{Gal}(x|x_{0})$
and $\mathcal{F}[Z]_{x}$ is invariant under both $\operatorname{Gal}(x|x_{0})$
and $\pi_{1}(X^{\prime},x)$, it is invariant under
$\pi_{1}(X_{0}^{\prime},x)$. It is then easy to check that
$\mathcal{F}[Z]_{x}$ is a continuous
$\pi_{1}(X_{0}^{\prime},x)$-representation. Therefore $\mathcal{F}[Z]_{x}$
gives rise to a saturated arithmetic lisse subsheaf
$\mathcal{F}[Z]\subseteq\mathcal{E}_{\mathbf{I},X^{\prime}}$. By construction,
$\mathcal{F}[Z]$ satisfies (1).

Since the irreducible components of $\mathfrak{Z}_{X^{\prime}}$ are in
bijection with the irreducible components of $\mathfrak{Z}_{\eta^{\prime}}$,
$\mathcal{F}_{Z}$ is contained in the restriction to $Z$ of one of the
geometric lisse sheaves $\mathcal{H}_{1},\mathcal{H}_{2},...,\mathcal{H}_{m}$.
Say, it is contained in $\mathcal{H}_{1,Z}$. It follows from definition that
$\mathcal{F}[Z]\subseteq\mathcal{H}_{1}$. But then, (8.2.3) implies that
$\mathcal{F}[Z]_{x}\otimes\mathbb{G}_{m}^{\wedge}\subseteq(X^{\prime}\times
X)^{/\Delta}$. So we have (2).

If the projection of $Z$ to $\mathscr{S}_{\mathbf{J},\mathbb{F}}$ is a single
point, then $\mathcal{F}[Z]\subseteq\mathcal{E}_{\mathbf{I}-\mathbf{J},Z}$. As
a result, for the base point $x\in Z_{0}(\mathbb{F})$ picked above and any
$g\in\pi_{1}(X^{\prime},x)$, we have $g\cdot\mathcal{F}[Z]_{x}\subseteq
g\cdot\mathcal{E}_{\mathbf{I}-\mathbf{J},x}\subseteq\mathcal{E}_{\mathbf{I}-\mathbf{J},x}$.
It follows that
$\mathcal{F}[Z]\subseteq\mathcal{E}_{\mathbf{I}-\mathbf{J},X^{\prime}}$. This
implies (3).$\hfill\square$

###  8.3. The case of GSpin Shimura varieties

In this section we prove mod $p$ André–Oort conjecture for GSpin Shimura
varieties, which is technically easier than the case of modular curves. Note
that we only need to prove that $X$, as a subvariety containing a Zariski
dense collection of special subvarieties, is itself special.
Proof of Theorem 4.13 for GSpin Shimura varieties. Let $X^{\prime}$ be an
étale open subset of $X$ that satisfies Proposition 8.3. We can assume that
$G(\mathbb{L}_{p,X^{\prime}},x)$ is connected for any $x\in
X^{\prime}(\mathbb{F})$. Let $Z\in\bm{A}_{X^{\prime}}$ and $x\in
X^{\prime}(\mathbb{F})$. Recall that
$G(\mathbb{D}(\Psi_{X^{\prime}}^{{\text{\'{e}t}}}),x)$ acts on
$U_{\operatorname{GL},\overline{\mu}}\simeq\omega_{x}(\mathbb{D}(\Psi^{{\text{\'{e}t}}}))$
(6.4.1). The arithmetic lisse sheaf $\mathcal{F}[Z]$ of Proposition 8.3
corresponds to a
$G(\mathbb{D}(\Psi_{X^{\prime}}^{{\text{\'{e}t}}}),x)$-subrepresentation of
$\omega_{x}(\mathbb{D}(\Psi^{{\text{\'{e}t}}}))$. Let
$f:X^{\prime}\rightarrow\mathscr{S}_{\mathbb{F}}$ be the composition of the
étale morphism $X^{\prime}\rightarrow X$ and the immersion
$X\rightarrow\mathscr{S}_{\mathbb{F}}$, and let $\mathscr{T}_{f,x}$ and
$T_{f,x}$ be objects defined in §6.1. We have by Proposition 8.3(2),

