# Frobenius trace fields of cohomologically rigid local systems

Raju Krishnamoorthy<EMAIL_ADDRESS>and  Yeuk Hay Joshua Lam
<EMAIL_ADDRESS>Humboldt Universität Berlin, Institut für Mathematik-
Alg.Geo., Rudower Chaussee 25 Berlin, Germany

###### Abstract.

Let $X/\mathbb{C}$ be a smooth projective variety and let $L$ be an
irreducible $\overline{\mathbb{Q}}_{\ell}$-local system on $X$ with torsion
determinant. Suppose $L$ is cohomologically rigid. The pair $(X,L)$ may be
spreaded out to a finitely generated base, and therefore reduced modulo $p$
for almost all $p$; the Frobenius traces of this mod $p$ reduction lie in a
number field $F_{p}$, by a theorem of Deligne. We investigate to what extent
$F_{p}$ depends on $p$. We prove that for a positive density of primes $p$,
the $F_{p}$’s are contained in a fixed number field. More precisely, we prove
that $F_{p}$ is unramified at primes $\ell$ such that $\ell\neq p$ and $\ell$
large, where the largeness condition is uniform and does not depend on $p$,
and also that $F_{p}$ is unramified at $p$ assuming a further condition on
$p$. We also speculate on the relation between the uniform boundedness of the
$F_{p}$’s, and the local system $L$ being _strongly_ of geometric origin, a
notion due to Langer-Simpson.

###### Contents

  1. 1 Introduction
  2. 2 Spreading out
  3. 3 Companions
  4. 4 Degree of trace fields
  5. 5 Proof of Theorem 1.1.7 away from $\text{char}(s)$
  6. 6 $F$-isocrystals and proof of Theorem 1.1.7
  7. 7 The adjoint, stable trace fields, and a question

##  1\. Introduction

In this paper, we investigate an arithmetic property of rigid local systems on
a complex variety $X$. More precisely, to such a local system $L$ and a prime
number $p$, there is a naturally associated number field, which is by
definition the field generated by the traces of Frobenius elements at $p$. Our
motivating question is whether these number fields are independent of $p$.

###  1.1. Results

For a smooth complex variety $Y$, we denote by $Y^{an}$ the associated complex
manifold. To state our main results, we first need the following definition,
due to Esnault-Groechenig.

######  Definition 1.1.1.

Let $Y/k$ be a smooth projective variety over a perfect field. Let
$\ell\neq\text{char}(k)$ be a prime number. Let $L$ be a lisse
$\overline{\mathbb{Q}}_{\ell}$-sheaf on $Y$ that is geometrically irreducible
and set $\text{End}^{0}(L)\subset\text{End}(L)$ to be the subsheaf of
traceless endomorphisms. We say $L$ is _cohomologically rigid_ if:

$h^{1}_{\text{\'{e}t}}(Y_{\bar{k}},\text{End}^{0}(L))=0.$

If $k=\mathbb{C}$, we may make the same definition for a $\mathbb{C}$-local
system on $Y^{\text{an}}$. If $\text{char}(k)=p>0$, and $\mathcal{E}$ is an
$F$-isocrystal on $Y$ with coefficients in $\overline{\mathbb{Q}}_{p}$, we say
that $\mathcal{E}$ is cohomologically rigid if
$h^{1}_{\text{crys}}(Y_{\bar{k}},\text{End}^{0}(\mathcal{E}))=0.$

Let $X/\mathbb{C}$ be a smooth connected projective complex variety, with
associated complex manifold $X^{an}$. Fix a positive integer $n\geq 1$. We
consider the finite collection
$\text{CohRig}_{d}^{n}(X^{\text{an}},\mathbb{C})$ of irreducible
cohomologically rigid $\mathbb{C}$ local systems on $X^{\text{an}}$ of rank
$n$ and determinant of order dividing $d$. Fixing an auxiliary prime $\ell$
and a (abstract, non-continuous) field isomorphism
$\iota\colon\mathbb{C}\rightarrow\overline{\mathbb{Q}}_{\ell}$, this set is
identified with
$\text{CohRig}_{d}^{n}(X^{\text{an}},\overline{\mathbb{Q}}_{\ell})$.

It is a theorem of Esnault-Groechenig that for every triple $(\ell,n,d)$ as
above, every element
$L\in\text{CohRig}_{d}^{n}(X^{an},\overline{\mathbb{Q}}_{\ell})$ is integral,
i.e. after picking a basepoint $x$, $L$ corresponds to a representation

$\pi_{1}(X^{\text{an}},x)\rightarrow GL_{n}(\mathcal{O}_{E}),$

where $\mathcal{O}_{E}$ is the ring of integers of an $\ell$-adic local field
$E$. Therefore it naturally extends to a _continuous_ representation of the
profinite completion: $\rho\colon\pi_{1}^{\text{\'{e}t}}(X,x)\rightarrow
GL_{n}(\mathcal{O}_{E})$. Abusing notation, we refer to this $\mathcal{O}_{E}$
local system also by $L$.

######  Definition 1.1.2.

Let $X/\mathbb{C}$ be a smooth projective variety, let $\ell$ be a prime
number, and let $L$ be an étale $\overline{\mathbb{Q}}_{\ell}$ local system on
$X$ whose determinant has order dividing $d$. A _spreading-out_ of the pair
$(X,L)$ is a quadruple: $(\mathfrak{X},\mathcal{L},S,\iota)$ where:

  * •

$S={\rm Spec}(R)$ is an integral affine scheme, with
$\mathbb{Z}[1/\ell]\subset R\subset\mathbb{C}$, whose structure morphism
$S\rightarrow{\rm Spec}(\mathbb{Z}[1/\ell])$ is smooth,

  * •

$\mathfrak{X}\rightarrow S$ is a smooth projective morphism, $\mathcal{L}$ is
a lisse $\overline{\mathbb{Q}}_{\ell}$ sheaf on $\mathfrak{X}$ whose
determinant is isomorphic to an integral power of the cyclotomic character
tensored with a $d$-torsion character, i.e.,

$\det(\mathcal{L})^{\otimes d}\cong\overline{\mathbb{Q}}_{\ell}(m)$

for some integer $m$,

  * •

$\iota$ is an isomorphism $\iota\colon\mathfrak{X}_{\mathbb{C}}\rightarrow X$
such that $\iota^{*}L\cong\mathcal{L}|_{\mathfrak{X}_{\mathbb{C}}}$.

We comment on the condition on $\det(\mathcal{L})$ in Remark 1.1.9. We now
come to the second main player of our work.

######  Definition 1.1.3 (Frobenius trace fields).

Let $(\mathfrak{X},\mathcal{L},S,\iota)$ be a spreading out of $(X,L)$, as in
Definition 1.1.2. For any closed point $s$ of $S$, let $\mathfrak{X}_{s}$
denote the fiber of $\mathfrak{X}\rightarrow S$ over $s$, and write $\rho_{s}$
for the representation of $\pi_{1}^{\text{\'{e}t}}(\mathfrak{X}_{s})$
corresponding to $\mathcal{L}|_{\mathfrak{X}_{s}}$ (after picking a
basepoint).

  * •

Let $F_{s}\subset\overline{\mathbb{Q}}_{\ell}$, the _Frobenius trace field at
$s$_, be the field generated (over $\mathbb{Q}$) by the traces of $\rho_{s}$
applied to conjugacy classes of Frobenius elements at closed points of
$\mathfrak{X}_{s}$.

  * •

Let $\tilde{F}_{s}\subset F_{s}\subset\overline{\mathbb{Q}}_{\ell}$, the
_stable trace field at $s$_, be the intersection of the Frobenius trace fields
of $\mathcal{L}|_{\mathfrak{X}_{s^{\prime}}}$, as ${\rm
Spec}(\mathbb{F}_{q^{\prime}})=s^{\prime}\rightarrow s={\rm
Spec}(\mathbb{F}_{q})$ ranges over all finite extensions of $s$ (equivalently,
as $q^{\prime}$ ranges over powers of $q$).

We say that $(\mathfrak{X},\mathcal{L},S,\iota)$ has _bounded Frobenius
traces_ if there exists a number field $F\subset\overline{\mathbb{Q}}_{\ell}$
such that $F_{s}\subset F$ for all closed points $s$ and _bounded stable
Frobenius traces_ if there exists a number field $\tilde{F}$ such that
$\tilde{F}_{s}\subset\tilde{F}$ for all closed points $s$.

That $F_{s}$ itself is a number field follows from works of Lafforgue [Laf02]
and Deligne [Del12]. We view the number fields $F_{s}$ as interesting and
concrete arithmetic invariants of a local system; one important feature is
that $F_{s}$ controls the _companions_ of the local system
$\mathcal{L}|_{\mathfrak{X}_{s}}$. In this paper, we are interested in whether
rigid local systems admit spreading outs with bounded Frobenius traces.

We first give the following simple-to-state results to give a flavor of the
statements, since they do not involve various quantifiers appearing in our
main result Theorem 1.1.7. For simplicity, for the third part of the result,
we have chosen to state it for $X$ which may be defined over a number field;
the version over a finitely generated base is analogous.

