######  Definition (Arithmetic models for $f$).

Let $f:X\rightarrow S$ be projective smooth morphism of $K$-varieties with $S$
a curve, as above. We say $f$ has a good arithmetic model over
$\mathcal{O}_{K}$ if there exists a good arithmetic model for the triple
$(S^{\prime},S,\Sigma_{S})$ over $\mathcal{O}_{K}$, such that for each $L/K$
finite and each $v\in\Sigma_{L,f}$ we have that

  1. 1.

there exists an $\mathcal{O}_{M}$-scheme $\mathcal{X}_{v}$ such that
$(\mathcal{X}_{v})_{L^{ur}_{v}}=X_{L^{ur}_{v}}$, and

  2. 2.

there exists a smooth proper morphism
$\tilde{f}_{v}:\mathcal{X}_{v}\rightarrow\mathcal{C}_{\mathcal{O}_{M}}$ of
$\mathcal{O}_{M}$-schemes whose generic fiber is the morphism $f_{v}$(see
below).

Let us fix a finite extension $L/K$ and a place $v\in\Sigma_{L,f}$. Assume the
existence of such a pair $(\mathcal{X}_{v},\tilde{f}_{v})$ and let $f_{v}$ be
the base change of $f$ via the morphism
$\operatorname{Spec}L^{ur}_{v}\rightarrow\operatorname{Spec}K$. We then define
$\widetilde{\mathbb{V}}_{l}$ to be the $l$-adic étale sheaf on
$\mathcal{C}_{\mathcal{O}_{M}}$ given by

$R^{n}(\tilde{f}_{v})_{*}(\mathbb{Z}_{l})=\underset{\leftarrow}{\lim}\text{
}R^{n}(\tilde{f}_{v})_{*}(\mathbb{Z}/l^{m}\mathbb{Z})$.

######  Lemma 11.2.

Let $f:X\rightarrow S$ over $K$ be as above. Assume that there exists a good
arithmetic model $(\mathcal{X},\tilde{f})$ for the morphism $f$ over
$\mathcal{O}_{K}$. Then the $l$-adic étale sheaf
$\widetilde{\mathbb{V}}_{l}:=R^{n}(\tilde{f}_{v})_{*}(\mathbb{Z}_{l})$ on
$\mathcal{C}_{\mathcal{O}_{M}}$ is an $l$-adic étale local system that extends
the $l$-adic étale local system $R^{n}(f_{v})_{*}(\mathbb{Z}_{l})$ on $S_{M}$.

###### Proof.

From the proper base change theorem for lisse $l$-adic sheaves we have that
$R^{n}(\tilde{f}_{v})_{*}(\mathbb{Z}_{l})$ is an extension of the sheaf
$R^{n}(f_{v})_{*}(\mathbb{Z}_{l})$.

The fact that $\widetilde{\mathbb{V}}_{l}$ is also an $l$-adic étale local
system on $\mathcal{C}_{\mathcal{O}_{M}}$ follows from the smooth proper base
change theorem for lisse $l$-adic sheaves. ∎

###  §12 A comparison of monodromy operators

The crux of our argument rests on a comparison between the action of inertia
and that of the local monodromy around the degeneration $s_{0}$. To be able to
establish this connection we need to assume the existence of the
aforementioned good arithmetic models.

####  §12.1 Review on the monodromy weight filtration

We present a short review of the two monodromy operators we want to compare.

##### The local monodromy operator

Let $f:X\rightarrow S$ be a G-admissible variation defined over the number
field $K$ and, as usual, $\Delta^{*}\subset S^{an}_{\mathbb{C}}$ a small disk,
relative to a fixed embedding $K\hookrightarrow\mathbb{C}$, centered at the
point of degeneration $s_{0}\in S^{\prime}(K)$.

We have by our assumptions that the local monodromy around $s_{0}$ on
$\mathbb{V}_{z}:=H^{n}(X^{an}_{z},\mathbb{Q})$ is in fact unipotent for any
$z\in\Delta^{*}$. We denote by $N_{z}:\mathbb{V}_{z}\rightarrow\mathbb{V}_{z}$
the corresponding nilpotent endomorphism that is the logarithm of the
unipotent endomorphism that defines this action.

We know that $N_{z}$ has degree of nilpotency $\leq n+1$ and hence, by $(6.4)$
of [Sch73], we get an ascending filtration

$0\subset W^{B}_{0}\subset W^{B}_{1}\subset\ldots\subset W^{B}_{2n-1}\subset
W^{B}_{2n}=\mathbb{V}_{z}.$ (57)

The filtration $W^{B}_{\bullet}$ is called the weight monodromy filtration. By
Lemma $(6.4)$ of [Sch73], we know that this is the unique ascending filtration
characterized by the properties

  1. 1.

$N_{z}(W^{B}_{i})\subset W^{B}_{i-2}$ for all $i$, and

  2. 2.

for all $1\leq i\leq n$ the endomorphism $N_{z}^{i}$ defines an isomorphism
$\operatorname{Gr}_{n+i}^{W^{B}_{\bullet}}\rightarrow\operatorname{Gr}_{n-i}^{W^{B}_{\bullet}}$.

We define from now on
$h^{B}_{i}:=\dim_{\mathbb{Q}}\operatorname{Gr}_{i}^{W^{B}_{\bullet}}$ and let
$h^{B}_{\max}:=\max\\{h_{j}:0\leq j\leq 2n\\}$. We note that the number $h$,
the dimension of the local system
$M_{0}R_{n}(f^{an}_{\mathbb{C}})_{*}\mathbb{Q}_{X^{an}_{\mathbb{C}}}|_{\Delta^{*}}$,
is equal to $h^{B}_{0}$.

We note that while the matrix $N_{z}$ depends on the point $z$ the numbers
$h^{B}_{j}$ do not.

##### The monodromy operator from inertia

Let $s\in S(L)$ with $L/K$ finite, be a point of the G-admissible variation of
$\mathbb{Q}$-HS given by the morphism $f:X\rightarrow S$.

Let $v\in\Sigma_{L,f}$ be a finite place of $L$. We then have the nilpotent
endomorphism $N_{v}$, given by the action of some subgroup of finite index of
the inertia group $I_{L_{v}}$, acting on the étale cohomology group
$H^{n}_{\text{\'{e}t}}(\bar{X}_{s,v},\mathbb{Q}_{l})$, where $l\neq p(v)$.

Let $W^{\text{\'{e}t }}_{\bullet}$ be the monodromy filtration defined on
$H^{n}_{\text{\'{e}t}}(\bar{X}_{s,v},\mathbb{Q}_{l})$ via the action of the
above $N_{v}$. We also let
$h^{\text{\'{e}t}}_{i}:=\dim_{\mathbb{Q}_{l}}\operatorname{Gr}_{i}^{W^{\text{\'{e}t}}_{\bullet}}$.

####  §12.2 $v$-adic proximity and comparison of operators

Consider $f:X\rightarrow S$ some G-admissible variation of Hodge structures.
Throughout this subsection we consider $s\in S(L)$ a fixed point, where $L/K$
is some finite extension, and $v\in\Sigma_{L,f}$ some fixed finite place of
$L$. We let $p=p(v)$ be the characteristic of the finite field
$\mathcal{O}_{L_{v}}/m_{v}$ and we fix some $l\neq p$.

We assume that a good arithmetic model, in the sense of §11,
$(\tilde{S},\mathcal{C},D)$ over $\mathcal{O}_{K}$ exists for $S$ and that a
good arithmetic model for the morphism $f$ exists over $\mathcal{O}_{K}$ with
respect to this triple.

Motivated by the exposition in §$4$ of [PST+21] we prove the following lemma.

######  Lemma 12.1.

Under the above assumptions, we have that if $s$ is $v$-adically close to
$s_{0}$ we have $h^{\text{\'{e}t}}_{i}=h^{B}_{i+n}$ for all $i$, where
$h^{B}_{j}$ and $h^{\text{\'{e}t}}_{j}$ are as in §12.1.

###### Proof.

We start with a bit of notation following the exposition in §$4$ of [PST+21].
First of all, for any group $G$ we will denote by $G_{(l)}$ its maximal
pro-$l$ quotient. As always we fix a punctured unit disk $\Delta^{*}\subset
S^{an}$ centered at $s_{0}$.

We let $y$ and $y_{0}$ be the sections of the arithmetic pencil
$\tilde{S}\times_{\mathcal{O}_{K}}\mathcal{O}_{L}\rightarrow\mathcal{O}_{L}$
whose generic fibers are the points $s$ and $s_{0}$ respectively. By 11.1 we
may assume without loss of generality that
$y_{\bar{\mathbb{F}}_{p}}=y_{0,\bar{\mathbb{F}}_{p}}\in
D(\bar{\mathbb{F}}_{p})$. We also let $t$ be such that it cuts out
$\tilde{s}_{0}\subset D$ in $\tilde{S}$. In particular, for this choice we
get, by our assumption that for all $s_{i}\in\Sigma_{S}$ and all primes $q$ we
have that
$(\tilde{s}_{i})_{\bar{\mathbb{F}}_{q}}\neq(\tilde{s}_{0})_{\bar{\mathbb{F}}_{q}}$
when $i\neq 0$, that

$\widehat{\mathcal{O}}_{\tilde{S}_{v},y_{0,\bar{\mathbb{F}}_{p}}}\cong\mathcal{O}_{L_{v}}[[t]]$.

We have, by our assumptions, that the local monodromy $\pi_{1}(\Delta^{*},z)$
acts unipotently on the fiber $(R^{n}f^{an}_{*}\mathbb{Z})_{z}$ for all
$z\in\Delta^{*}$. Letting $\gamma_{0}\in\pi_{1}(\Delta^{*},z)$ be a generator,
we denote by $U_{0}\in\operatorname{GL}(H^{n}(X^{an}_{z},\mathbb{Z}))$ the
unipotent endomorphism it maps to via the local monodromy representation. We
define $N_{z}$ to be the nilpotent logarithm of $U_{0}$.

By Grothendieck’s quasi-unipotent action theorem we know that there exists a
finite extension $F/L_{v}$ such that the Galois representation
$\rho_{l}:G_{L_{v}}\rightarrow\operatorname{GL}(H^{n}_{\acute{e}t}(\bar{X}_{s,v},\mathbb{Z}_{l}))$
restricts to a unipotent representation of the inertia group $I_{F}$. In other
words $\rho_{l}|_{I_{F}}$ is given by $\sigma\mapsto\exp(N_{v}t_{l}(\sigma))$
with $N_{v}$ nilpotent with degree of nilpotency $\leq n+1$.

We let $M:=F^{ur}$ and $\mathcal{O}_{M}$ be its ring of integers. We consider
the rings $R_{1}:=\mathcal{O}_{M}[[t]]\big{[}\frac{1}{t}\big{]}$,
$R_{2}:=M[[t]]\big{[}\frac{1}{t}\big{]}$, and
$R_{3}:=\mathbb{C}[[t]]\big{[}\frac{1}{t}\big{]}$. We note that, after fixing
an inclusion $\mathcal{O}_{M}\hookrightarrow\mathbb{C}$, these define the
following commutative diagram

${\operatorname{Spec}R_{3}}$${\operatorname{Spec}R_{2}}$${\operatorname{Spec}R_{1}}$${S_{\mathbb{C}}}$${S_{M}}$${\mathcal{C}_{\mathcal{O}_{M}}}$$\scriptstyle{g_{3}}$$\scriptstyle{g_{2}}$$\scriptstyle{g_{1}}$

with $g_{1}$, $g_{2}$ and $g_{3}$ being étale. In fact, note that the
$\operatorname{Spec}R_{i}$ are étale neighborhoods of the geometric point
$\bar{y}_{M}=\bar{s}$. Originally, $\operatorname{Spec}{R_{1}}$ is an étale
neighborhood of $\bar{y}_{M}$ in $\tilde{S}_{M}$ but it will not intersect the
divisor $D$, by construction and our assumption that for all
$s_{j},\>s_{i}\in\Sigma_{S}$ and all primes $q$ we have that
$(\tilde{s}_{i})_{\bar{\mathbb{F}}_{q}}\neq(\tilde{s}_{0})_{\bar{\mathbb{F}}_{q}}$
when $i\neq 0$.