(8.3.1)
$\mathcal{F}[Z]_{x}\otimes\mathbb{G}_{m}^{\wedge}\subseteq{X^{\prime}}^{/x}\subseteq\mathscr{T}_{f,x}.$

This soon implies that
$\mathcal{F}[Z]_{x}\otimes\mathbb{Q}_{p}\subseteq{T}_{f,x}$. By Remark 6.9 (or
by Theorem 4.12 and Proposition 7.3), the projection of
$G(\mathbb{D}(\Psi_{X^{\prime}}^{{\text{\'{e}t}}}),x)$ to
$\operatorname{GL}(T_{f,x})$ is either $\operatorname{SO}(T_{f,x})$, or
$\operatorname{U}(\varphi_{f,x,0})$, depending on whether $T_{f,x}$ is
nondegenerate or totally isotropic.

We now deduce from these facts that $\mathscr{T}_{f,x}=X^{\prime/x}$. Note
that the case $\dim X^{\prime}=1$ is trivial. So, in the following we assume
$\dim X^{\prime}\geq 2$. If $\dim{T}_{f,x}=2$, then it follows from dimension
reason that $\mathscr{T}_{f,x}=X^{\prime/x}$. If $\dim T_{f,x}\geq 3$, then
the action of $\operatorname{SO}(T_{f,x})$ or
$\operatorname{U}(\varphi_{f,x,0})$ on $T_{f,x}$ is irreducible. This forces
$\mathcal{F}[Z]_{x}\otimes\mathbb{Q}_{p}=T_{f,x}$. As a result, (8.3.1) is a
equality and $\mathscr{T}_{f,x}=X^{\prime/x}$. Since $X^{/x}=X^{\prime/x}$,
Theorem 4.10 implies that $X$ is special. $\hfill\square$

###  8.4. The case of products of modular curves

We now treat mod $p$ André-Oort conjecture for products of modular curves. We
begin by a simple case:

######  Lemma 8.4.

If $\bm{A}$ is simple, then $\overline{X}$ is the product of a special
subvariety in $\mathscr{S}_{\mathbf{I}_{\bm{A}},\mathbb{F}}$ with a subvariety
in $\mathscr{S}_{\mathbf{I}-\mathbf{I}_{\bm{A}},\mathbb{F}}$.

###### Proof.

Let $X^{\prime}$ be an étale open subset of $X$ that satisfies Proposition
8.3. We can assume that $G(\mathbb{L}_{p,X^{\prime}},x)$ is connected for any
$x\in X^{\prime}(\mathbb{F})$. Pick $x\in X^{\prime}(\mathbb{F})$. Let
$f:X^{\prime}\rightarrow\mathscr{S}_{\mathbb{F}}$ be the composition of the
étale morphism $X^{\prime}\rightarrow X$ and the immersion
$X\rightarrow\mathscr{S}_{\mathbb{F}}$, and let $\mathscr{T}_{f,x}$ and
$T_{f,x}$ be objects defined in §6.1. Let $Z\in\bm{A}_{X^{\prime}}$ and
consider the arithmetic lisse sheaf $\mathcal{F}[Z]$ of Proposition 8.3, we
then have

(8.4.1)
$\displaystyle\mathcal{F}[Z]_{x}\otimes\mathbb{G}_{m}^{\wedge}\subseteq
X^{\prime/x}\subseteq\mathscr{T}_{f,x}.$ (8.4.2)
$\displaystyle\mathcal{F}[Z]_{x}\subseteq\mathcal{E}_{\mathbf{I}_{\bm{A}},X^{\prime}}.$