######  Theorem 1.1.4.

Let $X/\mathbb{C}$ be a smooth projective variety and $n,d\geq 1$. Let
$L\in\text{CohRig}_{d}^{n}(X,\overline{\mathbb{Q}}_{\ell})$.

  1. (1)

There exists a spreading out $(\mathfrak{X},\mathcal{L},S,\iota)$ such that
the associated projective local system has bounded Frobenius traces;
concretely, the adjoint local system ${\rm
ad}(\mathcal{L})\vcentcolon=\mathcal{L}\otimes\mathcal{L}^{\vee}$ has bounded
Frobenius traces.

  2. (2)

There exists a spreading out $(\mathfrak{X},\mathcal{L},S,\iota)$ with bounded
stable Frobenius traces.

  3. (3)

Suppose that $X$ may be defined over $\bar{\mathbb{Q}}$, and let
$L\in\text{CohRig}_{d}^{n}(X,\overline{\mathbb{Q}}_{\ell})$. Then there exists
a spreading out $(\mathfrak{X},\mathcal{L},S,\iota)$, where $S={\rm
Spec}(\mathcal{O}_{K}[1/N])$, with $K$ a number field and $N$ an integer, and
another number field $E$, such that for a positive density111in the sense of
Cebotarev density of points $s$ of $S$, the local system
$\mathcal{L}|_{\mathfrak{X}_{s}}$ has Frobenius trace field contained in $E$.

######  Remark 1.1.5.

For both points (1) and (2), we can be more explicit about the number field
containing all Frobenius traces: it is not difficult to see that we can take
the number field where all cohomologically rigid local systems (of rank $n$
and determinant of order dividing $d$) on $X^{an}$ are defined. Moreover, with
slightly more work, one can show that (1) and (2) hold for _any_ spreading
out.

######  Remark 1.1.6.

In fact, points (1) and (2) of Theorem 1.1.4 follow from only small parts of
our arguments for our main result Theorem 1.1.7 below. Indeed, in some sense,
the main point of our paper is to study the difference in Frobenius trace
fields between a local system and its projectivized local system, with the
former being much more delicate.

There is very interesting recent work of Klevdal-Patrikis [KP23] which shows
that, for Shimura varieties in the superrigid regime and the natural local
systems on them, the Frobenius trace fields of the adjoint local system is
independent of $p$; in fact they prove something much stronger. It would be
interesting to prove the boundedness of Frobenius trace fields for the local
systems themselves on such Shimura varieties.

We now state our main result:

######  Theorem 1.1.7.

Suppose $L\in\text{CohRig}_{d}^{n}(X,\overline{\mathbb{Q}}_{\ell})$. Then
there exists a spreading out $(\mathfrak{X},\mathcal{L},S,\iota)$. For any
such spreading out, there exists integers $N$ (depending on $X,n,d$),
$\ell_{0}$ (depending on $\mathfrak{X},n,d$), and a number field $E$
(depending on $\mathfrak{X},n,d$) such that the following hold:

  1. (1)

For any closed point $s$ of $S$, $[F_{s}:\mathbb{Q}]\leq N$;

  2. (2)

For any closed point $s$ of $S$, and for any prime $m>\ell_{0}$ different from
the characteristic of $s$, $F_{s}$ is unramified over $m$;

  3. (3)

replacing $S$ by a scheme étale over it if necessary, there exists a number
field $E\subset\overline{\mathbb{Q}}_{\ell}$, and
$\alpha=\alpha(X,n,d)\in\mathbb{Z}_{\geq 1}$, such that for any prime number
$p$ and any $s\colon{\rm Spec}(\mathbb{F}_{q})\rightarrow S$ with
$\mathbb{F}_{q}$ containing $\mathbb{F}_{p^{\alpha}}$, the trace field of
$\mathcal{L}|_{\mathfrak{X}_{s}}$ is contained in $E$.

As may be seen, point (2) of the theorem is about the ramification of $F_{s}$
away from $\text{char}(s)$, the characteristic of $s$, and the proof of (3)
will proceed by analyzing the ramification of $F_{s}$ _at_ $\text{char}(s)$;
for this reason we sometimes refer to (2) as the theorem away from
$\text{char}(s)$ and to (3) as the theorem at $\text{char}(s)$.

We now make several remarks on Theorem 1.1.7.

######  Remark 1.1.8.

One important remark is that point (3) of Theorem 1.1.7 is certainly not true
for arbitrary $\mathbb{F}_{q}$ points of $S$, i.e. it is possible that $F_{s}$
is ramified at $p=\text{char}(\kappa(s))$ for infinitely many $s$. Let $K$ be
any number field such that $G_{K}$ has a character $\chi$ which is a square
root of the $\ell$-cyclotomic character. Consider the Legendre family of
elliptic curves on
$X_{K}\vcentcolon=\mathbb{P}_{K}^{1}\setminus\\{0,1,\infty\\}$, with
associated $\ell$-adic local system $L_{K}^{\prime}$ on $X_{K}$; now define
$L_{K}\vcentcolon=L_{K}^{\prime}\otimes\chi$ admits a spreading out
$\mathcal{L}$ to some $\mathfrak{X}$, and for any $s:{\rm
Spec}(\mathbb{F}_{p})\rightarrow\mathfrak{X}$, the Frobenius trace field of
$\mathcal{L}|_{s}$ is $\mathbb{Q}(\sqrt{p})$.

Indeed, the unramifiedness of $F_{s}$ at $p$ for arbitrary ${\rm
Spec}(\mathbb{F}_{q})$ points $s$ would imply that the Frobenius traces are
bounded222there are only finitely many number fields with bounded degree and
ramification.

######  Remark 1.1.9.

In Definition 1.1.2, we have specified that $\det(\mathcal{L})$ has a power
which is a power of the cyclotomic character. In some sense this is a somewhat
arbitrary condition: in principle one should be able to relax this condition,
for example to requiring that $\det(\mathcal{L})$ has a power which is _de
Rham_ at all places above $\ell$, or simply that $\det(\mathcal{L})$ has
bounded Frobenius traces. However, these conditions seem equally unnatural to
us: a priori, in light of Remark 1.1.8, one might hope that putting a
condition on $\det(\mathcal{L})$ (e.g. a condition on the Hodge-Tate weghts)
might pick out a twist of $\mathcal{L}$ with bounded Frobenius traces;
however, it is not even clear to us whether such a twist should exist (see
Section 1.2 for more on this), even assuming all conjectures.

We now give the proof of Theorem 1.1.4, which follows straightforwardly from
Theorem 1.1.7.

###### Proof of Theorem 1.1.4 given Theorem 1.1.7.

Point (1) is given in Section 7.1. Point (2) is immediate from point (3) of
Theorem 1.1.7.

We now give the proof of point (3) of Theorem 1.1.4. The spreading out
guaranteed in Theorem 1.1.7 only requires a finite extension of a field of
definition of $X$. Then Theorem 1.1.4 follows from Theorem 1.1.7 and the
following corollary of Cebotarev: for any number field $K$, finitely generated
subring $\mathcal{O}\subset K$, and positive integer $\alpha$, there exists a
finite extension $K^{\prime}/K$ and a finitely generated subring
$\mathcal{O}^{\prime}\subset K^{\prime}$ that contains $\mathcal{O}$, such
that for a positive proportion of the prime ideals $\mathfrak{p}^{\prime}$ of
$\mathcal{O}^{\prime}$, the residue field
$\mathcal{O}^{\prime}/\mathfrak{p}^{\prime}$ has degree over
$\mathbb{Z}/(\mathfrak{p}^{\prime}\cap\mathbb{Z})$, the underlying prime
field, divisible by $\alpha$. ∎

###  1.2. Discussions

The motivicity conjecture of Simpson [Sim92, p.9] states that each
$L\in\text{CohRig}_{d}^{n}(X^{\text{an}},\mathbb{C})$ is _of geometric origin_
(also referred to as _motivic_ in the following): this roughly means that
after restricting to a dense open subset $U\subset X$, it arises as a
subquotient (or even summand) in the cohomology of a family of smooth
projective varieties. Note that if $L$ is of geometric origin, with family
$\pi\colon\mathcal{Y}\rightarrow U$ and $i\geq 0$ such that $L|_{U}$ is a
summand of $R^{i}\pi_{*}\mathbb{C}$, then it is very well possible that
$L|_{U}$ is _not cut out by an algebraic cycle_.

Therefore, there is an a priori more restrictive notion, namely that $L$ is
cut out by algebraic cycles from the cohomology of a family of varieties; this
notion was introduced by Langer-Simpson [LS18] (see the end of Section 2.5 of
loc.cit.), and we follow their terminology and call such local systems
strongly of geometric origin, or strongly motivic.

######  Question 1.2.1.

Suppose $X$ is a smooth complex variety, and $L$ a local system on $X$. Does
the existence of a spreading out $(\mathfrak{X},S,\mathcal{L},\iota)$ with
bounded Frobenius traces imply that $L$ is strongly of geometric origin?