Since $y_{\bar{\mathbb{F}}_{p}}=y_{0,\bar{\mathbb{F}}_{p}}$, we get that $t$
pulls back to an element $t_{M}$ of the maximal ideal of $\mathcal{O}_{M}$ via
$\operatorname{Spec}(\mathcal{O}_{M})\overset{y_{\mathcal{O}_{M}}}{\rightarrow}\tilde{S}_{\mathcal{O}_{M}}$.
From this we get a morphism
$f_{1}:\mathcal{O}_{M}[[t]]\big{[}\frac{1}{t}\big{]}\rightarrow M$, since
$v_{M}(t_{M})>0$.

By Lemma $4.2$ of [PST+21] we get an induced map

$\phi_{1}:G_{M}^{(p)}\rightarrow\pi_{1}^{\acute{e}t}(\operatorname{Spec}R_{1},\bar{y}_{M})^{(p)}$,

between the prime-to-$p$ Galois groups, whose image is completely determined
by $v_{M}(t_{M})>0$. In particular we get a map

$\phi_{1,l}:(G_{M})_{(l)}\rightarrow\pi_{1}^{\acute{e}t}(\operatorname{Spec}R_{1},\bar{y}_{M})_{(l)}$.

With respect to the above embedding $\mathcal{O}_{M}\hookrightarrow\mathbb{C}$
we get a map $f_{2}:R_{1}\rightarrow R_{3}$ which induces an isomorphism of
prime-to-$p$ Galois groups, and hence of their maximal pro-$l$ quotients,
which we denote by

$\phi_{2,l}:\pi_{1}^{\acute{e}t}(\operatorname{Spec}R_{3},\bar{y}_{M})_{(l)}\rightarrow\pi_{1}^{\acute{e}t}(\operatorname{Spec}R_{1},\bar{y}_{M})_{(l)}$.

We also consider the composition
$\phi_{l}:=\phi_{2,l}^{-1}\circ\phi_{1,l}:(G_{M})_{(l)}\rightarrow\pi_{1}^{\acute{e}t}(\operatorname{Spec}R_{3},\bar{y}_{M})_{(l)}$.

Finally, we let $F:R_{3}\rightarrow R_{3}$ be the map defined by $t\mapsto
t^{v_{M}(t_{M})}$, by abuse of notation we also let
$F:\operatorname{Spec}(R_{3})\rightarrow\operatorname{Spec}(R_{3})$ be the
étale cover induced from $F$. Letting $\bar{y}_{1}$ be any geometric point in
the fiber of $\bar{y}_{M}=\bar{s}$ over $F$, we then get from $F$ an induced
morphism

$\psi_{l}:\pi_{1}^{\acute{e}t}(\operatorname{Spec}R_{3},\bar{y}_{1})_{(l)}\rightarrow\pi_{1}^{\acute{e}t}(\operatorname{Spec}R_{3},\bar{y}_{M})_{(l)}$.

We also note that by construction we have that $\psi_{l}$ has the same image
as $\phi_{l}$.

On $S_{M}$ we have the lisse $l$-adic sheaf
$\mathbb{V}_{l}:=(\underset{\leftarrow}{\lim}R^{n}(f_{v})_{*}(\mathbb{Z}/l^{i}\mathbb{Z}))$.
Its analytification $\mathbb{V}_{l}^{an}$ is nothing but the local system
$R^{n}f^{an}_{*}\mathbb{Z}_{X^{an}_{\mathbb{C}}}\otimes_{\mathbb{Z}}\mathbb{Z}_{l}$
on $S^{an}_{\mathbb{C}}$, which is the $l$-adic completion of the
$\mathbb{Z}$-local system that underlies the variation of $\mathbb{Z}$-HS we
are studying. The assumption that good arithmetic models for the morphism $f$
exist over $\mathcal{O}_{K}$ then implies, see 11.2, that we have that
$\mathbb{V}_{l}$ extends to a lisse $l$-adic sheaf
$\widetilde{\mathbb{V}}_{l}$ on $\mathcal{C}_{\mathcal{O}_{M}}$. We let
$\widetilde{\mathbb{V}}_{l,1}$ be the pullback of this sheaf via $g_{1}$. Note
that the pullback of $\mathbb{V}_{l}$ via the morphism
$\operatorname{Spec}(R_{2})\overset{g_{2}}{\rightarrow}S_{M}$, which we will
denote by $\mathbb{V}_{l,1}$, is nothing but the generic fiber of the lisse
$l$-adic sheaf $\widetilde{\mathbb{V}}_{l,1}$. By abuse of notation we also
denote by $\mathbb{V}_{l,1}$ the pullback of this last sheaf via the morphism
$\operatorname{Spec}(R_{3})\rightarrow\operatorname{Spec}(R_{2})$ above.
Finally, we denote by $\mathbb{V}_{l,2}$ the pullback of $\mathbb{V}_{l,1}$
via the map $F$.

For any $z\in\Delta^{*}$ we have the local monodromy representation given by
$\rho:\pi_{1}(\Delta^{*},z)\rightarrow\operatorname{GL}((\mathbb{V}_{l}^{an})_{z})$.
By our assumptions this action is unipotent given by $\gamma_{0}\mapsto
U_{0}$, where $\gamma_{0}$ a generator of $\pi_{1}(\Delta^{*},z)$. From the
tower of étale covers
$(\operatorname{Spec}R_{3})^{an}\overset{F^{an}}{\rightarrow}(\operatorname{Spec}R_{3})^{an}\overset{g_{3}^{an}}{\rightarrow}S^{an}_{\mathbb{C}}$
we get an inclusion
$\pi_{1}(\operatorname{Spec}(R_{3})^{an},{z}_{2})\hookrightarrow\pi_{1}(\operatorname{Spec}(R_{3})^{an},{z}_{1})\hookrightarrow\pi_{1}(\Delta^{*},z)$
where $z_{1}\in(g_{3}^{an})^{-1}(z)$ and $z_{2}\in(F^{an})^{-1}(z_{1})$.
Letting $\gamma_{i}\in\pi_{1}(\operatorname{Spec}(R_{3})^{an},{z}_{i})$ for
$i=1,2$ be generators of these groups we can identify these with
$\gamma_{0}^{a_{i}}$, where $a_{i}\in\mathbb{Z}$ and $a_{1}|a_{2}$. Without
loss of generality we may and do assume that $a_{i}>0$.

By construction we then have the following commutative diagrams for $i=1,2$

${\pi_{1}(\operatorname{Spec}(R_{3})^{an},z_{i})}$${\operatorname{GL}((\mathbb{V}_{l,i})^{an}_{z_{i}})}$${\pi_{1}(\Delta^{*},z)}$${\operatorname{GL}((\mathbb{V}_{l})_{z})}$$\scriptstyle{\rho_{i}}$$\scriptstyle{\rho}$

from which we see that $\gamma_{i}$ maps to $U_{i,z}:=U_{0}^{a_{i}}$. In
particular, the nilpotent logarithm $N_{i,z}$ of $U_{i,z}$ is $a_{i}N_{z}$.

Now, let $\gamma_{3}$ be a generator of the group
$\pi_{1}(\operatorname{Spec}(R_{3})^{an},\bar{y}_{M})\simeq\mathbb{Z}$.
Considering the representation
$\pi_{1}(\operatorname{Spec}(R_{3})^{an},\bar{y}_{M})\rightarrow\operatorname{GL}((\mathbb{V}_{l,1}^{an})_{\bar{y}_{M}})$
with $\gamma_{3}\mapsto U_{3}$. We get that $U_{3}$ is conjugate to $U_{1,z}$,
i.e. there exists some $P$ with $U_{3}=PU_{1,z}P^{-1}$. So that $U_{3}$ is
also unipotent and we may write $N_{3}:=PN_{1,z}P^{-1}$ for its nilpotent
logarithm.

Putting everything together we get that

$N_{3}={a_{1}}PN_{z}P^{-1}.$ (58)

We also have a commutative diagram

${\pi_{1}(\operatorname{Spec}(R_{3})^{an},\bar{y}_{M})}$${\operatorname{GL}((\mathbb{V}_{l,1})^{an}_{\bar{y}_{M}})}$${\pi_{1}^{\text{\'{e}t}}(\operatorname{Spec}(R_{3}),\bar{y}_{M})}$${\operatorname{GL}((\mathbb{V}_{l,1})_{\bar{y}_{M}})}$$\scriptstyle{\rho_{1}}$$\scriptstyle{\rho_{1}^{\text{\'{e}t}}}$

We note that
$\pi_{1}(\operatorname{Spec}(R_{3})^{an},\bar{y}_{M})\simeq\mathbb{Z}$ and
that it is dense in the étale fundamental group
$\pi_{1}^{\text{\'{e}t}}(\operatorname{Spec}R_{3},\bar{y}_{M})$. Thus, the
continuity of $\rho_{1}^{\acute{e}t}$ implies that the image of
$\rho_{1}^{\acute{e}t}$ is
$P_{1}U_{3}^{\mathbb{Z}_{l}}P_{1}^{-1}:=\\{P_{1}\exp(\alpha
N_{3})P_{1}^{-1}:\alpha\in\mathbb{Z}_{l}\\}$. This is a pro-$l$ group and we
therefore get that $\rho_{1}^{\acute{e}t}$ factors through
$\pi_{1}^{\text{\'{e}t}}(\operatorname{Spec}(R_{3}),\bar{y}_{M})_{(l)}$.

We know turn our attention to the action of inertia. We know that the
following diagram is commutative

${G_{M}}$${\pi_{1}^{\acute{e}t}(\operatorname{Spec}R_{3},\bar{y}_{M})}$${\operatorname{GL}(H^{n}_{\text{\'{e}t}}(\bar{X}_{s,v},\mathbb{Z}_{l})),}$$\scriptstyle{\rho_{l}}$$\scriptstyle{c}$

where $c$ is the homomorphism induced by the composition of the identification
of $(\mathbb{V}_{l,1})_{\bar{y}_{M}}\simeq
H^{n}_{\text{\'{e}t}}(\bar{X}_{s,v},\mathbb{Z}_{l})$ and the action of
$\pi_{1}^{\acute{e}t}(\operatorname{Spec}R_{3},\bar{y}_{M})$ on
$(\mathbb{V}_{l,1})_{\bar{y}_{M}}$.

By construction, the image of
$\rho_{l}:G_{M}\rightarrow\operatorname{GL}(H^{n}_{\text{\'{e}t}}(\bar{X}_{s,v},\mathbb{Z}_{l}))$
above is a unipotent group. This implies that $Im(\rho_{l})$ is in fact a
pro-$l$ group. In more detail, we have $Im(\rho_{l})=\\{\exp(\alpha
N_{v}):\alpha\in\mathbb{Z}_{l}\\}$ which is a pro-$l$ group.

From our earlier comments we get that the image of $c$ is a pro-$l$ group as
well. We thus have that the above diagram induces a commutative diagram

${(G_{M})_{(l)}}$${\pi_{1}^{\acute{e}t}(\operatorname{Spec}R_{3},\bar{y}_{M})_{(l)}}$${\operatorname{GL}(H^{n}_{\text{\'{e}t}}(\bar{X}_{s,v},\mathbb{Z}_{l})),}$$\scriptstyle{\phi_{l}}$$\scriptstyle{\bar{\rho}_{l}}$$\scriptstyle{c}$

where $\phi_{l}$ is as above.

In particular we get that $\exp(N_{v})\in Im(c)$. In more detail this implies
that $\exp(N_{v})=QP_{1}\exp(\alpha N_{3})P_{1}^{-1}Q^{-1}$ for some
$\alpha\in\mathbb{Z}_{l}$. Taking logarithms and combining this with (58) this
translates to

$N_{v}=\alpha a_{1}P_{0}N_{z}P_{0}^{-1},$ (59)

where $P_{0}\in\operatorname{GL}_{\mu}(Z_{l})$, upon considering
$\mathbb{Z}_{l}$-bases of the fibers of the various lisse $l$-adic sheaves.