Recall that for $\mathbf{J}\subseteq\mathbf{I}$, $f_{\mathbf{J}}$ is the
composition of $f$ with the projection
$\mathscr{S}_{\mathbf{I},\mathbb{F}}\rightarrow\mathscr{S}_{\mathbf{J},\mathbb{F}}$.
Theorem 4.10 implies that there is a special subvariety
$\mathcal{X}_{f_{\mathbf{I}_{\bm{A}}}}$ of $\mathcal{S}_{\mathbf{I}_{\bm{A}}}$
such that
$\mathscr{X}_{f_{\mathbf{I}_{\bm{A}}},\mathbb{F},\operatorname{red}}^{/x}=\mathcal{T}_{f_{\mathbf{I}_{\bm{A}}},x}$.
Let $Y$ be the Zariski closure of the projections of the elements in $\bm{A}$
to $\mathscr{S}_{f_{\mathbf{I}-\mathbf{I}_{\bm{A}}},\mathbb{F}}$, and let
$\mathscr{X}_{f_{\mathbf{I}_{\bm{A}}},\mathbb{F},\operatorname{red}}^{+}$ be
the unique irreducible component of
$\mathscr{X}_{f_{\mathbf{I}_{\bm{A}}},\mathbb{F},\operatorname{red}}$ passing
through $x$. Then

(8.4.3)
$\overline{X}\subseteq\mathscr{X}_{f_{\mathbf{I}_{\bm{A}}},\mathbb{F},\operatorname{red}}^{+}\times
Y.$

By definition, the projection of $\overline{X}$ to $Y$ is surjective. It
suffices to show that this is actually an equality.

Recall that the torus
$G(\operatorname{gr}\mathbb{D}(\Psi_{\mathbf{I},X^{\prime}}),x)=G(\mathbb{D}(\operatorname{Br}_{\mathbf{I},X^{\prime}}),x)$
acts on $U_{\operatorname{GL}^{\prime},\overline{\mu}_{\mathbf{I}}}$ by
Proposition 6.3. Since
$G(\mathbb{D}(\operatorname{Br}_{\mathbf{I}_{\bm{A}},X^{\prime}}),x)$ fixes
the subspace $\mathcal{F}[Z]_{x}\otimes\mathbb{Q}_{p}$, and the projection of
$\mathcal{F}[Z]_{x}\otimes\mathbb{Q}_{p}$ to each
$U_{\operatorname{GL},\overline{\mu}_{i}}$ for $i\in\mathbf{I}_{\bm{A}}$ is
surjective, we have
$\operatorname{rk}G(\mathbb{D}(\operatorname{Br}_{\mathbf{I}_{\bm{A}},X^{\prime}}),x)\leq\operatorname{rk}\mathcal{F}[Z]_{x}$.
On the other hand, from Theorem 4.12 and Proposition 7.1, we see that
$\operatorname{rk}G(\mathbb{D}(\operatorname{Br}_{\mathbf{I}_{\bm{A}},X^{\prime}}),x)=\dim
T_{f_{\mathbf{I}_{\bm{A}}},x}$. Finally, it follows from (8.4.1) and (8.4.2)
that $\mathcal{F}[Z]_{x}\subseteq\mathcal{T}_{f_{\mathbf{I}_{\bm{A}}},x}$.
Combining these, we have

(8.4.4)
$\displaystyle\mathcal{F}[Z]_{x}=\mathcal{T}_{f_{\mathbf{I}_{\bm{A}}},x},$
(8.4.5)
$\displaystyle\mathcal{T}_{f,x}=\mathcal{F}[Z]_{x}\oplus\mathcal{T}_{f_{\mathbf{I}-\mathbf{I}_{\bm{A}},x}}.$

Let $y$ be the projection of $x$ to
$\mathscr{S}_{\mathbf{I}-\mathbf{I}_{\bm{A}},\mathbb{F}}$. The results
(8.4.4),(8.4.5) and (8.4.1) show that ${X}_{y}^{/x}$, the completion of the
fiber $X_{y}$ at $x$, is also $\mathcal{T}_{f_{\mathbf{I}_{\bm{A}}},x}$.
Therefore

$\dim\overline{X}_{y}=\operatorname{rk}\mathcal{T}_{f_{\mathbf{I}_{\bm{A}}},x}=\dim\mathscr{X}_{f_{\mathbf{I}_{\bm{A}}},\mathbb{F},\operatorname{red}}^{+}.$