######  Remark 1.2.2.

It is straightforward to see that if $L$ is strongly motivic, then a spreading
out with bounded Frobenius traces does exist. Unfortunately, it seems hard to
probe Question 1.2.1 since there are no known examples of local systems of
geometric origin which are not strongly so. We note that local systems which
are strongly motivic admit spreading outs satisfying other properties, such as
being _de Rham_ at all places above $\ell$, which a priori seems independent
from having bounded Frobenius traces. Indeed, the relative Fontaine-Mazur
conjecture [LZ17] together with the Tate conjecture imply that a local system
is strongly motivic if it admits a spreading out which is de Rham at $\ell$.

Note also that, conjecturally, the subtlety between geometric and strongly
geometric local systems disappear once we consider the projectivized local
system. Indeed, in his beautiful work, Petrov [Pet23] shows that the adjoint
of any geometrically irreducible arithmetic
$\overline{\mathbb{Q}}_{\ell}$-local system is de Rham at places above $\ell$,
and hence this adjoint local system satisfies the assumptions of the relative
Fontaine-Mazur conjecture, and therefore is conjecturally strongly motivic.

For the case of rank two local systems, many cases of Question 1.2.1 have been
answered in the positive by [ST18] and [KYZ22]. However, this case is not
representative since motivic rank two local systems arise in the cohomology of
a family of abelian varieties, and in this case motivic implies strongly
motivic, by the work of Moonen [Moo98, Theorem 6.4]. To illustrate the
subtlety, note that if $X/\mathbb{C}$ is smooth projective, $\pi\colon
A\rightarrow X$ is an abelian scheme, and $L$ is an irreducible summand of
$R^{1}\pi_{*}\mathbb{C}$, then the claim is that there exists _another abelian
scheme_ $B\rightarrow X$, together with endomorphisms
$R\hookrightarrow\text{End}^{0}(B)$, such that $L$ is a component under the
decomposition of the first cohomology under $R$.333The arithmetic analog of
this discussion is the following. An _arithmetic local system_ is an étale
$\overline{\mathbb{Q}}_{\ell}$-local system $L$ on $X=X_{\mathbb{C}}$ that is
a sub-quotient of a local system $M$ that extends to a local system
$\mathbb{M}$ on $X_{K}$, where $K$ is a finitely generated field. (For
simplicity, we will assume that $L$ is a sub-representation of $M$.) It turns
out that, after extending $K$ by a finitely extension, $L$ also extends to an
étale local system $\mathbb{L}$ on $X_{K}$. However, there is _no reason_ that
there is an arithmetization $\mathbb{L}$ for which the map $L\rightarrow M$ is
$G_{K}$-equivariant.

###  1.3. Sketch of proof of Theorem 1.1.7

For this proof sketch we assume $d=1$, i.e. that our local systems, when
restricted to the geometric fundamental group, have trivial determinant. The
overall strategy is to analyze the ramification behavior of the trace field
$F_{s}$ at each prime $m$ by looking at the $m$-adic companions.

For point (1) note that the degree of the trace field is the number of
$\ell$-companions; that $\ell$-companions of cohomologically rigid local
systems are again cohomolgically rigid was proven by Esnault-Groechenig, and
we may conclude since there are only finitely many such local systems with
fixed rank and trivial determinant.

Let us write $p=\text{char}(s)$ and $k(s)$ for the residue field at $s$. For
the proof of point (2), we let $\ell_{0}=\ell_{0}(X,n,d)$ be a prime such that
for all prime numbers $m>\ell_{0}$, the topological, cohomologically rigid,
local systems on $X^{an}$ all have coefficients unramified at $m$. For a
spreading out $(\mathfrak{X},\mathcal{L},S,\iota)$, a point $s\in S$, and
$m\neq p$, let $\mathcal{L}_{m}$ be any $m$-companion of
$\mathcal{L}|_{\mathfrak{X}_{s}}$, which is an $m$-adic local system. Then
$\mathcal{L}_{m}|_{\mathfrak{X}_{\bar{s}}}$ also has unramified coefficients.
We then show that, after possibly discarding finitely many $m$’s, we may
descend this local system to $\mathfrak{X}_{s}$ itself, with _unramified
coefficients_ : indeed, it is immediate that there is a descent as a Weil
sheaf on $\mathfrak{X}_{s}$ (i.e. a representation of the Weil group), but to
get an étale sheaf we need an argument involving absolute irreducibility of
residual representations. This descent will not, in general, be the local
system we started with: rather, it is a _twist_ of the original by a character
of the base ${\rm Spec}(k(s))$; finally, since they are both étale sheaves,
one sees that the twisting cannot add ramification to the trace field at
$\ell$, as desired. The details are given at the end of Section 5.

For point (3) of Theorem 1.1.7, as mentioned above, we proceed by showing that
$F_{s}$ is also unramified at $m=\text{char}(s)=p$, at least under some
assumptions on the point $s$, which we now explain. We again analyze the
$p$-companions of $\mathcal{L}|_{\mathfrak{X}_{s}}$, which in this case means
$F$-isocrystals on $\mathfrak{X}_{s}$: these were shown to exist by Esnault-
Groechenig [EG20]. Let $\mathcal{F}$ be such a $p$-companion, and in the ideal
world one would want to show that $\mathcal{F}$ also has unramified
coefficients: this cannot be true in this generality, as Remark 1.1.8 shows,
and we will instead show that it is true as long as $s$ is an $\mathbb{F}_{q}$
point with $q$ divisible by some fixed integer $\alpha=\alpha(X,n,d)$, i.e.
after extending $s$. After extending $s:{\rm Spec}(\mathbb{F}_{q})\rightarrow
S$ to some $s^{\prime}:{\rm Spec}(\mathbb{F}_{q^{\prime}})\rightarrow S$, the
work of Esnault-Groechenig provides us with a collection of cohomologically
rigid $F$-isocrystals on $\mathfrak{X}_{s}$ which have unramified coefficients
by construction. The restriction $\mathcal{F}|_{\mathfrak{X}_{s^{\prime}}}$ is
isomorphic to one of these $F$-isocrystals up to twisting by a $F$-isocrystal
from the base ${\rm Spec}(\mathbb{F}_{q^{\prime}})$. Assuming $q^{\prime}$ is
sufficiently divisible, one can show that this twisting does not affect the
unramifiedness of the coefficients, which lets us conclude.

###  1.4. Related works

As mentioned above, the work most closely related to ours is perhaps that of
Klevdal-Patrikis [KP23]. The works of Snowden-Tsimerman [ST18] and
Krishnamoorthy-Yang-Zuo [KYZ22] prove motivicity of rank two local systems
given boundedness of Frobenius traces, under a mild condition on local
monodromies. In [Shi20], Shimizu proved that Frobenius traces are bounded for
local systems satisfying the assumptions of the Fontaine-Mazur conjecture,
assuming the latter as well as the Generalized Riemann Hypothesis.

###  1.5. Outline

In Section 2 we collect together results of various authors about spreading
out rigid local system. Section 4 proves point (1) of Theorem 1.1.7 concering
the degree of Frobenius trace fields. Sections 5 and 6, which make up the core
of the proof of Theorem 1.1.7, handle points (2) and (3) of Theorem 1.1.7
respectively: the former concerns ramification away from $p$ and the latter at
$p$. Finally, in Section 7 we deduce Theorem 1.1.4, which concern the adjoint
local system and the stable trace field, from our main theorem. We conclude
with discussions on trace fields of geometric local systems and a question
about weights of VHS associated to Galois conjugates of rigid local systems.

###  1.6. Notations and conventions

For any field $k$, we denote by $G_{k}$ its absolute Galois group, and for a
scheme $Z$ we write $\pi_{1}^{\text{\'{e}t}}(Z)$ for its étale fundamental
group. For maps of schemes $Z\rightarrow S$ and $S^{\prime}\rightarrow S$, we
denote by $Z_{S^{\prime}}$ the basechange $Z\times_{S}S^{\prime}$. The symbol
$q$ will always denote a power of a prime number $p$, and $\mathbb{F}_{q}$
denotes the finite field with $q$ elements. The notation
$N=N(A,B,C,\cdots)\in\mathbb{Z}$ means that the integer $N$ depends only on
$A,B,C,\cdots$. For a point $s:{\rm Spec}(\mathbb{F}_{q})\rightarrow S$,
$\bar{s}$ denotes a geometric point $\bar{s}:{\rm
Spec}(\overline{\mathbb{F}}_{q})\rightarrow S$ over $s$.

### Acknowledgments

Most of this work builds on the ground-breaking works of Esnault-Groechenig
[EG20, EG18]. We are grateful to Hélène Esnault, Bruno Klingler, Daniel Litt,
and Sasha Petrov for many enlightening discussions. Lam was supported by a
Dirichlet Fellowship during the course of this work. Krishnamoorthy was
supported by the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation program, grant agreement no. 101020009,
project TameHodge.