Note that $\alpha a_{1}\neq 0$ since $Im(\phi_{l})=Im(\psi_{l})$. Indeed, in
this case we would have that $Im(\psi_{l})=0$ and, hence, that the monodromy
representation
$\rho_{2}:\pi_{1}(\operatorname{Spec}(R_{3}),z_{2})\rightarrow\operatorname{GL}((\mathbb{V}_{l,2}^{an})_{z_{2}})$
is trivial. This would imply that $N_{2,z}=0$ and hence that $N_{z}=0$ which
is impossible, since we have a non-isotrivial singularity at $s_{0}$ by
assumption of having a G-admissible variation

Combining this with 12.2, our result follows. ∎

######  Lemma 12.2.

Let $k$ be a field with $char(k)=0$ and let $N_{1}$ and $N_{2}$ be two
nilpotent elements of of $\operatorname{End}(V)$, where $V$ is some
$\mu$-dimensional $k$-vector space. Assume that $N_{1}$ is conjugate to
$aN_{2}$ for some $a\in k^{\times}$. Let $W^{1}_{\bullet}$ and
$W^{2}_{\bullet}$ be the ascending filtrations of $V$ associated to $N_{1}$
and $N_{2}$ respectively, and let
$h^{j}_{i}:=\dim_{k}\operatorname{Gr}^{W^{j}_{\bullet}}_{i}$ for $j=1$, $2$.
Then $h^{1}_{i}=h^{2}_{i}$ for all $i$.

###### Proof.

The easiest way to see this is via the explicit formulas for the filtration
$W_{\bullet}$ associated to a nilpotent endomorphism $N$, see Remark $(2.3)$
of [SZ85]. The fact that $N_{1}=PaN_{1}P^{-1}$ for some
$P\in\operatorname{GL}(V)$ and some $a\in k^{\times}$ defines, using the
aforementioned formulas, isomorphisms
$\operatorname{Gr}^{W^{1}_{\bullet}}_{i}\simeq\operatorname{Gr}^{W^{2}_{\bullet}}_{i}$
for all $i$ and the result follows.

∎

###  §13 Conditions for $v$-adic proximity

Motivated by 10.2 we make the following definition.

######  Definition.

Let $f:X\rightarrow S$ be a G-admissible variation defined over the number
field $K$. Let $s\in S(\bar{\mathbb{Q}})$ be a point of the variation whose
associated algebra of Hodge endomorphism is
$D_{s}=M_{m_{1}}(D_{1})\oplus\ldots\oplus M_{m_{r}}(D_{r})$. Let us also
define $F_{i}:=Z(D_{i})$, $d_{i}^{2}:=[D_{i}:F_{i}]$, and
$f_{i}:=[F_{i}:\mathbb{Q}]$.

We say that the point $s$ satisfies condition $\star$ if it satisfies any of
the following

  1. $\star_{1}$

if we have that

$\displaystyle\sum_{i=1}^{r}m_{i}f_{i}>\mu-\dim_{\mathbb{Q}}Im(N^{B}).$
($\star_{1}$)

  2. $\star_{2}$

if there exists $i$ such that for the set

$\Pi^{i}_{D_{s}}:=\\{l\in\Sigma_{\mathbb{Q},f}:\exists
w\in\Sigma_{F_{i},f},\text{ with }w|l\text{ and
}[F_{i,w}:\mathbb{Q}_{l}]>\frac{h^{B}_{\max}}{m_{i}}\\}$

we have that

$|\Pi^{i}_{D_{s}}|\geq 2.$ ($\star_{2}$)

  3. $\star_{3}$

if we have that $\exists i\in\\{1,\ldots,r\\}$ such that

    1. (a)

$d_{i}m_{i}\geq h^{B}_{\max}$, and

    2. (b)

for the sets $P^{i}_{D_{s}}:=\\{l\in\Sigma_{\mathbb{Q},f}:l\text{ is totally
split in }F_{i}\\}$ and $Q^{i}_{D_{s}}:=\\{l\in\Sigma_{\mathbb{Q},f}:\exists
w\in\Sigma_{F_{i},f}\text{, }w|l\text{ and
}\operatorname{inv}_{w}(D_{i})\notin\mathbb{Z}\\}$ we have that
$|P^{i}_{D_{s}}\cap Q^{i}_{D_{s}}|\geq 2$.

  4. $\star_{4}$

if for some $i$ we have that $D_{i}$ is a quaternion algebra, and letting
$R^{i}_{D_{s}}:=\\{l\in\Sigma_{\mathbb{Q},f}:\exists
w\in\Sigma_{F_{i},f}\text{, }w|l\text{,
}\operatorname{inv}_{w}(D_{i})\not\in\mathbb{Z}\text{,and
}m_{i}4[F_{i,w},\mathbb{Q}_{l}]\not{|}h^{B}_{j}\text{ }\forall j\\}$, we have
that $|R^{i}_{D_{s}}|\geq 2$.

  5. $\star_{5}$

if for some $i$ we have that $D_{i}$ is a quaternion algebra, and letting
$S^{i}_{D_{s}}:=\\{l\in\Sigma_{\mathbb{Q},f}:\exists
w\in\Sigma_{F_{i},f},\text{ }w|l\text{,
}\operatorname{inv}_{w}(D_{i})\in\mathbb{Z}\text{,and
}m_{i}2[F_{i,w},\mathbb{Q}_{l}]\not|h^{B}_{j}\text{ }\forall j\\}$, we have
that $|S^{i}_{D_{s}}|\geq 2$.

  6. $\star_{6}$

if for some $i$ we have that $D_{i}$ is a quaternion algebra and for the above
sets $|R^{i}_{D_{s}}\cup S^{i}_{D_{s}}|\geq 2$.

  7. $\star_{7}$

if for some $i$ we have that $D_{i}$ is of Type IV with
$d_{i}^{2}=\dim_{F_{i}}[D_{i}:F_{i}]$ and for the set

$T^{i}_{D_{s}}:=\\{l\in\Sigma_{\mathbb{Q},f}:\exists
w\in\Sigma_{F_{i},f},\text{ }w|l\text{ and
}m_{i}d_{i}[F_{i,w},\mathbb{Q}_{l}]\not|h^{B}_{j}\text{ }\forall j\\}$,

we have that $|T^{i}_{D_{s}}|\geq 2$.

######  Remark.

1\. We remark that all conditions above only depend on data coming from the
original G-admissible variation $f:X\rightarrow S$. These should be thought of
as demanding that we have “a lot of endomorphisms”. This “a lot” here should
be contrasted with the fact that the generic algebra of Hodge endomorphisms is
just $\mathbb{Q}$.

2\. The necessity for demanding that several of the sets of primes in these
conditions have at least two elements arises from 12.1. As we will see in the
proof of 13.1, once we fix the place $v$, and by extension the prime
$p\in\mathbb{Q}$ with $v|p$, we need some condition as those in
$\star_{2}$-$\star_{7}$ to hold for some prime $l\neq p$ to be able to apply
the aforementioned result.

3\. We note that the sets in conditions $\star_{2}$, $\star_{5}$, $\star_{6}$,
and $\star_{7}$ will potentially have infinitely many elements, while the sets
in $\star_{3}$ and $\star_{4}$ will have at most finitely many elements. This
rests on the fact that $\operatorname{inv}_{w}(D)\notin\mathbb{Z}$ only for
finitely many primes.

4\. We note that, assuming that the extension $F_{i}/\mathbb{Q}$ is cyclic,
condition $\star_{5}$ is equivalent to the existence of some $i$ for which
$m_{i}2[F_{i}:\mathbb{Q}]\not|h^{B}_{j}$ for all $j$.

5\. The different treatment needed for general type IV algebras, addressed in
$\star_{7}$, stems from the fact that even if we know that that the
restriction of the division algebra at some place $w$ is not of the form
$M_{t}(F_{i,w})$ for some, it might still be of the form $M_{t}(D^{\prime})$
with $D^{\prime}$ a central division algebra $D^{\prime}$ with
$[D^{\prime}:F_{i,w}]>1$ and $t>1$. This issue does not appear in the case of
quaternion algebras since the only two possibilities are
$D_{i}\otimes_{F_{i}}F_{i,w}\simeq M_{2}(F_{i,w})$ or
$D_{i}\otimes_{F_{i}}F_{i,w}$ is a quaternion algebra over $F_{i,w}$.

####  §13.1 Some linear algebraic lemmas

To prove the result we want, first we will need some elementary lemmas from
linear algebra and the theory of linear algebraic groups.

######  Lemma 13.1.

Let $N\in M_{n}(k)$, where $k$ is a field with $char(k)=0$, be a non-zero
nilpotent upper-triangular matrix with $\dim_{k}Im(N)=r$. Then, there exist
unipotent upper triangular matrices $Q_{L},Q_{R}\in GL_{n}(k)$ such that the
matrix $N_{red}:=Q_{L}NQ_{R}$ is strictly upper triangular with at most one
non-zero entry in each row and column.

###### Proof.

We employ row and column reduction together with induction. ∎

######  Remark.

Another way of phrasing the above lemma is that $N_{red}=(\epsilon_{i,j})$
such that there exist exactly $r$ entries $\epsilon_{i,j}\neq 0$, which we can
take without loss of generality to be equal to $1$. These entries are all in
distinct rows and columns, i.e. if
$\epsilon_{i,j},\epsilon_{i^{\prime},j^{\prime}}\neq 0$ then $i\neq
i^{\prime}$ and $j\neq j^{\prime}$.

######  Lemma 13.2.

Let $V$ be an $n$-dimensional vector space over an algebraically closed field
$k$ of characteristic $0$ and let $N\in\operatorname{End}(V)$ be a nilpotent
linear operator with $r:=\dim_{k}Im(N)>0$. Let $T$ be a sub-torus of the
algebraic group $\operatorname{GL}(V)^{N}$ of automorphisms of $V$ commuting
with $N$. Then $\dim_{k}T\leq n-r$.

###### Proof.

We may assume that the sub-torus $T$ is maximal in
$G_{N}:=\operatorname{GL}(V)^{N}$. The torus $T$ will be contained in a
maximal sub-torus $T_{m}$ of the group $\operatorname{GL}(V)$. From the theory
of linear algebraic groups777The results we used from the theory of Linear
algebraic groups can be found in Chapter 6 of [Spr98]. all maximal tori of
$GL(V)$ are conjugate. Hence, there exists $P\in\operatorname{GL}(V)(k)$ such
that $PT_{m}P^{-1}$ is the torus of diagonal matrices in
$\operatorname{GL}(V)$, with respect to a fixed basis $\\{\vec{e}_{i}:1\leq
i\leq n\\}$.

We notice that, setting $N_{P}:=PNP^{-1}$, we have that
$P\operatorname{GL}(V)^{N}P^{-1}=\operatorname{GL}(V)^{N_{P}}$, and the sub-
torus $T$ will be isomorphic to the sub-torus $PTP^{-1}$. For the element
$N_{P}$ we will have that it is nilpotent and that
$r=\dim_{k}Im(N)=\dim_{k}Im(N_{P})$. We may thus assume from now on, replacing
$T$ by $PTP^{-1}$ and $N$ by $N_{P}$, that $T$ is comprised of diagonal
matrices and is contained in the subgroup of diagonal matrices
$\mathbb{G}_{m}^{n}\subset\operatorname{GL}_{n}$. Let $A\in T(k)$ be generic
and write $A=\operatorname{diag}(a_{1},\ldots,a_{n})$. Since $A$ and $N$
commute they are simultaneously upper triangularizable, so there exists
$P_{1}\in GL_{n}(k)$ such that $A_{1}=P_{1}AP_{1}^{-1}$ and
$N_{1}:=P_{1}NP_{1}^{-1}$ are both upper triangular and
$A_{1}N_{1}=N_{1}A_{1}$.

We note that we can write

$A_{1}=\operatorname{diag}(a^{\prime}_{1},\ldots,a^{\prime}_{n})+A^{\prime}$
(60)

where $A^{\prime}$ is some strictly upper triangular matrix. We note that
$(a^{\prime}_{1},\ldots,a^{\prime}_{n})=(a_{(r_{1})},\ldots,a_{(r_{n})})$
where $i\mapsto r_{i}$ defines a permutation of the set $\\{1,\ldots n\\}$.

We apply 13.1 to the matrix $N_{1}$ and let $Q_{L},Q_{R}$ be unipotent
matrices as in the lemma. We then have

$(Q_{L}N_{1}Q_{R})(Q_{R}^{-1}A_{1}Q_{R})=(Q_{L}A_{1}Q_{L}^{-1})(Q_{L}N_{1}Q_{R}).$
(61)

Let $B=(b_{i,j}):=(Q_{L}N_{1}Q_{R})(Q_{R}^{-1}A_{1}Q_{R})$ and
$Q_{L}N_{1}Q_{R}=(\epsilon_{i,j})=N_{red}$. Note that, since the matrices
$Q_{L}$ and $Q_{R}$ are unipotent and upper triangular, the matrices
$(Q_{R}^{-1}A_{1}Q_{R})$, $(Q_{L}^{-1}A_{1}Q_{L})$, and $A_{1}$ are all upper
triangular matrices that have the same diagonal.