This implies that

(8.4.6)
$\overline{X}_{y}=\mathscr{X}_{f_{\mathbf{I}_{\bm{A}}},\mathbb{F},\operatorname{red}}^{+}\times\\{y\\}.$

Let $x^{\prime}$ be a point of $X^{\prime}(\mathbb{F})$. The same argument as
above shows that, if we replace $x$ by $x^{\prime}$ in (8.4.1) (8.4.2),
(8.4.4) and (8.4.5), they remain true. Write $y^{\prime}$ for the projection
of $x^{\prime}$ to $\mathscr{S}_{\mathbf{I}-\mathbf{I}_{\bm{A}},\mathbb{F}}$,
we see that

$\dim\overline{X}_{y^{\prime}}=\operatorname{rk}\mathcal{T}_{f_{\mathbf{I}_{\bm{A}}},x^{\prime}}=\operatorname{rk}\mathcal{T}_{f_{\mathbf{I}_{\bm{A}}},x}=\dim\mathscr{X}_{f_{\mathbf{I}_{\bm{A}}},\mathbb{F},\operatorname{red}}^{+},$

where the equality in the middle follows from (8.4.5) or Corollary 6.4. As a
result, (8.4.6) holds when we replace $y$ by $y^{\prime}$. Let $x^{\prime}$
run over $X^{\prime}(\mathbb{F})$, we see that (8.4.6) holds for Zariski dense
$y^{\prime}\in Y(\mathbb{F})$. As a result, (8.4.3) is an equality.
$\hfill\square$
Proof of Theorem 4.13 for products of modular curves. We do induction on
$\\#\mathbf{I}$. If $\\#\mathbf{I}=1$ there is nothing to prove. Suppose the
theorem is true for $\\#\mathbf{I}<n$. Let $\\#\mathbf{I}=n$. There is always
a nonempty subset $\mathbf{J}\subseteq\mathbf{I}_{\bm{A}}$, and a Zariski
dense sub-collection $\bm{B}\subseteq\bm{A}$, such that each special
subvariety in $\bm{B}$ has positive dimensional projection to
$\mathscr{S}_{i,\mathbb{F}}$, where $i\in\mathbf{J}$, while having zero
dimensional projection to $\mathscr{S}_{\mathbf{I}-\mathbf{J},\mathbb{F}}$. In
other words, $\bm{B}$ is simple and $\mathbf{I}_{\bm{B}}=\mathbf{J}$. Lemma
8.4 then implies that $\overline{X}$ is the product of a special subvariety of
$\mathscr{S}_{\mathbf{I}_{\bm{B}},\mathbb{F}}$ and a subvariety
$Y\subseteq\mathscr{S}_{\mathbf{I}-\mathbf{I}_{\bm{B}},\mathbb{F}}$. If
$\mathbf{I}_{\bm{B}}=\mathbf{I}_{\bm{A}}$, then we are already done. If
$\mathbf{I}_{\bm{B}}\subsetneq\mathbf{I}_{\bm{A}}$, we can project all special
subvarieties in $\bm{A}$ down to
$\mathscr{S}_{\mathbf{I}-\mathbf{I}_{\bm{B}},\mathbb{F}}$. The images form a
collection of special subvarieties of $Y$, which we denote by
$\overline{\bm{A}}$. Then
$\mathbf{I}_{\overline{\bm{A}}}=\mathbf{I}_{\bm{A}}-\mathbf{I}_{\bm{B}}>0$. By
induction hypothesis, $Y$ is the product of a special subvariety of
$\mathscr{S}_{\mathbf{I}_{A}-\mathbf{I}_{B},\mathbb{F}}$ and a subvariety
$Y^{\prime}\subseteq\mathscr{S}_{\mathbf{I}-\mathbf{I}_{A},\mathbb{F}}$. As a
result $\overline{X}$ is a product of a special subvariety of
$\mathscr{S}_{\mathbf{I}_{A},\mathbb{F}}$ and a subvariety
$Y^{\prime}\subseteq\mathscr{S}_{\mathbf{I}-\mathbf{I}_{A},\mathbb{F}}$.
Therefore the theorem holds for $\\#\mathbf{I}=n$. $\hfill\square$

## References