##  2\. Spreading out

As defined in the introduction,
$\text{CohRig}_{d}^{n}(X^{\text{an}},\overline{\mathbb{Q}}_{\ell})$ denotes
the collection of rank $n$ cohomologically rigid local systems whose
determinant is torsion of order dividing $d$. For any variety $X$ over a
perfect field $k$ and prime $\ell$ different from the characteristic of $k$,
we define the slight variant
$\text{CohRig}_{d}^{n}(X,\overline{\mathbb{Q}}_{\ell})$, which is the
collection of rank $n$, cohomologically rigid, étale
$\overline{\mathbb{Q}}_{\ell}$-local systems, whose determinant is torsion of
order dividing $d$; for a $\ell$-adic local field $E$, we denote by
$\text{CohRig}_{d}^{n}(X,E)$ the collection of $E$-local systems with the same
adjectives as above.

### 2.1.

The goal of this section is to prove the following

######  Lemma 2.1.1.

Let $X/\mathbb{C}$ be a smooth projective variety and fix integers $n,d\geq
1$. Then there exists a spreading out $\mathfrak{X}\rightarrow S$ such that
for every $L\in\text{CohRig}_{d}^{n}(X,\overline{\mathbb{Q}}_{\ell})$, there
is an extension of $L$ to an étale local system $\mathcal{L}$ on
$\mathfrak{X}$ with the further property that $\det(\mathcal{L})$ is torsion
of order $d$.

This result is certainly known to experts, and we merely collect together the
arguments of various authors, as we will indicate. As
$\text{CohRig}_{d}^{n}(X,\overline{\mathbb{Q}}_{\ell})$ is a finite set, it
suffices to show that for each $L$, there exists such a spreading out: this is
done in Lemma 2.3.1. In Section 2.2 we first show that we can spread out $X$
and $L$ to a finitely generated field, and then in Section 2.3 we show that
this can be extended to a finite type base.

###  2.2. Descent

######  Proposition 2.2.1.

Let $X/K$ be a smooth variety over a field of characteristic 0, let $E_{\ell}$
be an $\ell$-adic local field, and let $\mathcal{O}$ be the ring of integers
in $E_{\ell}$. Let $\rho\colon\pi_{1}(X_{K})\rightarrow{\rm
GL}_{n}(\mathcal{O})$ be an _abstract_ representation (i.e. not assumed to be
continuous), where $n$ is coprime to $\ell$, such that the restriction to
$\pi_{1}(X_{\bar{K}})$ has determinant of order $d\geq 1$. Then there exists a
finite extension $K^{\prime}/K$ and a representation
$\rho^{\prime}\colon\pi_{1}(X_{K^{\prime}})\rightarrow{\rm GL}_{n}(O)$ such
that

$\rho|_{\pi_{1}(X_{\bar{K}})}\cong\rho^{\prime}|_{\pi_{1}(X_{\bar{K}})}$

and $\rho^{\prime}$ has determinant of order $d$.

###### Proof.

First of all, by extending $K$, we may assume that $X$ has a $K$-point $x$.

Consider the representation $\rho|_{x}\colon G_{K}\rightarrow{\rm
GL}_{n}(\mathcal{O})$. We claim that after another finite extension of $K$,
the character $\det(\rho|_{x})\colon G_{K}\rightarrow\mathcal{O}^{\times}$ has
an $n^{\text{th}}$ root. Indeed, replacing $K$ by a finite extension if
necessary, we may assume that $\det(\rho_{x})$ is trivial mod $\pi$, where
$\pi$ is a uniformizer of $\mathcal{O}$. Then, as $(n,\ell)=1$, it follows
from Hensel’s lemma/the binomial theorem that there is a well-defined
(continuous) group homomorphism:

$\sqrt[n]{-}\colon 1+\pi\mathcal{O}\rightarrow\mathcal{O}^{\times}$

The tensor product $\rho\otimes\big{(}\sqrt[n]{\det(\rho|_{x})}\big{)}^{-1}$
will therefore have determinant order $d$, as required. ∎

######  Proposition 2.2.2.

Let $X/\mathbb{C}$ be a smooth projective variety. Let $L$ be an element of
$\text{CohRig}_{d}^{n}(X,\overline{\mathbb{Q}}_{\ell})$. Then there exists a
finitely generated subfield $K\subset\mathbb{C}$ and a variety $\mathcal{X}/K$
such that $\mathcal{X}\otimes\mathbb{C}\simeq X$, and $L$ extends to a local
system $\mathbb{L}$ on $\mathcal{X}/K$ with determinant of order dividing $d$.

As mentioned before, this is a slight enhancement of an argument of Simpson’s,
and very similar arguments have appeared in [EG18, Lit21]: see for example
[Lit21, Proposition 3.1.1] and the references therein.

###### Proof.

For ease of notation we will give the proof in the case $d=1$. The case of
general $d$ is essentially identical.

Suppose that $L$ has coefficients in the ring of integers $\mathcal{O}\subset
E_{\ell}$; we view $L$ equivalently as a free $\mathcal{O}$-module $M$ of rank
$n$ with an action of $\pi_{1}^{\text{\'{e}t}}(X)$ via
$\pi_{1}^{\text{\'{e}t}}(X)\rightarrow{\rm GL}(M)$.

Let $K\subset\mathbb{C}$ be a finitely generated field, and $\mathcal{X}/K$
such that $\mathcal{X}\otimes\mathbb{C}\simeq X$.

For each $g\in G_{K}$, let $\ {}^{g}M$ denote the representation $\rho\circ
Ad(g)$. As in [Sim92, Proof of Theorem 4], we assume that there exists
$c(g)\in{\rm GL}_{n}(E_{\ell})$ such that $M\otimes\mathbb{Q}$ and $\
{}^{g}M\otimes\mathbb{Q}$ are conjugate by $c(g)$; the element $c(g)$ is well
defined up to a scalar in $E_{\ell}^{\times}$.

As $\rho$ is irreducible, it follows from a Jordan-Zassenhaus type theorem
that there are only finitely many homothety classes of
$\pi_{1}^{\text{\'{e}t}}(X)$-invariant lattices inside $M\otimes\mathbb{Q}$
[Suh21, Theorem 1.1]. We claim that $G_{K}$ acts on this finite set of
homothety classes. Indeed, let $N\subset M\otimes\mathbb{Q}$ be a
$\pi_{1}^{\text{\'{e}t}}(X)$-stable lattice. Then we have an integral
representation $\pi_{1}^{\text{\'{e}t}}(X)\rightarrow{\rm GL}(N)$. Conjugating
by each $g\in G_{K}$, we obtain a (potentially non-isomorphic) integral
representation; however, by our assumption, the rational isomorphism class of
this new representation is simply $M\otimes\mathbb{Q}$. Therefore $G_{K}$ acts
on the set of _isomorphism classes_ of integral
$\pi_{1}^{\text{\'{e}t}}(X)$-subrepresentations of $M\otimes\mathbb{Q}$, i.e.
the set of homothety classes of stable lattices. By taking a finite extension
of $K$ we now assume the action is trivial, and by scaling each $c(g)$
appropriately we may therefore assume

$c(g):M\xrightarrow{\simeq}\ ^{g}M.$

as $\pi_{1}^{\text{\'{e}t}}(X)$-representations.

We claim that from the $c(g)$, we may construct a class $\xi\in
H^{2}(G_{K},\mathcal{O}^{\times})$, given by the following formula:

$\xi(g,h)=c(gh)^{-1}\circ\ ^{g}c(h)\circ c(g)\colon G_{K}\times
G_{K}\rightarrow\text{Aut}_{\pi_{1}(X_{\bar{K}})}(M)\cong\mathcal{O}^{\times}.$

(Again, here $\mathcal{O}^{\times}$ is equipped with the _trivial_ $G_{K}$
action.)

For the moment, let us assume that $\xi=0$ and see how to conclude. Indeed,
$\xi=0$ implies that the representation extends to a _not necessarily
continuous_ representation $\pi_{1}^{\text{\'{e}t}}(X_{K})\rightarrow GL(M)$.
(This is a general feature of semi-direct products.) Note that the restriction
to the geometric fundamental group is continuous, and moreover, the
projectivization is continuous by Simpson’s argument.

Using Proposition 2.2.1, by increasing $K$, we may modify our representation
such that its determinant has order $d$. Call this representation $\rho$. Let
$\mu_{d}{\rm GL}(M)$ denote the subgroup of elements of ${\rm GL}(M)$ with
$d$-torsion determinant, so that $\rho$ has image in $\mu_{d}{\rm GL}(M)$.
Then the natural map of $\ell$-adic groups:

$\pi\colon\mu_{d}{\rm GL}(M)\rightarrow PGL(M)$

is finite-to-one, and moreover is a local homeomorphism by [Con12, Lemma 5.3]
(as the valuation ring in any algebraic extension of $\mathbb{Q}_{\ell}$ is
Henselian.). As we know that the composition $\pi\circ\rho$ is continuous, it
follows that $\rho$ is continuous. Therefore, the $\rho$ we have constructed
satisfies the conclusion of the Proposition.