Let $I:=\\{(i_{t},j_{t}):1\leq t\leq r\\}$ denote the set of indices with
$\epsilon_{i,j}=1$, note that the rest of the entries of the matrix $N_{red}$
are zero. For each pair $(i_{t},j_{t})\in I$ we compute via the left hand side
of (61)

$b_{i_{t},j_{t}}=a^{\prime}_{j_{t}},$ (62)

while computing these using the right hand side of (61) we have

$b_{i_{t},j_{t}}=a^{\prime}_{i_{t}},$ (63)

Combining the above we find that for $1\leq t\leq r$ we have

$a^{\prime}_{i_{t}}=a^{\prime}_{j_{t}}.$ (64)

These imply that the Zariski closure of
$\operatorname{diag}(a^{\prime}_{1},\ldots,a^{\prime}_{n})$ in
$\mathbb{G}_{m}^{n}$ is contained in the subvariety defined by the ideal

$J_{T}:=\langle X_{i_{t}}-X_{j_{t}}:(i_{t},j_{t})\in I\rangle$ (65)

of $k[X_{1},\ldots,X_{n}]$.

Let $f_{t}=X_{i_{t}}-X_{j_{t}}$ be the above polynomials generating the ideal
$J_{T}$. Note that each indeterminant $X_{i}$ appears at most twice among the
polynomials $f_{t}$, at most once as one of the $X_{i_{t}}$ and at most once
as one of the $X_{j_{t}}$. We also note that by construction we have
$i_{t}<j_{t}$ for all $t$.

We start by proving the following claim:

Claim: The polynomials $f_{t}$ are linearly independent over $k$.

###### Proof of Claim.

We proceed by induction on $r$. The cases $r=1,2,$ are trivial. So let us
assume that $r=3$.

Assume, without loss of generality, that

$f_{3}=\lambda_{1}f_{1}+\lambda f_{2}.$ (66)

Since $X_{i_{3}}$ appears on the left it must appear on the right, so either
$i_{3}=j_{1}$ or $i_{3}=j_{2}$, due to the above remarks. Assuming without
loss of generality that $i_{3}=j_{1}$ we get that $\lambda_{1}=-1$. Similarly
we get that $j_{3}=i_{1}$ or $j_{3}=i_{2}$. If $j_{3}=i_{1}$ it would
contradict the fact that $i_{t}<j_{t}$ for all $t$, so $j_{3}=i_{2}$.

This implies $\lambda_{2}=-1$ and (66) becomes

$X_{i_{3}}-X_{j_{3}}=-(X_{i_{1}}-X_{j_{1}})-(X_{i_{2}}-X_{j_{2}}).$ (67)

Applying the above we get $X_{j_{2}}-X_{i_{1}}=0$ which implies $i_{1}=j_{2}$.

Putting everything together we get

$i_{3}<j_{3}=i_{2}<j_{2}=i_{1}<j_{1}=i_{3},$ (68)

which is obviously a contradiction.

Now assume that $r\geq 4$. To reach a contradiction we assume that there exist
$\lambda_{i}\in k$ such that

$X_{i_{r}}-X_{j_{r}}=\lambda_{1}f_{1}+\ldots+\lambda_{r-1}f_{r-1}.$ (69)

Again, since $j_{r}$ appears on the left there exists a unique $i_{t}$ such
that $i_{t}=j_{r}$. We assume, without loss of generality that $t=r-1$ so that
$\lambda_{r-1}=-1$. Similarly without loss of generality we may assume that
$i_{r}=j_{r-2}$ and $\lambda_{r-2}=-1$.

Using the above with (69) we get

$X_{i_{r}}-X_{j_{r}}=-(X_{j_{r}}-X_{j_{r-1}})-(X_{i_{r-2}}-X_{i_{r}})+\lambda_{1}(X_{i_{1}}-X_{j_{1}})+\ldots+\lambda_{r-3}(X_{i_{r-3}}-X_{j_{r-3}}),$
(70)

which, after canceling, becomes

$X_{i_{r-2}}-X_{j_{r-1}}=\lambda_{1}(X_{i_{1}}-X_{j_{1}})+\ldots+\lambda_{r-3}(X_{i_{r-3}}-X_{j_{r-3}}).$
(71)

This gives a contradiction by the inductive step as follows:

First, note that $i_{r-2}<j_{r-2}=i_{r}<j_{r}=i_{r-1}<j_{r-1}$. Now, consider
the following set of pairs of indices

$I^{\prime}=\\{(i_{1},j_{1}),\ldots(i_{r-3},j_{r-3}),(i_{r-2},j_{r-1})\\}$.

Notice that this satisfies all the properties of the original set of pairs of
indices, namely $i_{t}<j_{t}$ for all $t$, where we can “rename” $j_{r-2}$ as
$j_{r-1}$, each $i_{t}$ appears at most once in the first coordinate of the
pairs and the same hold for the $j_{t}$ in the second coordinate of the pairs.

∎

Since the $f_{t}$ are linearly independent linear polynomials it follows that
$Z(J_{T})\subset\mathbb{A}_{k}^{n}$ has dimension $n-r$ and the result
follows. ∎

####  §13.2 Ruling out $v$-adic proximity

With an eye towards “globality” of relations among the values of G-functions
at a certain point, we establish the following proposition.

######  Proposition 13.1.

Let $f:X\rightarrow S$ be a G-admissible variation of $\mathbb{Q}$-HS defined
over some number field $K$. Let $s\in S(L)$, where $L/K$ is some finite
extension, and let $v\in\Sigma_{L,f}$ be a finite place of $L$.

If $s$ satisfies condition $\star$ then, assuming the Hodge conjecture holds
and that a good arithmetic model exists for the morphism $f$, the point $s$
cannot be $v$-adically close to the degeneration $s_{0}$.

###### Proof.

Step 1: Assume that $s$ is $v$-adically close to $s_{0}$. We then get by 12.1
that $h^{\text{\'{e}t}}_{i}=h^{B}_{i+n}$, assuming the existence of a good
arithmetic model for $f$ over $\mathcal{O}_{K}$.

From 10.1, assuming the validity of the Hodge conjecture, we get that

$D_{s}\otimes_{\mathbb{Q}}\mathbb{Q}_{l}\hookrightarrow\operatorname{End}(H^{n}_{\text{\'{e}t}}(\bar{X}_{s,v},\mathbb{Q}_{l}))^{N_{v}},$
(72)

where $l\neq p(v)$. We are thus in a position to use 10.2 for the algebra
$D_{s}\otimes\mathbb{Q}_{l}$ and the nilpotent endomorphism $N_{v}$ on the
space $H^{n}_{\text{\'{e}t}}(\bar{X}_{s,v},\mathbb{Q}_{l})$.

From this we get that we must have that for all $1\leq i\leq r$ and all
$w\in\Sigma_{F_{j},f}$ with $w|l$ there exists $j(i,w)$ such that

$M_{m_{i}}(D_{i}\otimes_{F_{i}}F_{i,w})\hookrightarrow
M_{h^{B}_{j(i,w)}}(\mathbb{Q}_{l}).$ (73)

Step 2: We start by ruling out points that satisfy $\star_{1}$. This follows
from 13.2. Indeed the dimension of the maximal subtorus of
$(D_{s}\otimes\bar{\mathbb{Q}}_{l})^{\times}$ is
$\displaystyle\sum_{i=1}^{r}m_{i}f_{i}$. On the other hand, the maximal
subtorus of
$\operatorname{GL}(H^{n}_{\text{\'{e}t}}(\bar{X}_{s,v},\mathbb{Q}_{l}))^{N_{v}}$
has dimension $\leq\mu-\dim_{\mathbb{Q}_{l}}N_{v}$. The result then follows
from 12.1, which implies that
$\dim_{\mathbb{Q}_{l}}Im(N_{v})=\dim_{\mathbb{Q}}Im(N_{B})>0$.

Assume now that $s$ satisfies condition $\star_{2}$. We then get that there
exists $i$, a prime $l\neq p(v)$, and a place $w\in\Sigma_{F_{i},f}$ with
$w|l$ for which $[F_{i,w}:\mathbb{Q}_{l}]m_{i}>h^{B}_{\max}$. This contradicts
the validity of (73). Indeed the maximal commutative semisimple algebra of
$M_{m_{i}}(D_{j}\otimes_{F_{i}}F_{i,w})$ has dimension
$\geq[F_{i,w}:\mathbb{Q}_{l}]m_{i}$ over $\mathbb{Q}_{l}$, while that of
$M_{h^{B}_{\max}}(\mathbb{Q}_{l})$ has dimension $h^{B}_{\max}$ over
$\mathbb{Q}_{l}$.

Assume that $s$ satisfies condition $\star_{3}$ and choose $i$ as in
$\star_{3}$. Then we have that there exists $l\in\Sigma_{\mathbb{Q},f}$ that
is totally split in $F_{i}$ with $l\neq p(v)$. Therefore we have that
$F_{i,w}=\mathbb{Q}_{l}$ for all $w\in\Sigma_{F_{i},f}$ with $w|l$.

Also by $\star_{3}$ we know that we can find $w\in\Sigma_{F_{i},f}$ with $w|l$
such that $\operatorname{inv}_{v}(D_{i})\notin\mathbb{Z}$. Since once again by
assumption we have $(m_{i}d_{i})\geq h^{B}_{\max}$ we get that (73) is an
isomorphism in this case, with $h^{B}_{j(i,w)}=h^{B}_{\max}$. This would imply
that $\operatorname{inv}_{w}(D_{i})=0\in\mathbb{Q}/\mathbb{Z}$ contradicting
the above assumption.

Assume that $s$ satisfies condition $\star_{4}$. Then by assumption there
exists a prime $l\neq p(v)$ such that there exists $w\in\Sigma_{F_{i},f}$ for
which $D_{i,w}=D_{i}\otimes_{F_{i}}F_{i,w}$ is a quaternion algebra over
$F_{i,w}$. If a $j(i,w)$ satisfying (73) existed for the simple summand
$M_{m_{i}}(D_{i,w})$ of $D_{s}\otimes_{\mathbb{Q}}\mathbb{Q}_{l}$, we would
have $m_{i}\dim_{\mathbb{Q}_{l}}D_{i,w}|h^{B}_{j(i,w)}$ which contradicts our
assumptions. Indeed, such an embedding would imply an isomorphism of
$M_{m_{i}}(D_{i,w})$-modules
$\mathbb{Q}_{l}^{h^{B}_{j(i,w)}}\cong(D_{i,w}^{m_{i}})^{r}$ for some $r$.
Comparing $\mathbb{Q}_{l}$-dimensions the contradiction follows.

The argument for $\star_{5}$ and $\star_{6}$ is practically identical to that
of condition $\star_{4}$.

Finally, the proof in the case where $s$ satisfies condition $\star_{7}$ is
practically identical to that of $\star_{4}$ but has a small twist. Let us
assume that $|T^{i}_{D_{s}}|\geq 2$ and let $l\in T^{i}_{D_{s}}$ and $w|l$ be
such that $l\neq p(v)$ and $m_{i}d_{i}[F_{i,w}:\mathbb{Q}_{l}]\not|h^{B}_{j}$
for all $j$.

If $\operatorname{inv}_{w}(D)\neq 0$ we have that
$D_{i}\otimes_{F_{i}}F_{i,w}\simeq M_{r}(D^{\prime})$ with $D^{\prime}$ a
division algebra with center $F_{i,w}$. Let
$d^{\prime}:=\sqrt{[D^{\prime}:F_{i,w}]}$. If an index $j(i,w)$ satisfying
(73) existed, by the same argument as above, we would have that
$m_{i}r\dim_{\mathbb{Q}_{l}}(D^{\prime})|h^{B}_{j(i,w)}$. Since
$rd^{\prime}=d_{i}$ and $\dim_{\mathbb{Q}_{l}}(D^{\prime})=d^{\prime
2}[F_{i,w}:\mathbb{Q}_{l}]$ we get the contradiction we wanted.

The case $\operatorname{inv}_{w}(D)=0$ follows from the same argument above,
though we do not have to introduce a new division algebras $D^{\prime}$ since
$D_{i}\otimes_{F_{i}}F_{i,w}\simeq M_{d_{i}}(F_{i,w})$. ∎

######  Remark.

The proof in the case where condition $\star_{7}$ is satisfied shows that
$\star_{7}$ is weaker than the strongest condition we can actually impose on
these algebras.

In fact the proof shows that we would need to guarantee the impossibility of
the occurrence of

$rd^{\prime 2}\cdot[F_{i,w}:\mathbb{Q}_{l}]\>|\>h^{B}_{j}$ for some $j$,

where $r$ and $d^{\prime}$ are as in the proof of $\star_{7}$.

###  §14 Proof of 1.1

Finally, we combine all parts of our exposition to prove 1.1.

###### Proof of 1.1:.

Let $s\in S(L)$ be a point satisfying the conditions in 1.1 for the variation
$\mathbb{V}=R^{n}f^{an}_{*}\mathbb{Q}_{X^{an}_{\mathbb{C}}}$ where $L/K$ is
some finite extension. We let $\xi:=x(s)$, where $x$ is the local parameter of
$S^{\prime}$ at $s_{0}$ with respect to which the $y_{i}$ are written as power
series.

By 5.1 there exists a finite extension $\hat{L}/L$ such that $D_{s}$ acts on
$H^{n}_{DR}(X_{s}\times_{L}\hat{L}/\hat{L})$. From 5.2 we also know that
$\hat{L}$ may be chosen so that $[\hat{L}:L]$ is bounded only in terms of
$\mu:=\dim_{\mathbb{Q}}H^{n}(X^{an}_{s},\mathbb{Q})$.

Let $y_{1},\ldots,y_{h\mu}$ be the G-functions that comprise the first $h$
columns of the relative $n$-period matrix associated to the morphism $f$. We
then have the polynomials (53) with coefficients in $\hat{F}_{s}$ and degree
$\leq[\hat{L}:\mathbb{Q}]$. By 9.2 we know that these polynomials define
relations among the values of the G-functions in question at $\xi$ that are
non-trivial, as long as there exists at least one archimedean place
$v\in\Sigma_{\hat{L},\infty}$ for which $s$ is $v$-adically close to $s_{0}$.

Consider now $v\in\Sigma_{\hat{L},f}$ to be any finite place of $\hat{L}$. By
13.1 we know that $s$ cannot be $v$-adically close to $s_{0}$, in other words
that $|\xi|_{v}\geq\min\\{1,R_{v}(\vec{y})\\}$, where $R_{v}(\vec{y})$ is the
local radius of convergence of the $y_{i}$.

Now we split into two cases.

Case 1: For all archimedean places $v\in\Sigma_{\hat{L},\infty}$ we have that
$|\xi|_{v}\geq\min\\{1,R_{v}(\vec{y})\\}$.

Combining this assumption with the above result we get that

$h(\xi^{-1})\leq\rho(\vec{y})$,

where $\rho(\vec{y})$ is the global radius of the collection of power series
$y_{i}$. Combining Lemma 2 of I.$\S 2.2$ of [And89] with the Corollary of
VI.$\S 5$ of loc.cit., we get that
$h(s)=h(\xi)=h(\xi^{-1})\leq\rho(\vec{y})<\infty$.

This concludes this case.

Case 2:  There exists at least one archimedean place
$v\in\Sigma_{\hat{L},\infty}$ for which $s$ is $v$-adically close to $s_{0}$.

In this case the relation defined by the polynomials in (53) among the values
at $\xi$ of the $y_{i}$ is by construction global, since there are no other
places $v\in\Sigma_{\hat{L},f}$ for which $s$ is $v$-adically close to
$s_{0}$.

Since we know that the relation (53) is both non-trivial and global we get
from 2.1 that

$h(\xi)\leq c_{1}(\vec{y})\delta^{3\mu h-1}(\log\delta+1),$ (74)

where $\delta$ is the degree of the polynomial (53) in
$\bar{\mathbb{Q}}[x_{1},\ldots x_{h\mu}]$.

By construction of (53) we know that
$\delta\leq[\hat{L}:\mathbb{Q}]=[\hat{L}:L]\cdot[L:\mathbb{Q}]$. On the other
hand 5.2 gives the bound $[\hat{L}:L]\leq((6.31)\mu^{2})^{\mu^{2}}$. Combining
these remarks with (74) we get that there exist positive constants
$C_{1},C_{2}$, independent of the point $s$, such that

$h(\xi)\leq C_{1}([L:\mathbb{Q}]+1)^{C_{2}},$ (75)

as we wanted.

The result follows by combining the two above cases, or simply by replacing
$C_{1}$ in (75) by $\max\\{C_{1},\rho(\vec{y})\\}$. ∎

###  §15 Complex Multiplication and other examples

We construct examples of possible algebras of endomorphisms of Hodge
structures where the conditions for points $s$ to be in the set $\Sigma$ of
1.1 are easy to check. We start with the case of CM-Hodge structures and then
construct infinite families of possible such algebras of all possible types,
i.e. type I-IV in Albert’s classification 3.1.

####  §15.1 Hodge Structures with complex multiplication

Hodge structures with complex multiplication, or simply CM Hodge structures,
play the same role for Hodge structures with weight $\geq 2$ that abelian
varieties with complex multiplication play for the weight $1$ Hodge
structures. For an introduction to these we point the interested reader to
[GGK12] and [Moo99].

The main ingredient we will need about CM Hodge structures is the following
lemma that describes their algebra of Hodge endomorphisms. Following [GGK12]
we write “CMpHS” as an abbreviation of the term “polarized Hodge structure
with complex multiplication”.

######  Lemma 15.1.

Let $V$ be a CMpHS then there is a unique decomposition of $V$ into simple
Strongly CMpHS’s $V=V_{1}^{\oplus m_{1}}\oplus\cdots\oplus V_{l}^{\oplus
m_{r}}$. In particular for the algebra $D$ of Hodge endomorphisms of $V$ we
have that

$D\cong\displaystyle\bigoplus_{i=1}^{r}M_{m_{i}}(K_{i})$,

where $K_{i}\cong D_{i}$, the algebra of Hodge endomorphisms of $V_{i}$, is a
CM field with $[K_{i}:\mathbb{Q}]=\dim_{\mathbb{Q}}V_{i}$.

###### Proof.

See [GGK12] Ch.V and especially the facts on page $195$. For a proof see
[Moo99] where the necessary machinery is developed. ∎

Motivated by this we make the following definition.

######  Definition.

Let $\mathbb{V}$ be a polarized variation of $\mathbb{Q}$-HS over some base
$T$. We say that the point $t\in T(\mathbb{C})$ is a CM point of the
variation, or a special point of the variation, if the Hodge structure
$\mathbb{V}_{t}$ is a CMpHS.

######  Remarks.

1\. Consider a CMpHS $(V,\phi)$ as in 15.1. Let $E$ be the maximal commutative
semi-simple algebra of the algebra $D$. We have by 15.1 that

$E=K_{1}^{m_{1}}\times\cdots\times K_{r}^{m_{r}}.$ (76)

We also have that
$\dim_{\mathbb{Q}}E=\displaystyle\sum_{j=1}^{r}m_{j}[K_{j}:\mathbb{Q}]$.
Noting that
$\dim_{\mathbb{Q}}V=\displaystyle\sum_{j=1}^{r}m_{j}\dim_{\mathbb{Q}}V_{j}$
and that $\dim_{\mathbb{Q}}V_{j}=[K_{j}:\mathbb{Q}]$ we get that

$\dim_{\mathbb{Q}}V=\dim_{\mathbb{Q}}E.$ (77)

2\. The above property guarantees that CM-points satisfy the conditions of
8.3. Indeed, from the above and 8.4 we have that CM-points satisfy (49) and so
we get non-trivial relations among the values of the G-functions we study at
$\xi=x(s)$.

The globality of these relations follows from the fact that CM-points satisfy
condition $\star_{2}$.

#####  §15.1.1 CM-points have potentially good reduction

Given a CM abelian variety $A$ defined over a number field $K$ it is well
known, see [ST68], that $A$ will have potentially good reduction at each
finite place of $K$. The linear algebraic lemma 13.2 has an interesting
consequence. Namely, it shows that a similar picture holds true for smooth
projective varieties with CM polarized Hodge structure defined over a number
field $K$. Even though the term “good reduction” or “potentially good
reduction” cannot be expected to hold in general, as in the existence of Néron
models, the term still makes sense from the point of view of Galois
representations.

######  Proposition 15.1.

Let $f:X\rightarrow S$ be a G-admissible variation of $\mathbb{Q}$-HS defined
over the number field $K$. Let $L/K$ be a finite extension, let $s\in S(L)$ be
a CM point for this variation, and let $v\in\Sigma_{L,f}$ be a finite place of
$L$.

Assume the Hodge conjecture holds. Then the $l$-adic Galois representation of
$G_{K_{v}}$ on $H^{n}_{\text{\'{e}t}}(\bar{X}_{s,v},\mathbb{Q}_{l})$ has
potentially good reduction for all $l\neq p(v)$.

###### Proof.

Let $D_{s}$ be the algebra of Hodge endomorphisms at $s$. We set
$V_{s}:=H^{n}(\bar{X}_{s,\mathbb{C}}^{an},\mathbb{Q})$ to be the
$\mathbb{Q}$-HS corresponding to $s$, with $\dim_{\mathbb{Q}}V_{s}=2\mu$.

From 15.1 we know that $V_{s}$ decomposes as
$V_{s}=\displaystyle\bigoplus_{i=1}^{r}V_{i}^{\oplus m_{i}}$, so that
$D_{s}\cong\displaystyle\bigoplus_{i=1}^{r}M_{m_{i}}(K_{i})$, where $V_{i}$
are irreducible CM polarized $\mathbb{Q}$-HS with algebras of Hodge
endomorphisms $K_{i}$ being CM fields with
$n_{i}:=[K_{i}:\mathbb{Q}]=\dim_{\mathbb{Q}}V_{i}$. We also note that,
trivially from the above, we have that
$\mu=\displaystyle\sum_{i=1}^{r}m_{i}n_{i}$, where
$\mu:=\dim_{\mathbb{Q}}H^{n}(X^{an}_{s},\mathbb{Q})$.

Let $p=p(v)$ be the characteristic of the residue field
$\kappa(v):=\mathcal{O}_{L_{v}}/m_{L_{v}}$. We fix $l\in\mathbb{N}$ a prime
with $l\neq p$. We then notice that the $\bar{\mathbb{Q}}_{l}$-algebra
$\bar{D}_{s,l}:=D_{s}\times_{\mathbb{Q}}\bar{\mathbb{Q}}_{l}\cong(D_{s}\times_{\mathbb{Q}}\mathbb{Q}_{l})\otimes_{\mathbb{Q}_{l}}\bar{\mathbb{Q}}_{l}$
is such that $\bar{D}_{s,l}^{\times}$ contains all of the closed
$\bar{\mathbb{Q}}_{l}$-points of a $\mu$-dimensional torus.

From 10.1 we know that the inertia group $I_{L_{v}}$ acts quasi-unipotently on
$H^{n}_{\text{\'{e}t}}(\bar{X}_{s,v},\mathbb{Q}_{l})$. Therefore up to a
finite extension of $L_{v}$ we may and do assume that it in fact acts
unipotently. In this case we get an associated nilpotent endomorphism $N_{v}$,
as we saw in our earlier discussion, whose exponential determines the action
of the inertia group.

In this case, from 10.1, we have that

$\bar{D}_{s,l}^{\times}\hookrightarrow\operatorname{GL}(H^{n}_{\text{\'{e}t}}(\bar{X}_{s,v},\mathbb{Q}_{l})\otimes_{\mathbb{Q}_{l}}\bar{\mathbb{Q}}_{l})^{N_{v}}$,

where $N_{v}$ is the above nilpotent matrix associated to the action of the
inertia group. The result now follows from 13.2. Indeed, if $N_{v}\neq 0$ we
get that
$\dim_{\bar{\mathbb{Q}}_{l}}\bar{D}^{\times}_{s,l}\leq\mu-\operatorname{rank}(N_{v})<\mu$
which contradicts the above. ∎

####  §15.2 Some more complicated examples

We start with highlighting some examples where 1.1 applies. We fix for the
remainder some general notation as in the main part of our exposition.