Furthermore, note that the injection of trivial $G_{K}$-modules:
$\mathcal{O}\hookrightarrow\bar{\mathbb{Z}}_{\ell}$ splits, and hence $\xi$
vanishes if and only if the image class in
$H^{2}(G_{K},\bar{\mathbb{Z}}_{\ell}^{\times})$ vanishes. Abusing notation, we
call this image class $\xi$.

Finally, we show that we can kill the Brauer class $\xi$ by passing to a
finite extension. Let us denote by $PL$ the projective local system associated
to $L$.

By [Sim92, Proof of Theorem 4], $PL$ extends to a local system on
$\mathcal{X}/K$; let us denote this by $P\mathbb{L}$. Moreover, by enlarging
$K$ if necessary, we may assume that the torsion character $\det(L)$ extends
to a rank one local system on $\mathcal{X}/K$.

On the other hand, the obstruction to $\det(L)$ extending is $c^{n}$, where
the latter denotes the image of $c$ under the map

$H^{2}(G_{K},\bar{\mathbb{Z}}_{\ell}^{\times})\rightarrow
H^{2}(G_{K},\bar{\mathbb{Z}}_{\ell}^{\times})$

induced by the short exact sequence

$0\rightarrow\mu_{n}\rightarrow\bar{\mathbb{Z}}_{\ell}^{\times}\xrightarrow[]{x\mapsto
x^{n}}\bar{\mathbb{Z}}_{\ell}^{\times}\rightarrow 0.$

By the previous discussion we have $c^{n}=0$, and hence $c$ comes from a class
$b$ in $H^{2}(G_{K},\mu_{n})$. Now, by [Sha, Proposition 2.2.16], $b$ lies in
the image of the inflation map
$\mathrm{Infl}:H^{2}(G_{K}/N,\mu_{n})\rightarrow H^{2}(G_{K},\mu_{n})$, for
some finite index normal subgroup $N$; finally, the composition
$H^{2}(G_{K}/N,\mu_{n})\xrightarrow[]{\mathrm{Infl}}H^{2}(G_{K},\mu_{n})\xrightarrow[]{\mathrm{Res}}H^{2}(N,\mu_{n})$
is zero, and so we may replace $K$ by the finite extension defined by $N$ to
kill the class $c$, as required. ∎

###  2.3. Spreading out over a finite type base

The following is a slight generalization of [Pet23, Proposition 6.1], which is
attributed to Litt, as well as to Liu-Zhu [LZ17].

######  Lemma 2.3.1.

Let $X/\mathbb{C}$ be a smooth projective variety, and
$L\in\text{CohRig}_{d}^{n}(X,\overline{\mathbb{Q}}_{\ell})$. Then there exists
a smooth $\mathbb{Z}$-algebra $A\subset\mathbb{C}$ and a spreading-out
$\mathfrak{X}\rightarrow{\rm Spec}(A)$ such that $L$ extends to a local system
$\mathcal{L}$ on all of $\mathfrak{X}$; moreover, we may choose
$\mathfrak{X}\rightarrow{\rm Spec}(A)$ such that $\mathcal{L}$ has torsion
determinant of order $d$.

###### Proof.

We denote by $\rho^{\text{geo}}:\pi_{1}(X)\rightarrow{\rm
GL}_{n}(\mathcal{O})$ the representation corresponding to $L$, where
$\mathcal{O}$ is the ring of integers of an $\ell$-adic local field
$E_{\ell}$; here we have used the integrality of rigid local systems.

By Proposition 2.2.2, there exists a finitely generated field
$K\subset\mathbb{C}$ such that $X$ descends to $K$ and moreover
$\rho^{\text{geo}}$ descends to a representation $\rho$ of $\pi_{1}(X_{K})$,
whose determinant has order $d$. By enlarging $E_{\ell}$ if necessary, we
assume that $\rho$ still takes values in ${\rm GL}_{n}(\mathcal{O})$. There
exists an integrally closed, finitely generated, subring $A^{\prime}\subset K$
such that $X$ extends to a smooth proper scheme $\mathfrak{X}/{\rm
Spec}(A^{\prime})$, and $\det(\rho)$ extends to $\mathfrak{X}$.

By enlarging $K$, we may assume we have a point $x\in X(K)$, and moreover that
$\bar{\rho}:\pi_{1}^{\text{\'{e}t}}(X_{K})\rightarrow{\rm
GL}_{n}(\mathcal{O}/\ell^{2})$ is trivial. Let $\rho_{x}:G_{K}\rightarrow{\rm
GL}_{n}(\mathcal{O})$ denote the corresponding representation.

Let $\mathcal{T}$ be the union of the set of prime numbers dividing $|{\rm
GL}_{n}(\mathcal{O}/\ell^{2})|$ and the set of height one primes where
$\det{\rho}$ is ramified. (Note that this means $\mathcal{T}$ contains the set
of primes that divide $|{\rm GL}_{n}(\mathcal{O}_{E_{\ell}}/\ell^{N})|$ for
all $N\geq 1$.)

Let $A\vcentcolon=A^{\prime}[\frac{1}{\mathcal{T}}]$, and denote by
$\mathfrak{X}_{A}$ the basechange of $\mathfrak{X}$ to ${\rm Spec}(A)$. Let
$\mathfrak{p}$ be a height one prime ideal of $A$, with associated discrete
valuation ring $A_{\mathfrak{p}}$, whose fraction field we denote by
$K_{\mathfrak{p}}$; let $p$ denote the characteristic of the residue field of
$A_{\mathfrak{p}}$. We first show that $\rho_{x}$ is unramified at
$\mathfrak{p}$.

The representation $\rho\colon\pi_{1}^{\text{\'{e}t}}(X_{K},x)\rightarrow{\rm
GL}_{n}(\mathcal{O})$ factors through the prime-to-$p$ quotient
$\pi_{1}^{\text{\'{e}t}}(X_{K},x)^{(p^{\prime})}$. By local constancy of the
prime-to-$p$ fundamental group, $G_{K_{\mathfrak{p}}}$ acts on
$\pi_{1}^{\text{\'{e}t}}(X_{K},x)^{(p^{\prime})}$ through the unramified
quotient. By the same argument as in [Pet23, Proof of Proposition 6.1], we
deduce that $\rho_{x}$ is unramified at $v$.

We now show that $\rho$ extends to a local system on $\mathfrak{X}_{A}$. For
each $j\geq 1$, let $K_{j}/K$ be the extension trivializing
$\rho_{x}\bmod\ell^{j}$, and let $A_{j}$ be the integral closure of $A$ in
$K_{j}$; the above discussion shows that $A_{j}$ is étale over $A$. As in
loc.cit., to show that $\rho$ extends to $\mathfrak{X}_{A}$, it suffices to
show that it extends to $\mathfrak{X}_{A_{j}}$: indeed, this follows from the
fact that $A\rightarrow A_{j}$ is an étale ring map. That $\rho$ extends to
$\mathfrak{X}_{A_{j}}$ follows directly from [Sta22, Tag 0EYJ]. ∎

##  3\. Companions

In this brief section, we recall what we need from the theory of $\ell$-adic
companions. For more details, see [Ked22]. Let $Y/\mathbb{F}_{q}$ be a smooth,
geometrically connected variety over a finite field of characteristic $p$.

###  3.1. $\ell$-adic companions

######  Definition 3.1.1.

Let $L$ be an lisse $\overline{\mathbb{Q}}_{\ell}$ Weil sheaf on $Y$. We say
that $L$ is _algebraic_ if for all closed points $y$, the characteristic
polynomial of the Frobenius conjugacy class of $y$ acting on $L$,
$P_{y}(L,t)\in\overline{\mathbb{Q}}_{\ell}[t]$, has coefficients in
$\bar{\mathbb{Q}}\subset\overline{\mathbb{Q}}_{\ell}$.

######  Definition 3.1.2.

Let $L$ be an algebraic lisse $\overline{\mathbb{Q}}_{\ell}$ Weil sheaf on
$Y$. Let $\ell^{\prime}\neq p$ be a prime number (possibly equal to $\ell$).
Let
$\tau\colon\overline{\mathbb{Q}}_{\ell}\rightarrow\overline{\mathbb{Q}}_{\ell^{\prime}}$
be a (possibly non-continuous) field isomorphism; abusing notation, let $\tau$
also denote the induced isomorphism
$\overline{\mathbb{Q}}_{\ell}[t]\rightarrow\overline{\mathbb{Q}}_{\ell^{\prime}}[t]$.
A _$\tau$ -companion to $L$_ is a lisse
$\overline{\mathbb{Q}}_{\ell^{\prime}}$-Weil sheaf $L^{\prime}$ on $Y$, such
that for all closed points $y$, we have:

$\tau(P_{y}(L,t))=P_{y}(L^{\prime},t)\in\overline{\mathbb{Q}}_{\ell^{\prime}}[t].$

The mathematical content of the following theorem is due to Lafforgue (for
curves) and Deligne and Drinfeld (for higher dimensional varieties).

######  Theorem 3.1.3.

Let $L$ be an irreducible lisse $\overline{\mathbb{Q}}_{\ell}$-Weil sheaf on
$Y$.

  1. (1)

Suppose $L$ has algebraic determinant (as a rank 1 Weil sheaf). Then for all
primes $\ell^{\prime}\neq p$ and for all field isomorphisms
$\tau\colon\overline{\mathbb{Q}}_{\ell}\rightarrow\overline{\mathbb{Q}}_{\ell^{\prime}}$,
$\tau$-companions to $L$ exist and are unique up to isomorphism.

  2. (2)

Suppose there exist integers $k,m$, with $k$ positive, such that
$\det(L)^{\otimes k}\cong\overline{\mathbb{Q}}_{\ell}(m)$. Then $L$ extends to
an étale sheaf on $Y$ and so do all $\tau$-companions.

###### Proof.

We provide references to [Ked22] for these results. The first follows from
Corollary 3.10 and Theorem 4.28 of _loc. cit._ The second follows from the
fact that companions is preserved under tensorial operations, together with
the fact that there exists a $p$-plain character $\chi$ such that
$L\otimes\chi$ has finite order determinant, and is hence $p$-plain by Theorem
1.3. (For the definition of $p$-plain, see Conjecture 1.1(iii) of _loc.
cit._.) ∎

###  3.2. $p$-adic companions

Let $\textbf{F-Isoc}^{\dagger}(Y)_{\overline{\mathbb{Q}}_{p}}$ denote the
category of overconvergent $F$-isocrystals on $Y$ with coefficients in
$\overline{\mathbb{Q}}_{p}$. As $F$-isocrystals over finite fields have a
notion of “characteristic polynomial of Frobenius”, we may simply copy the
above definition of companions. For more details, see [Ked22].

##  4\. Degree of trace fields

### 4.1.

Colloquially, the following lemma says the following: every cohomologically
rigid $\overline{\mathbb{Q}}_{\ell}$-local system on $X$ with fixed $(n,d)$
may be defined over a finite extension $E_{\ell}/\mathbb{Q}_{\ell}$, and
moreover, we may relate these local systems to local systems in characteristic
$p$ for any spreading out of $X$. This lemma follows follows from the
technique of Esnault-Groechenig.

######  Lemma 4.1.1.

Fix $(X,n,d)$ as in the introduction.

  1. (1)

For any prime number $\ell$, there exists an $\ell$-adic local field
$E_{\ell}$, equipped with an embedding
$E_{\ell}\xhookrightarrow{}\overline{\mathbb{Q}}_{\ell}$, such that for any
spreading out $\mathfrak{X}/S$ of $X$ and for any closed point $s$ of $S$
there is the following natural diagram:

$\textstyle{\text{CohRig}_{d}^{n}(\mathfrak{X}_{\bar{s}},\overline{\mathbb{Q}}_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{CohRig}_{d}^{n}(X,\overline{\mathbb{Q}}_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$EG$\textstyle{\text{CohRig}_{d}^{n}(X^{\text{an}},\overline{\mathbb{Q}}_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{CohRig}_{d}^{n}(\mathfrak{X}_{\bar{s}},E_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{CohRig}_{d}^{n}(X,E_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{CohRig}_{d}^{n}(X^{\text{an}},E_{\ell}),}$

where the horizontal arrows are injective and vertical arrows are equalities.
Moreover, there exists an $N_{1}=N_{1}(X,n,d)$ such that for all $p>N_{1}$ and
all closed points $s$ of $S$ whose residue characteristic is $p$, all of the
above maps are isomorphisms.

  2. (2)

Furthermore, there exists $N_{2}=N_{2}(X,d,n)$, such that for $\ell>N_{2}$, we
may take $E_{\ell}$ to be unramified over $\mathbb{Q}_{\ell}$.

###### Proof.

  1. (1)

The set $\text{CohRig}_{d}^{n}(X^{an},\overline{\mathbb{Q}}_{\ell})$ is
finite, and we may take a local field $E_{\ell}$, equipped with
$E_{\ell}\xhookrightarrow{}\overline{\mathbb{Q}}_{\ell}$ such that all the
elements of $\text{CohRig}_{d}^{n}(X^{an},\overline{\mathbb{Q}}_{\ell})$ have
coefficients in $E_{\ell}$, i.e. the natural map
$\text{CohRig}_{d}^{n}(X^{an},E_{\ell})\rightarrow\text{CohRig}_{d}^{n}(X^{an},\overline{\mathbb{Q}}_{\ell})$
is bijective. By [EG18, Theorem 1.1], we may assume that each element of
$\text{CohRig}_{d}^{n}(X^{an},E_{\ell})$ is in fact integral, so that the
natural map
$\text{CohRig}_{d}^{n}(X,E_{\ell})\rightarrow\text{CohRig}_{d}^{n}(X^{an},E_{\ell})$
is also bijective.

Given a spreading-out $\mathfrak{X}\rightarrow S$ of $X$, a point $\eta\in S$,
and a specialization of points $\eta\leadsto s$, for a point $s\in S$, there
is a surjective specialization map [Sta22, 0C0P]

(4.1.1)
$\text{sp}\colon\pi_{1}^{\text{\'{e}t}}(\mathfrak{X}_{\eta})\rightarrow\pi_{1}^{\text{\'{e}t}}(\mathfrak{X}_{s}),$

giving rise to a surjective map
$\pi_{1}^{\text{\'{e}t}}(X)\cong\pi_{1}^{\text{\'{e}t}}(\mathfrak{X}_{\bar{\eta}})\rightarrow\pi_{1}^{\text{\'{e}t}}(\mathfrak{X}_{\bar{s}}).$
This yields the left horizontal arrows (for both
$\overline{\mathbb{Q}}_{\ell}$-and $E_{\ell}$-coefficients), which are
manifestly injective. It remains to show that these arrows are isomorphisms
for large $p$. Recall that the specialization map (4.1.1) induces an
isomorphism on maximal prime-to-$p$ quotients [Sta22, 0BUQ].

Let $N_{1}$ be such that, for all primes $p>N_{1}$, every finite quotient of
${\rm GL}_{n}(\mathcal{O}_{E_{\ell}})$ has order prime to $p$. Then for all
$p>N_{1}$, the induced map
$\text{CohRig}_{d}^{n}(\mathfrak{X}_{\bar{s}},E_{\ell})\rightarrow\text{CohRig}_{d}^{n}(X,E_{\ell})$
is a bijection, as required.

  2. (2)

We may simply take $N_{2}$ such that the field of definition of every element
of $\text{CohRig}_{d}^{n}(X^{an},\overline{\mathbb{Q}}_{\ell})$ is unramified
at all $\ell>N_{2}$.

∎

######  Lemma 4.1.2.

Let $Y/k$ be a smooth projective variety over a finite field. Then the
collection of $\overline{\mathbb{Q}}_{\ell}$-local systems on $Y$ with the
following properties:

  * •

rank $n$,

  * •

determinant of order $d$,

  * •

geometrically irreducible, and

  * •

cohomologically rigid

is finite and closed under $\tau$-companions for any abstract field
isomorphism
$\tau\colon\overline{\mathbb{Q}}_{\ell}\rightarrow\overline{\mathbb{Q}}_{\ell}$.

###### Proof.

The finiteness of this set follows from [EK12, Theorem 1.1]. The fact that it
is closed under $\tau$-companions follows from the $L$-function argument in
[EG18, Proof of Theorem 1.1] (see also [EG20, Proposition 7.4]). In greater
detail: irreducibility is preserved under $\tau$-companions by the usual
$L$-function argument, which implies that geometric irreducibility is as well
as the $\tau$-companions relation is compatible with extension of base field.
Similarly, $\tau$-companions preserves the rank of cohomology (again via
$L$-functions), hence sends cohomologically rigid local systems to
cohomologically rigid local systems. ∎

###### Proof of Theorem 1.1.7(1).

As in the statement of Theorem 1.1.7, suppose we have a spreading out
$(\mathfrak{X},\mathcal{L},S,\iota)$, and a closed point $s\in S$. Let
$L_{s}\vcentcolon=\mathcal{L}|_{\mathfrak{X}_{s}}$. Note that $L_{s}$ has
algebraic determinant and is absolutely irreducible, hence by a theorem of
Deligne [Del12, Théorème 3.1], the field $F_{s}$ generated by Frobenius traces
is a number subfield of $E_{\ell}$.

To prove the claim, it suffices to show that the number of $\ell$-companions
of $L_{s}$ is bounded by an integer $A=A(X,n,d)$. Indeed, the companions are
in bijection with embeddings
$F_{s}\hookrightarrow\overline{\mathbb{Q}}_{\ell}$.