We consider $f:X\rightarrow S$ a G-admissible variation of Hodge structures
with fibers of odd dimension $n$ and generic special Mumford-Tate group
$Sp(\mu,\mathbb{Q})$, where
$\mu:=\dim_{\mathbb{Q}}H^{n}(X_{z}^{an},\mathbb{Q})$ for any $z\in S^{an}$ and
let $h:=\dim_{\mathbb{Q}}Im((N^{*})^{n})$.

As before, for a point $s\in S(\bar{\mathbb{Q}})$ we let
$V_{s}:=H^{n}(X^{an}_{s},\mathbb{Q})$ and assume that the decomposition of
$V_{s}$ into simple polarized sub-$\mathbb{Q}$-HS is given by
$V_{1}^{m_{1}}\oplus\ldots\oplus V_{r}^{m_{r}}$. We also let
$D_{s}=M_{m_{1}}(D_{1})\oplus\ldots\oplus M_{m_{r}}(D_{r})$ be its algebra of
Hodge endomorphisms.

For our process to kick in we need to have that

$h>\frac{\dim_{\mathbb{Q}}V_{j}}{[Z(D_{j}):\mathbb{Q}]}\text{ for some
}j\text{, or that }$ (78)
$h\geq\min\\{\frac{\dim_{\mathbb{Q}}V_{i}}{[Z(D_{i}):\mathbb{Q}]}:i\text{ such
that }D_{i}=\operatorname{End}_{HS}(V_{i})\text{ is of type IV }\\}.$ (79)

##### Some examples of CM-fields and totally real fields

The fields $F_{0,\beta}$ and $F_{\beta}$: Let $\beta\in\mathbb{N}$ and let $p$
be a prime with $p\equiv 1\mod 2\beta$. It is well known that the field
$\mathbb{Q}(\zeta_{p})$ is a cyclic extension of $\mathbb{Q}$ with totally
real subfield $\mathbb{Q}(\zeta_{p}+\zeta_{p}^{-1})$. We can therefore find a
subfield $F_{0,\beta}$ of $\mathbb{Q}(\zeta_{p}+\zeta_{p}^{-1})$ with
$[F_{0,\beta}:\mathbb{Q}]=\beta$. Note that by construction this extension is
also cyclic and totally real.

By Frobenius density it follows888See [Jan73] Ch. IV, Corollary $5.4$.that the
set

$In(F_{0,\beta}):=\\{l\in\Sigma_{\mathbb{Q},f}:l\text{ is inert in
}F_{0.\beta}\\}$

is infinite. Fix $l_{1},l_{2}\in In(F_{\beta})$ with $l_{1}\neq l_{2}$. Then
from quadratic reciprocity we can find a prime $q=q(l_{1},l_{2})\in\mathbb{Q}$
such that in the extension $F_{\beta}:=F_{0,\beta}(i\sqrt{q})$ the prime
ideals $l_{j}\mathcal{O}_{F_{0,\beta}}$, $j=1$, $2$, split in $F_{\beta}$.

In conclusion we get that for the field $F_{\beta}$ constructed above there
are distinct primes $l_{1}$, $l_{2}\in\mathbb{Q}$ whose splitting in
$F_{\beta}$ is given by

$l_{j}\mathcal{O}_{F_{\beta}}=w_{j,1}\cdot w_{j,2}$.

In particular we get that $[(F_{\beta})_{w_{j,i}}:\mathbb{Q}_{l_{j}}]=n$ for
all $i,$ $j$ and that, trivially by construction, if $\sigma$ denotes complex
conjugation in $F_{\beta}/\mathbb{Q}$, we have $\sigma w_{j,1}=w_{j,2}$.

Note that the family of CM-fields constructed above is infinite, for fixed
$\beta$, by varying either the infinitely many pairs $(l_{1},l_{2})$ or the
infinite choices $q(l_{1},l_{2})$.

##### Examples of algebras of Type IV

Consider $F_{\beta}$ a field as those constructed above. Let $D$ be a division
algebra over $F_{\beta}$ and let $d^{2}=[D:F_{\beta}]$. For such an algebra to
be of type IV in Albert’s classification it needs to satisfy the following
conditions:

  1. 1.

for any finite place $v\in\Sigma_{F_{\beta},f}$ we have that
$\operatorname{inv}_{v}(D)+\operatorname{inv}_{\sigma(v)}(D)=0$, and

  2. 2.

$\operatorname{inv}_{v}(D)=0$ for all such places that satisfy $\sigma(v)=v$.

By construction of our fields $F_{\beta}$ we get places $w_{j,i}$, $1\leq
i,j\leq 2$, that come in pairs with $\sigma(w_{j,1})=w_{j,2}$ for $j=1,$ $2$.
We want, in view of the conditions $\star$ introduced in §13, to work with
algebras $D$ that ramify at these finite places. For that reason we consider
the following set of central division algebras, or CDA for short, with center
$F_{\beta}$

$\mathcal{D}_{IV}(\beta,d):=\\{D:CDA/F_{\beta},\text{
}[D:F_{\beta}]=d^{2},\text{
}\operatorname{inv}_{w_{j,i}}(D)\not\in\mathbb{Z}\\}$.

We note that the set $\mathcal{D}_{IV}(\beta,d)$ actually depends on the
numbers $\beta$, $l_{j}$, $q(l_{1},l_{2})$, and $d$ so it would be perhaps
more accurate to denote this by
$\mathcal{D}_{IV}(d,\beta,l_{1},l_{2},q(l_{1},l_{2}))$, though we avoid this
for notational brevity.

We note that for all choices of $\beta$, $l_{j}$, and $q(l_{1},l_{2})$ as
above the set $\mathcal{D}_{IV}(\beta,2)$, i.e. the set of quaternion algebras
satisfying the above conditions is non-empty. This follows from the
classification theorem of quaternion algebras over global fields, see [Vig80]
Ch. III, Theoreme $3.1$.

Let $f:X\rightarrow S$ be as above and let $s\in S(\bar{\mathbb{Q}})$ and
assume that one of the algebras $D_{k}$ that appear in the decomposition of
the algebra of Hodge endomorphisms $D_{s}$ at the point $s$ is an element of
$\mathcal{D}_{IV}(\beta,d)$ for some choice of
$(\beta,d,l_{1},l_{2},q(l_{1},l_{2}))$. Then we can find simple conditions to
check whether $s$ is in the set $\Sigma$ of 1.1.

Indeed, assuming that

$2\beta\geq\frac{\dim_{\mathbb{Q}}V_{k}}{h}$ (80)

is enough to guarantee the validity of (79). After this we just need to check
the validity of at least one of the conditions in §13. In our case, by
construction of the fields $F_{\beta}$, conditions $\star_{4}$ and $\star_{7}$
translate into easy to check conditions, mainly thanks to the fact that
$[(F_{\beta})_{w_{j,i}}:\mathbb{Q}_{l_{j}}]=\beta$.

In more detail, by our construction in the case $d=2$ condition $\star_{4}$
follows from

$4\beta m_{k}\not|h^{B}_{j}\text{ for all }j,$ (81)

where $h^{B}_{j}$ are as in §12.1. Condition $\star_{7}$, only applicable in
our case when $d=d_{k}\geq 3$, follows from

$m_{k}d\beta\not|h^{B}_{j}\text{ for all }j.$ (82)

Finally, for the case $d=1$ so that $D_{k}=F_{\beta}$ , or in other words the
case where $V_{k}$ is a CM-HS while the other $V_{t}$ are arbitrary polarized
sub-HS of $V_{s}$, it is easy to create a condition analogous to the
conditions $\star$ created in §13. Indeed, it is easy to check, using
arguments as in 13.1, that the the condition

$m_{k}\beta\not|h^{B}_{j}\text{ for all }j,$ (83)

is enough to guarantee the impossibility of $v$-adic proximity for all finite
places $v\in\Sigma_{L}$, where $L=K(s)$ with our usual notation.

With these observations in mind we define $\Sigma_{IV}\subset
S(\bar{\mathbb{Q}})$ to be the set that consists of the points $s\in
S(\bar{\mathbb{Q}})$ whose corresponding algebra of Hodge endomorphisms
satisfies the above hypothesis, i.e. for some $k$ we have that
$D_{k}\in\mathcal{D}_{IV}(\beta,d)$ for some choice of
$(\beta,d,l_{1},l_{2},q(l_{1},l_{2}))$, condition (80) holds, and condition
(81) holds if $d_{k}=2$, or (82) holds if $d_{k}\geq 3$, or (83) holds if
$d_{k}=1$.

For the points in $\Sigma_{IV}$ we get that they satisfy the conditions needed
so that they are in the set $\Sigma$ of 1.1. We thus have the following
corollary.

######  Corollary 15.1.

Let $f:X\rightarrow S$ be a morphism over $K$ defining a G-admissible
variation of $\mathbb{Q}$-HS satisfying the conditions of 1.1. Let
$\Sigma_{IV}$ be the above set of points.

Then, there exist constants $C_{1}$, $C_{2}>0$ such that for all
$s\in\Sigma_{IV}$ we have

$h(s)\leq C_{1}[K(s):K]^{C_{2}}$,

where $h$ is a Weil height on $S^{\prime}$.

######  Example.

The simplest example of this nature that we could find is the following:

Assume that $\mu=\dim_{\mathbb{Q}}V_{s}=16$ and that $h=h_{0}^{B}=4$ for a
G-admissible variation with $n=3$ satisfying the conditions of 1.1.

In this case we let $\beta=2$ and $d=2$ above. In other words we can consider
points for which $D_{s}$ is a quaternion algebra over a CM-field constructed
as above. Note that (80) is satisfied, leaving us with checking
$8\not|h^{B}_{j}$ for all $j$.

We have $h^{B}_{0}=h^{B}_{6}=4$ so all we need to check the validity of (81)
is to make sure that $8\not|h^{B}_{j}$ for $j=1,\>2$, and $3$. To establish
this we note that $W_{3}^{B}\neq W_{0}^{B}$ and $W_{3}^{B}\neq W_{5}^{B}$ in
the notation of (57), which can be shown for example using the description of
the $W_{i}^{B}$ in [SZ85] remark $(2.3)$. Since in our case
$\dim_{\mathbb{Q}}W_{5}^{B}=12$ and $\dim_{\mathbb{Q}}W_{0}^{B}=4$ the result
follows.

##### Algebras of types I-III

The totally real fields $F_{0,\beta}$ with $[F_{0,\beta}:\mathbb{Q}]=\beta$
that we created above, help us construct convenient examples to check the
conditions in §13, mainly since they are cyclic. In fact, every such field
constitutes a type I algebra in Albert’s classification.

By the aforementioned classification of quaternion algebras over number
fields, see [Vig80] or [Voi21], we have a bijection between the set

$\\{\text{Quaternion algebras over }F_{0,\beta}\text{ up to isomorphism}\\}$

and the set999Normally we would have to make sure that the the subsets $P$ in
question do not have any complex archimedean places. Since our fields are
totally real this condition is simply vacuous.

$\\{P\subset\Sigma_{F}:|P|\equiv 0\mod 2\\}$

given by $B\mapsto\operatorname{Ram}(B)$, with $\operatorname{Ram}(B)$ the set
of places over which the quaternion algebra $B$ ramifies.

With this in mind we define, in parallel to the type IV case above, the
following sets of quaternion algebras $D/F_{0,\beta}$:

$\mathcal{D}_{II}(\beta):=\\{D:\operatorname{Ram}(D)\cap\Sigma_{F,\infty}=\emptyset,\>w_{j,i}\in\operatorname{Ram}(D)\text{
for all }j,i\\}$

$\mathcal{D}_{III}(\beta):=\\{D:\Sigma_{F,\infty}\subset\operatorname{Ram}(D),\>w_{j,i}\in\operatorname{Ram}(D)\text{
for all }j,i\\}$.

######  Remark.

We note that by the aforementioned classification theorem it follows that
these sets are in fact infinite, for fixed $\beta$ and pair of primes
$(l_{1},l_{2})$.

Note that for these algebras we will have by construction the following:

  1. 1.

$[(F_{0,\beta})_{w_{j,i}}:\mathbb{Q}_{l_{j}}]=[F_{0,\beta}:\mathbb{Q}]=\beta$
for all $1\leq i,j\leq 2$, and

  2. 2.

for $D\in\mathcal{D}_{III}(\beta)\cup\mathcal{D}_{II}(\beta)$ we will have
that $\operatorname{inv}_{w_{j,i}}(D)\neq 0$.