Let $M$ be a $\ell$-companion of $L_{s}$. By Lemma 4.1.2,
$M|_{\mathfrak{X}_{\bar{s}}}$ is cohomologically rigid, and hence is an
element $\text{CohRig}_{d}^{n}(\mathfrak{X}_{\bar{s}},E_{\ell})$, where
$E_{\ell}$ is the field from the statement of Lemma 4.1.1. By Lemma 4.1.1, we
have
$|\text{CohRig}_{d}^{n}(\mathfrak{X}_{\bar{s}},E_{\ell})|\leq|\text{CohRig}_{d}^{n}(X,E_{\ell})|$,
and the latter depends only on $X,n,$ and $d$. Finally, we claim that the
fibers of the map

$\\{\ell\text{-companions of}\
L_{s}\\}\rightarrow\text{CohRig}_{d}^{n}(\mathfrak{X}_{\bar{s}},E_{\ell})$

have size at most $nd$; this finishes the proof since we can simply take
$A\vcentcolon=nd|\text{CohRig}_{d}^{n}(X,E_{\ell})|$. It remains to prove the
claim. Indeed, suppose $M_{1},M_{2}$ are companions of $L_{s}$ which are
isomorphic upon restriction to $\mathfrak{X}_{\bar{s}}$. Then $M_{1}\simeq
M_{2}\otimes\psi$ for some character $\psi$ of $G_{\kappa(s)}$; since
$M_{1},M_{2}$ both have determinants of order $d$, we have $\psi^{\otimes
nd}$, which gives the desired claim. ∎

##  5\. Proof of Theorem 1.1.7 away from $\text{char}(s)$

### 5.1.

As before, let $X/\mathbb{C}$ be a smooth projective variety. Let $\rho_{i}$
be a list of the elements of $\text{CohRig}_{d}^{n}(X^{an},\mathbb{C})$. There
exists a number field $E$, a set of places $\Sigma$ of $E$, such that each
$\rho_{i}$ comes from a local system of projective
$\mathcal{O}_{E,\Sigma}$-modules444where $\mathcal{O}_{E,\Sigma}$ denotes the
ring of $\Sigma$-integers of $E$. In fact, by [EG18] we may take $\Sigma$ to
be empty of rank $n$.

######  Lemma 5.1.1.

There exists $N=N(X,d,n)$ such that for all $\ell>N$ and every $i$, $\rho_{i}$
is absolutely irreducible mod $\lambda$, where $\lambda$ is a prime of
$\mathcal{O}_{E,\Sigma}$ lying above $\ell$.

###### Proof.

It suffices to prove the statement for a fixed $\rho=\rho_{i}$. Let us write
$\rho:\pi_{1}(X^{an})\rightarrow{\rm GL}(M)$ for $M$ a projective module
$\mathcal{O}_{\Sigma}$-module, which we may do by [EG18]. Suppose for the sake
of contradiction that for infinitely many $\lambda$’s, $M/\lambda
M\otimes\bar{\mathbb{F}}_{\lambda}$ has a $\pi_{1}(X^{an})$-invariant subspace
of rank $i$, with $0<i<n$. We fix generators $g_{1},\cdots,g_{m}$ of
$\pi_{1}(X^{an})$.

For any integer $1\leq i\leq n-1$, consider the
$\mathcal{O}_{E,\Sigma}$-scheme $Z\subset{\rm Gr}(i,M)$ defined by
$Z\vcentcolon=\cap_{j}{\rm Gr}(i,M)^{g_{j}}$, where ${\rm Gr}(i,M)^{g_{j}}$ is
the (scheme-theoretic) fixed points of $g_{i}$, and we are taking the scheme
theoretic intersection.

Then $Z$ is a closed subscheme of ${\rm Gr}(i,M)$, which is of finite type
over ${\rm Spec}(\mathbb{Z})$ and has closed points in arbitrarily large
characteristics, and therefore has a characteristic zero point, contradicting
the absolute irreducibility of $M$. ∎

The following follows straightforwardly from the fact that
$\pi_{1}(X^{\text{an}})\rightarrow\pi_{1}^{\text{\'{e}t}}(X)$ has dense image.

######  Proposition 5.1.2.

Let $X$ be a smooth proper variety over $\mathbb{C}$, and
$\rho:\pi_{1}^{\text{\'{e}t}}(X)\rightarrow{\rm
GL}_{n}(\overline{\mathbb{Q}}_{\ell})$ a continuous representation. Suppose
that the composition

$\pi_{1}(X^{\text{an}})\rightarrow\pi_{1}^{\text{\'{e}t}}(X)\xrightarrow{\rho}{\rm
GL}_{n}(\overline{\mathbb{Q}}_{\ell})$

has image in ${\rm GL}_{n}(\mathcal{O}_{E_{\ell}})$, where
$\mathcal{O}_{E_{\ell}}$ is the ring of integers of an $\ell$-adic local field
$E_{\ell}$. Then the same is true for $\rho$.

Let $Y/\mathbb{F}_{q}$ be a smooth variety, with $\bar{Y}$ the basechange to
${\overline{\mathbb{F}}_{q}}$. We set $F\colon\bar{Y}\rightarrow\bar{Y}$ to be
the ${\overline{\mathbb{F}}_{q}}$-morphism induced via base change from the
$q$-Frobenius on $Y$. There is an induced outer automorphism of
$\pi_{1}^{\text{\'{e}t}}(\bar{Y})$, which we denote by $F^{*}$. This map may
equivalently be described as follows: consider the homotopy exact sequence
$0\rightarrow\pi_{1}^{\text{\'{e}t}}(\bar{Y})\rightarrow\pi_{1}^{\text{\'{e}t}}(Y)\rightarrow
G_{\mathbb{F}_{q}}\rightarrow 0$, where the latter is topologically generated
by the $q$-Frobenius. For any element in $\pi_{1}^{\text{\'{e}t}}(Y)$ lifting
$q$-Frobenius, conjugation by this element gives the desired automorphism of
$\pi_{1}^{\text{\'{e}t}}(\bar{Y})$. In particular, if $V$ is equipped with a
continuous action of $\pi_{1}^{\text{\'{e}t}}(\bar{Y})$, then there is another
continuous action of $\pi_{1}^{\text{\'{e}t}}(\bar{Y})$ on $V$, which we
denote by $F^{*}V$, given by precomposing with this outer automorphism.

######  Proposition 5.1.3 ([Del80, § 1.1.14]).

Let $Y/\mathbb{F}_{q}$ be a smooth variety , and denote by $\bar{Y}$ its
basechange to an algebraic closure ${\overline{\mathbb{F}}_{q}}$. Let
$E_{\ell}$ be an $\ell$-adic local field with $\ell\neq p$. Then there is an
equivalence between the category of continuous, geometrically absolutely
irreducible $E_{\ell}$-representations of $\pi_{1}^{\text{\'{e}t}}(Y)$, and
the category of pairs $(V,\Phi)$, where $V$ is a continuous absolutely
irreducible $E_{\ell}$-representation of $\pi_{1}^{\text{\'{e}t}}(\bar{Y})$
with torsion determinant, and a $\pi_{1}^{\text{\'{e}t}}(\bar{Y})$-equivariant
map

$\Phi:F^{*}V\simeq V,$

whose determinant is an $\ell$-adic unit.

###### Proof.

By [Del80, § 1.1.14], each $E_{\ell}$-representation $V_{0}$ of
$\pi_{1}^{\text{\'{e}t}}(Y)$ is in particular a Weil sheaf on $Y$ and
therefore gives a pair $(V,\Phi)$, with $V$ a representation of
$\pi_{1}^{\text{\'{e}t}}(\bar{Y})$ and $\Phi\colon F^{*}V\simeq V$ a
$\pi_{1}^{\text{\'{e}t}}(\bar{Y})$-equivariant map. By [Del80, § 1.3.4] $\det
V$ is torsion, say of order $d$. To see that $\Phi$ has determinant a unit,
note that $(\det V_{0})^{\otimes d}$ is a rank one representation of ${\rm
Gal}({\overline{\mathbb{F}}_{q}}/\mathbb{F}_{q})$ where the action of a
generator is given by $(\det(\Phi))^{d},$ and hence a unit.

Conversely, given $(V,\Phi)$ satisfying the stated hypotheses, we obtain a
Weil sheaf $\mathcal{F}$ on $Y$, and it remains to show that it is an étale
sheaf. Since $\mathcal{F}$ is assumed irreducible, by [Del80, § 1.3.14] it
suffices to show that $L\vcentcolon=\det(\mathcal{F})$ is an étale sheaf. Let
$\bar{\mathcal{F}}$ denote the étale sheaf on $\bar{Y}$ corresponding to $V$,
whose determinant is of order $d$, say. Then $L^{\otimes d}$ is a Weil sheaf
pulled back from ${\rm Spec}(\mathbb{F}_{q})$, and in fact an étale sheaf from
the condition that $\det\Phi$ is a unit. Since the category of rank 1 étale
sheaves on ${\rm Spec}(\mathbb{F}_{q})$ is divisible, we may find a rank one
étale sheaf $\psi$ on ${\rm Spec}(\mathbb{F}_{q})$ such that
$(L\otimes\psi)^{\otimes d}$ is trivial; in other words
$L=(L\otimes\psi)\otimes\psi^{-1}$ is the tensor product of a Weil sheaf with
torsion determinant with an étale sheaf on ${\rm Spec}(\mathbb{F}_{q})$, and
hence is itself an étale sheaf–see [Del12, § 0.4]. ∎

We record the following immediate consequence of the Brauer-Nesbitt theorem:

######  Proposition 5.1.4.

Let $G$ be a group equipped with an action on finite dimensional
$E_{\ell}$-vector spaces $V$ and $W$ which are moreover absolutely
irreducible. Suppose $V$ and $W$ are isomorphic over
$\overline{\mathbb{Q}}_{\ell}$; then the same is true of $V$ and $W$.