Given these it is much easier to check for the validity of the conditions for
a point $s\in S(\bar{\mathbb{Q}})$ to be in the set $\Sigma$ of 1.1, assuming
that one of the algebras $D_{k}$ appearing in the decomposition of the algebra
of Hodge endomorphisms $D_{s}$ is an element of
$\mathcal{D}_{III}(\beta)\cup\mathcal{D}_{II}(\beta)$, or even that
$D_{k}=F_{0,\beta}$ for some $\beta$.

Indeed, assume that for some $k$ we have
$D_{k}\in\mathcal{D}_{III}(\beta)\cup\mathcal{D}_{II}(\beta)$. Then, condition
(78) translates to simply checking

$\beta>\frac{\dim_{\mathbb{Q}}V_{k}}{h}.$ (84)

On the other hand, checking $\star_{4}$, which is the strongest out of the
conditions in §13, becomes straightforward. Indeed, letting $h^{B}_{j}$ be the
dimensions of the quotients resulting from the weight monodromy filtration as
in §12.1, $\star_{4}$ in this case follows from

$4m_{k}\beta\not|h^{B}_{j}\text{ for all }j.$ (85)

In the case where $D_{k}=F_{0,\beta}$ it is easy to check, with the same
arguments as in the proof of 13.1, that the the condition

$m_{k}\beta\not|h^{B}_{j}\text{ for all }j,$ (86)

is enough to guarantee the impossibility of $v$-adic proximity for all finite
places $v\in\Sigma_{L}$, where $L=K(s)$ with our usual notation.

With the above in mind, for our fixed morphism $f:X\rightarrow S$, we consider
the set $\Sigma_{I-III}\subset S(\bar{\mathbb{Q}})$ that consists of the
points $s\in S(\bar{\mathbb{Q}})$ that are such that, if the corresponding
algebra of Hodge endomorphisms is given by
$D_{s}=M_{m_{1}}(D_{1})\oplus\ldots\oplus M_{m_{r}}(D_{r})$, we have that
there exists $1\leq k\leq r$ and $\beta,\>l_{1},\text{ and}l_{2}\in\mathbb{Q}$
as above such that (84) holds and either
$D_{k}\in\mathcal{D}_{III}(\beta)\cup\mathcal{D}_{II}(\beta)$ and (85) holds,
or $D_{k}=F_{0,\beta}$ and (86) holds.

In this case, 1.1 applies to such points and we have the following corollary.

######  Corollary 15.2.

Let $f:X\rightarrow S$ be a morphism over $K$ defining a G-admissible
variation of $\mathbb{Q}$-HS satisfying the conditions of 1.1. Let
$\Sigma_{I-III}$ be the above set of points.

Then, there exist constants $C_{1}$, $C_{2}>0$ such that for all
$s\in\Sigma_{I-III}$ we have

$h(s)\leq C_{1}[K(s):K]^{C_{2}}$,

where $h$ is a Weil height on $S^{\prime}$.

######  Remarks.

1\. We note that in both 15.1 and 15.2 the conditions imposed on the points
$s$ revolve around only one of the division algebras $D_{k}$ that appear in
the decomposition of the algebra of Hodge endomorphisms $D_{s}$. The rest of
the algebras $D_{t}$ with $t\neq k$ could have arbitrary properties.

2\. We could simplify the situation by considering points $s$ for which the
algebra $D_{s}$ is equal to one of the algebras constructed in the above
examples, i.e. we are in the case where $r=m_{1}=1$ and $D_{s}$ is a central
simple algebra itself.

## Appendix

###  Appendix A Some notes on polarizations

####  A.1 The non-relative case

Notation: Let $X/k$ be a smooth projective variety over a subfield $k$ of
$\mathbb{C}$ and let $n=\dim_{k}X$.

##### Short review on polarizing forms

For all $d\in\mathbb{N}$ there exist non-degenerate bilinear polarizing forms

$\langle,\rangle_{DR}:H^{d}_{DR}(X/k)\otimes_{k}H^{d}_{DR}(X/k)\rightarrow k$,
and

$\langle,\rangle_{B}:H^{d}(X^{an}_{\mathbb{C}},\mathbb{Q})\otimes_{\mathbb{Q}}H^{d}(X^{an}_{\mathbb{C}},\mathbb{Q})\rightarrow(2\pi
i)^{-d}\mathbb{Q}=\mathbb{Q}(-d)$,

on de Rham cohomology and Betti cohomology respectively. We also write
$\langle,\rangle_{B}=(2\pi i)^{-d}\langle,\rangle$, where $\langle,\rangle$
has values in $\mathbb{Q}$ and is of the same type, i.e. symmetric or skew-
symmetric, as $\langle,\rangle_{B}$.

These two are the polarizing forms of the corresponding cohomology group.
Their existence follows from the fact that $X$ is projective and smooth and
they are constructed via a very ample line bundle [Del71].

We also have that, via the two embeddings $k\hookrightarrow\mathbb{C}$ and
$(2\pi i)^{-d}\mathbb{Q}\hookrightarrow\mathbb{C}$, and the comparison
isomorphism

$P^{d}_{X}:H^{d}(X/k)\otimes_{k}\mathbb{C}\rightarrow
H^{d}(X^{an}_{\mathbb{C}},\mathbb{Q})\otimes_{\mathbb{Q}}\mathbb{C}$

the two bilinear forms $\langle,\rangle_{DR}$ and $\langle,\rangle_{B}$ are
compatible under $P^{d}_{X}$, meaning that

$\langle v,w\rangle_{DR}=\langle P^{d}_{X}(v),P^{d}_{X}(w)\rangle_{B},\forall
v,w\in H^{d}_{DR}(X/k)\otimes_{k}\mathbb{C}.$ (87)

##### Relations on periods-Notation

From now on we assume that $d=n=\dim_{k}X$. We can and do consider from now on
the above polarizing forms $\langle,\rangle_{DR}$, $\langle,\rangle_{B}$, and
the form $\langle,\rangle$ as vectors in the spaces
$H^{n}_{DR}(X/k)^{*}\otimes_{k}H^{n}_{DR}(X/k)^{*}$,
$(H^{n}(X^{an}_{\mathbb{C}},\mathbb{Q})^{*}\otimes_{\mathbb{Q}}H^{n}(X^{an}_{\mathbb{C}},\mathbb{Q})^{*})(-n)$,
and
$H^{n}(X^{an}_{\mathbb{C}},\mathbb{Q})^{*}\otimes_{\mathbb{Q}}H^{n}(X^{an}_{\mathbb{C}},\mathbb{Q})^{*}$
respectively.

In this case, i.e. $d=n$, via Poincaré duality, these forms will correspond to
elements $t_{DR}\in H^{n}_{DR}(X/k)\otimes_{k}H^{n}_{DR}(X/k)$,
$t_{B}\in(H^{n}(X^{an}_{\mathbb{C}},\mathbb{Q})\otimes_{\mathbb{Q}}H^{n}(X^{an}_{\mathbb{C}},\mathbb{Q}))(n)$,
and $t\in
H^{n}(X^{an}_{\mathbb{C}},\mathbb{Q})\otimes_{\mathbb{Q}}H^{n}(X^{an}_{\mathbb{C}},\mathbb{Q})$,
respectively.

The compatibility of the comparison isomorphism $P^{n}_{X}$ with Poincaré
duality implies that $P^{n}_{X}\otimes P^{n}_{X}(t_{DR})=t_{B}=(2\pi i)^{n}t$.
In particular $t_{DR}$ is a Hodge class defined over the field $k$. For cycles
such as this it is known101010See page $169$ of [And89]. that they impose
polynomial relations among the $n$-periods with coefficients in the field
$k((2\pi i)^{n})$.

In what follows we show that the relations constructed by $t_{DR}$ are in fact
the Riemann-relations, i.e. they are the equations imposed on the $n$-periods
by (87). This is used without proof by André in, essentially, the case were
$n=1$. The author is sure that this part is known to experts in the field and
includes it only for the sake of completeness of the exposition.

Notation: We consider from now on a fixed basis $\\{\gamma_{i}:1\leq
i\leq\mu:=\dim_{\mathbb{Q}}H^{n}(X^{an}_{\mathbb{C}},\mathbb{Q})\\}$ of
$H_{n}(X^{an}_{\mathbb{C}},\mathbb{Q})$ and we let $\gamma_{i}^{*}$ be the
elements of its dual basis, which constitutes a basis of
$H^{n}(X^{an}_{\mathbb{C}},\mathbb{Q})$. We also consider $\omega_{i}$, $1\leq
i\leq\mu$, a fixed $k$-basis of $H^{n}_{DR}(X/k)$.

With respect to these choices the isomorphism $P^{n}_{X}$ corresponds to the
matrix $(\int_{\gamma_{j}}\omega_{i})$. We denote this matrix also by
$P^{n}_{X}$ so that the isomorphism is nothing but
$P^{n}_{X}(v)=\prescript{t}{}{v}P^{n}_{X}$, were on the right we have the
matrix acting on the right. Vectors in the various spaces will be considered
as column vectors in the various bases. Finally, we denote the matrix of the
$n$-periods by $P:=(2\pi i)^{-n}P^{n}_{X}$.

With the above notation fixed we let
$\langle\omega_{i},\omega_{j}\rangle_{DR}=d_{i,j}$ and let
$M_{DR}=(d_{i,j})\in\operatorname{GL}_{\mu}(k)$, which will be the matrix
corresponding to the form $\langle,\rangle_{DR}$, i.e.

$\langle v,w\rangle_{DR}=\prescript{t}{}{v}M_{DR}w$.

Considering, alternatively as above, $\langle,\rangle_{DR}$ as an element of
the space $H^{n}_{DR}(X/k)^{*}\otimes_{k}H^{n}_{DR}(X/k)$, the above are
equivalent to

$\langle,\rangle_{DR}=\displaystyle\sum_{i,j=1}^{\mu}d_{i,j}\omega^{*}_{i}\otimes\omega^{*}_{j}$.

Similarly we let
$q_{i,j}=\langle\gamma^{*}_{i},\gamma^{*}_{j}\rangle\in\mathbb{Q}$ and set
$M_{B}=(q_{i,j})\in\operatorname{GL}_{\mu}(\mathbb{Q})$. This implies that
$\langle\gamma_{i}^{*},\gamma_{j}^{*}\rangle=(2\pi i)^{-n}q_{i,j}$. Same as
above these relations can be rewritten as

$\langle v,w\rangle=\prescript{t}{}{v}M_{B}w$ and $\langle
v,w\rangle_{B}=\prescript{t}{}{v}((2\pi i)^{-n}M_{B})w$,

for all $v,w\in H^{n}(X^{an}_{\mathbb{C}},\mathbb{C})$. Alternatively, if we
were to consider $\langle,\rangle$ and $\langle,\rangle_{B}$ as elements of
$H^{n}(X^{an}_{\mathbb{C}},\mathbb{C})^{*}\otimes_{\mathbb{C}}H^{n}(X^{an}_{\mathbb{C}},\mathbb{C})^{*}$
we can write these as
$\langle,\rangle=\sum_{i,j=1}^{\mu}q_{i,j}\gamma_{i}\otimes\gamma_{j}$, and
$\langle,\rangle_{B}=\sum_{i,j=1}^{\mu}(2\pi
i)^{-n}q_{i,j}\gamma_{i}\otimes\gamma_{j}$ respectively.

We now consider the Poincaré duality isomorphisms
$\Pi_{DR}:H^{n}_{DR}(X/k)\rightarrow H^{n}_{DR}(X/k)^{*}$ and
$\Pi_{B}:H^{n}(X^{an}_{\mathbb{C}},\mathbb{Q})\rightarrow
H^{n}(X^{an}_{\mathbb{C}},\mathbb{Q})$, which we have since $\dim_{k}X=n$.
With respect to the bases $\\{\omega_{i}\\}$ and $\\{\omega^{*}_{i}\\}$ the
isomorphism $\Pi_{DR}$ corresponds to an invertible matrix which we denote by
$A_{DR}\in\operatorname{GL}_{\mu}(k)$. Similarly, with respect to the bases
$\\{\gamma_{i}\\}$ and $\\{\gamma^{*}_{i}\\}$ we get the invertible matrix
$A_{B}$ corresponding to $\Pi_{B}$.

Finally, let us write
$t_{DR}=\sum_{i,j=1}^{\mu}\lambda_{i,j}\omega_{i}\otimes\omega_{j}$,
$t=\sum_{i,j=1}^{\mu}\tau_{i,j}\gamma_{i}^{*}\otimes\gamma_{j}^{*}$ and
$t_{B}=\sum_{i,j=1}^{\mu}(2\pi
i)^{n}\tau_{i,j}\gamma_{i}^{*}\otimes\gamma_{j}^{*}$, where $\lambda_{i,j}\in
k$ and $\tau_{i,j}\in\mathbb{Q}$. We also define
$\Lambda_{DR}=(\lambda_{i,j})$ and $\Lambda_{\mathbb{Q}}=(\tau_{i,j})$.

##### Classes and forms

With the above notation fixed from now on we turn to describing the relation
between the classes $t_{DR}$, and $t$ and the respective forms.

By definition we have $\Pi_{DR}^{\otimes 2}(t_{DR})=\langle,\rangle_{DR}$.
This implies that

$\sum_{i,j=1}^{\mu}\lambda_{i,j}\Pi_{DR}(\omega_{i})\otimes\Pi_{DR}(\omega_{j})=\sum_{i,j=1}^{\mu}d_{i,j}\omega_{i}^{*}\otimes\omega_{j}^{*}.$
(88)

We know that $\Pi_{DR}(\omega_{i})=\Sigma a_{i,j}\omega_{j}^{*}$, with
$A_{DR}=(a_{i,j})$ . Applying this to (88) its is easy to see, with a few
trivial computations, that $\prescript{t}{}{A}_{DR}\Lambda_{DR}A_{DR}=M_{DR}$,
or equivalently we get the equality

$\Lambda_{DR}=\prescript{t}{}{A}^{-1}_{DR}M_{DR}A^{-1}_{DR}.$ (89)

Similarly for the pair $t$ and $\langle,\rangle$ we find that

$\Lambda_{\mathbb{Q}}=\prescript{t}{}{A}_{B}^{-1}M_{B}A_{B}^{-1},$ (90)

coming from the equality $\Pi_{B}^{\otimes 2}(t)=\langle,\rangle$.

The relation given by $t_{DR}$.

We review how a relation on the $n$-periods is constructed from $t_{DR}$. We
start from the equality $(2\pi i)^{-n}(P_{X}^{n})^{\otimes 2}(t_{DR})=t$. This
in turn implies that for all $l,m$, with the notation as above, we have

$\sum_{i,j=1}^{\mu}\lambda_{i,j}((2\pi
i)^{-n}\int_{\gamma_{l}}\omega_{i})((2\pi
i)^{-n}\int_{\gamma_{m}}\omega_{j})=(2\pi i)^{-n}\tau_{l,m}.$ (91)

These equations are the relations between $n$-periods that we eluded to
earlier. Putting them altogether the previous relation is equivalent to the
equality

$\prescript{t}{}{P}\Lambda_{DR}P=(2\pi i)^{-n}\Lambda_{\mathbb{Q}}.$ (92)

Comparing the matrices $A_{B}$ and $A_{DR}$.

Earlier on we had the matrices $A_{DR}$ and $A_{B}$ corresponding to the
respective Poincaré duality isomorphisms. We saw in (89) and (90) how these
matrices relate the “$\Lambda$-matrices” and “$M$-matrices”. We would like to
replace the “$\Lambda$-matrices in (92) by the corresponding “$M$-matrices”,
showing thus that the relations created are nothing but the Riemann-
relations111111See §7.1 for a definition and the reason of why we needed
these. . The first step is to describe how the matrices $A_{DR}$ and $A_{B}$
relate to one another.

We had the isomorphisms $\Pi_{DR}$ and $\Pi_{B}$ and the matrices $A_{DR}$ and
$A_{B}$ that represented these with respect to the bases we have chosen. We
know that the comparison isomorphism $P^{n}_{X}$ respects Poincaré duality,
meaning that the following diagram commutes:

${H^{n}_{DR}(X/k)\otimes_{k}\mathbb{C}}$${H^{n}(X^{an}_{\mathbb{C}},\mathbb{Q})\otimes_{\mathbb{Q}}\mathbb{C}}$${H^{n}_{DR}(X/k)^{*}\otimes_{k}\mathbb{C}}$${H^{n}(X^{an}_{\mathbb{C}},\mathbb{Q})\otimes_{\mathbb{Q}}\mathbb{C}}$$\scriptstyle{\Pi_{DR}\otimes_{k}\mathbb{C}}$$\scriptstyle{P^{n}_{X}}$$\scriptstyle{\Pi_{B}\otimes_{\mathbb{Q}}\mathbb{C}}$$\scriptstyle{(P^{n}_{X})^{\vee}}$

where $(P^{n}_{X})^{\vee}(f)=f\circ P^{n}_{X}$ for all $f\in
H^{n}(X^{an}_{\mathbb{C}},\mathbb{Q})^{*}\otimes_{\mathbb{Q}}\mathbb{C}$.

Looking at what the relation of the above diagram, i.e.
$\Pi_{DR}\otimes_{k}\mathbb{C}=(P_{X}^{n})^{\vee}\circ(\Pi_{B}\otimes_{\mathbb{Q}}\mathbb{C})\circ
P^{n}_{X}$, does to the basis $\\{\omega_{i}\\}$, and using the fact that with
respect to the bases $\\{\gamma_{j}\\}$ and $\\{\omega_{i}^{*}\\}$ the matrix
representing $(P^{n}_{X})^{\vee}$ will be the matrix
$\prescript{t}{}{(\int_{\gamma_{j}}\omega_{i})}$, i.e. the transpose of
$P^{n}_{X}$, we conclude that

$A_{DR}=(\int_{\gamma_{j}}\omega_{i})\cdot
A_{B}\cdot\prescript{t}{}{(\int_{\gamma_{j}}\omega_{i})}.$ (93)

Conclusions

Combining (92) with (89) and (90) we get

$\displaystyle\prescript{t}{}{P}(\prescript{t}{}{A}^{-1}_{DR}M_{DR}A^{-1}_{DR})P=(2\pi
i)^{-n}(\prescript{t}{}{A}_{B}^{-1}M_{B}A_{B}^{-1}).$ (94)

From (93) we get

$\prescript{t}{}{P}\prescript{t}{}{A}^{-1}_{DR}=\frac{1}{(2\pi
i)^{n}}\prescript{t}{}{A}_{B}^{-1}P,\text{ and}$ (95)
$A^{-1}_{DR}P=\frac{1}{(2\pi i)^{n}}\prescript{t}{}{P}^{-1}A_{B}^{-1}.$ (96)

Using (95) and (96) together with (94) we get

$PM_{B}\prescript{t}{}{P}=(2\pi i)^{-n}M_{DR}.$ (97)

But, this is the relation we get between the above matrices by looking at the
equation (87) and translating it in terms of matrices. Indeed, (87) translates
to

$\prescript{t}{}{v}P^{n}_{X}((2\pi
i)^{-n}M_{B}){}\prescript{t}{}{(\prescript{t}{}{w}P^{n}_{X})}=\prescript{t}{}{v}M_{DR}w$
for all $v,w\in H^{n}_{DR}(X/k)\otimes_{k}\mathbb{C}$.

From this we recover (97) by multiplying on both sides by $(2\pi i)^{-n}$ and
noting that $P=(2\pi i)^{-n}P^{n}_{X}$.

What is actually of use to us is not exactly (97) but rather the same relation
for the transpose of $P$. To obtain this, first from (97) we get trivially

$(2\pi i)^{n}M_{B}=P^{-1}M_{DR}\prescript{t}{}{P}^{-1}$,

then taking inverses on both sides we get

$\prescript{t}{}{P}M_{DR}^{-1}P=(2\pi i)^{-n}M_{B}^{-1}.$ (98)

####  A.2 The relative case

Setting: We consider $f:X\rightarrow S$ a smooth projective morphism of
$k$-varieties itself defined over the same subfield $k$ of $\mathbb{C}$. We
also assume that $S$ is a smooth connected curve which is not necessarily
complete over $k$ and the dimension of the fibers of $f$ is $n$.

We then have, for all $d\in\mathbb{N}$, the relative version of the comparison
isomorphism between the algebraic de Rham and the Betti cohomology

$P^{d}_{X/S}:H^{d}_{DR}(X/S)\otimes_{\mathcal{O}_{S}}\mathcal{O}_{S^{an}_{\mathbb{C}}}\rightarrow
R^{d}f^{an}_{*}\mathbb{Q}_{X^{an}_{\mathbb{C}}}\otimes_{\mathbb{Q}_{S^{an}_{\mathbb{C}}}}\mathcal{O}_{S^{an}_{\mathbb{C}}}.$
(99)

Once again we let, in parallel to the non-relative case we studied earlier,
$\mu$ denote the rank of these sheaves.

We once again have the same picture, as far as polarizing forms are concerned,
as in the non-relative case. In other words we have $\langle,\rangle_{DR}$ a
polarizing form of the de Rham cohomology sheaves $H^{d}_{DR}(X/S)$ which is
defined over the field $k$, and a polarizing form $\langle,\rangle_{B}=(2\pi
i)^{-n}\langle,\rangle$ of the sheaves on the right of (99). These two forms
will be compatible with the relative isomorphism (99), meaning that we have

$\langle P^{d}_{X/S}(v),P^{d}_{X/S}(w)\rangle_{B}=\langle v,w\rangle_{DR},$
(100)

holds for all sections $v,w$ of the sheaf on the right of (99).

From now on we focus on the case $d=n$. We choose $U\subset S$ a non-empty
affine open subset. Then the form $\langle,\rangle_{DR}|_{U}$ will map, via
the relative version of the Poincaré duality isomorphism, to a class
$t_{DR}\in H^{n}_{DR}(X/S)\otimes_{\mathcal{O}_{S}}H^{n}_{DR}(X/S)|_{U}$.

Similarly we repeat this process for the forms $\langle,\rangle$ and
$\langle,\rangle_{B}$, over the analytification $U^{an}_{\mathbb{C}}$, to get
elements
$t\in(R^{n}f_{*}\mathbb{Q}\otimes_{\mathbb{Q}_{S^{an}_{\mathbb{C}}}}R^{n}f_{*}\mathbb{Q})|_{U^{an}_{\mathbb{C}}}$
and
$t_{B}\in(R^{n}f_{*}\mathbb{Q}\otimes_{\mathbb{Q}_{S^{an}_{\mathbb{C}}}}R^{n}f_{*}\mathbb{Q})(n)|_{U^{an}_{\mathbb{C}}}$
with $t_{B}=(2\pi i)^{n}t$.

Compatibility of Poincaré duality with the relative comparison isomorphism
shows that $P^{n}_{X/S}\otimes P^{n}_{X/S}|_{U}(t_{DR})=t_{B}$. In other words
the class $t_{DR}$ is a relative Hodge class thus defining polynomial
relations among the relative $n$-periods.

Now we can repeat the arguments we made in the non-relative case. First, we
choose $\\{\omega_{i}\\}$ a basis of section of $H^{n}_{DR}(X/S)$ over the
affine open subset $U\subset S$ and $\\{\gamma_{j}^{*}\\}$ a frame of
$R^{n}f_{*}^{an}\mathbb{Q}_{X^{an}_{\mathbb{C}}}|_{V}$, or equivalently a
frame $\\{\gamma_{j}\\}$ of the relative homology
$R_{n}f^{an}_{*}\mathbb{Q}_{X^{an}_{\mathbb{C}}}|_{V}$, where $V\subset
U^{an}_{\mathbb{C}}$ is some open analytic subset. We get that the matrix
$P_{X/S}=((2\pi i)^{-n}\int_{\gamma_{j}}\omega_{i})$ satisfies

$P_{X/S}M_{B}\prescript{t}{}{P}_{X/S}=(2\pi i)^{-n}M_{DR},$ (101)

where $M_{B}$ and $M_{DR}$ are the matrices of the forms $\langle,\rangle$ and
$\langle,\rangle_{DR}$ with respect to the basis of section and the frame
chosen above.

The same process as before shows us that (101) is equivalent to the validity
of the polynomial relations on the relative $n$-periods defined by the
relative Hodge class $t_{DR}$. Finally, the same elementary argument as before
shows the validity of the relative analogue of relation (98), i.e. the Riemann
relations that we use in §7.1.

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