###### Proof of Theorem 1.1.7(2).

Suppose we have a spreading out $\mathcal{L}$ as in the statement of the
theorem, corresponding to a representation $\rho^{\prime}$ of
$\pi_{1}^{\text{\'{e}t}}(\mathfrak{X})$. Let $N_{2}$ be as in part (2) of
Lemma 4.1.1. Let $\rho^{\prime}_{s}$ be the restriction of $\rho^{\prime}$ to
$\pi_{1}^{\text{\'{e}t}}(\mathfrak{X}_{s})$, and for each prime number
$m>N_{2}$ with $m\neq\text{char}(s)$, let $\rho_{s}$ be any $m$-companion of
$\rho^{\prime}_{s}$. To prove part (2) of Theorem 1.1.7, it suffices to show
that $\rho_{s}$ has coefficients in an unramified extension of
$\mathbb{Q}_{m}$. By Lemma 4.1.1 (2), we may assume that the restriction to
$\mathfrak{X}_{\bar{s}}$ takes the form
$\rho_{\bar{s}}:\pi_{1}^{\text{\'{e}t}}(\mathfrak{X}_{\bar{s}})\rightarrow{\rm
GL}_{n}(E_{m})$, with $E_{m}$ being an unramified extension of
$\mathbb{Q}_{m}$. Since $\rho_{\bar{s}}$ is integral, we will view it as a
representation on a free $\mathcal{O}_{E_{m}}$-module $\mathbb{M}$, and let
$V\vcentcolon=\mathbb{M}\otimes\mathbb{Q}$.

By Proposition 5.1.4, there exists a
$\pi_{1}^{\text{\'{e}t}}(\mathfrak{X}_{\bar{s}})$-equivariant map

$\Phi:F^{*}V\simeq V,$

where $F$ denotes $q$-Frobenius. Let $\pi\in E_{m}$ denote a uniformizer.

######  Claim 5.1.5.

Assuming $m>N$ where $N=N(X,d,n)$ as in Lemma 5.1.1, there exists
$M\in\mathbb{Z}$ such that $\pi^{M}\Phi$ has determinant a unit.

Before giving the proof, we see how Claim 5.1.5 finishes the proof of part (2)
of Theorem 1.1.7. Indeed, the claim implies, by Proposition 5.1.3, that we may
use $\pi^{M}\Phi$ to descend $V$ to a new representation
$\tau\colon\pi_{1}^{\text{\'{e}t}}(\mathfrak{X}_{s})\rightarrow{\rm
GL}_{n}(E_{m})$. Now, the geometric determinants of $\tau$ and $\rho_{s}$ are
isomorphic, and therefore $\det(\tau)^{\otimes-1}\otimes\det(\rho_{s})$ is a
character of $\pi_{1}^{\text{\'{e}t}}((){\rm Spec}(\mathbb{F}_{q}))$.
Moreover, by our assumption that $\det(\rho_{s})^{d}$ is an integral power of
the cyclotomic character, we have in fact

$\det(\tau)^{\otimes-1}\otimes\det(\rho_{s}):\pi_{1}^{\text{\'{e}t}}({\rm
Spec}(\mathbb{F}_{q}))\rightarrow E_{m}^{\prime\times}.$

for some finite unramified extension $E_{m}^{\prime}$ of $E_{m}$. Finally,
taking $\psi$ to be an $n$-th root of this character, we have that there
exists an unramified extension $F_{m}/E^{\prime}_{m}$, and a character
$\psi:\pi_{1}^{\text{\'{e}t}}({\rm Spec}(\mathbb{F}_{q}))\rightarrow
F_{m}^{\times}$, such that 555here we abuse notation and continue to denote
the representation by $\tau$ even after extending coefficients via
$E_{m}\subset F_{m}$ $\det(\tau\otimes\psi)\simeq\det(\rho_{s})$. Here $F_{m}$
is unramified since it is obtained by taking an $n$-th root of
$(\det(\tau)^{\otimes-1}\otimes\det(\rho_{s}))(\text{Frob}_{q})$, which is an
$m$-adic unit.

Now the two representations $\tau\otimes\psi$ and $\rho_{s}$ of
$\pi_{1}^{\text{\'{e}t}}(\mathfrak{X}_{s})$ are geometrically isomorphic and
have the same determinant. Therefore they differ by twisting by an order $n$
character, and hence $\rho_{s}$ has coefficients in $F_{m}(\zeta_{n})$, which
is unramified at $\ell$, as desired.

This implies that $\rho_{s}$ has coefficients in $F_{m}(\zeta_{n})$, which is
unramified at $m$, as desired.

###### Proof of Claim 5.1.5.

Without loss of generality, by scaling $\Phi$ by powers of $\pi$, we may
assume that $\pi\mathbb{M}\subsetneq\Phi(F^{*}\mathbb{M})\subset M$. Then the
image

$\Phi(F^{*}\mathbb{M})/\pi\mathbb{M}\subset\mathbb{M}/\pi\mathbb{M}$

is stable under the action of
$\pi_{1}^{\text{\'{e}t}}(\mathfrak{X}_{\bar{s}})$. By Lemma 5.1.1,
$\Phi(F^{*}\mathbb{M})/\pi\mathbb{M}$ must be the entirety of
$\mathbb{M}/\pi\mathbb{M}$ or zero, and the assumption
$\pi\mathbb{M}\subsetneq\Phi(F^{*}\mathbb{M})$ in fact implies that we must be
in the former case. Therefore $\Phi$ is an isomorphism, and hence its
determinant is a unit, as claimed. ∎

∎

##  6\. $F$-isocrystals and proof of Theorem 1.1.7

In this section, we address the $\text{char}(s)$ part of Theorem 1.1.7. For
any smooth scheme $X/k$ over a perfect field $k$, let $\textbf{F-Isoc}(X)$
denote the category of $F$-isocrystals on $X$, which is
$\mathbb{Q}_{p}$-linear. For any extension $K/\mathbb{Q}_{p}$, let
$\textbf{F-Isoc}(X)_{K}$ denote the basechanged category, which is $K$-linear:
for a detailed discussion we refer the reader to [Kri22, § 3].

### 6.1.

We recall the set-up of Esnault-Groechenig [EG20]. In this section, we
consider _rigid irreducible flat connections_ of fixed rank $n$ and
determinant $d$-torsion, i.e., simple flat connections $(V,\nabla)$ of rank
$n$ on $X$ with determinant of order $d$ that are rigid in the appropriate de
Rham moduli space. Fixing $X$ and the numerical constraints $(n,d)$, there are
only finitely many such simple flat connections.

######  Setup 6.1.1.

Let $\mathfrak{X}\rightarrow S$ be a spreading out as in [EG20, Proposition
4.10] or [Esn22, 8.5]. In particular, every rigid stable flat connection
$(V_{i},\nabla_{i})$ of rank $n$ and determinant of order $d$ has a canonical
spreading-out to an $S$-relative flat connection
$(\mathcal{V}_{i},\nabla_{i})$ on $\mathfrak{X}$ whose fibers are all stable.

######  Theorem 6.1.2.

[Esnault-Groechenig] Setup as in 6.1.1. There exists a positive integer $f$,
depending only on $X$, $n$, and $d$ with the following property. Let
$k\supset\mathbb{F}_{p^{f}}$ be a finite field whose ring of $p$-typical Witt
vectors is denoted by $W$. Let $\tilde{s}\colon{\rm Spec}(W)\rightarrow S$ be
a $W$-point of $S$, with $s$ the corresponding mod-$p$ point. Then for any
rigid de Rham bundle $(V,\nabla)$ on $X$, and with canonical spreading out
$(\mathcal{V},\nabla)$ to $\mathfrak{X}\rightarrow S$, the $\tilde{s}$-adic
completion of $(\mathcal{V},\nabla)\mid_{\mathfrak{X}_{\tilde{s}}}$ a formal
flat connection on the $p$-adic formal scheme $\mathfrak{X}_{\tilde{s}}$,
satisfies the following:

  * •

$(\mathcal{V},\nabla)\mid_{\mathfrak{X}_{\tilde{s}}}$ is the evaluation of a
crystal on the absolute crystalline site
$\textrm{CRYS}(\mathfrak{X}_{s}/\mathbb{Z}_{p})$ (i.e., $\nabla$ is
topologically quasi-nilpotent); and

  * •

$(\mathcal{V},\nabla)\mid_{\mathfrak{X}_{\tilde{s}}}$ admits an $F^{f}$
Frobenius structure.666In fact,
$(\mathcal{V},\nabla)\mid_{\mathfrak{X}_{\tilde{s}}}$ will underlie a filtered
$F^{f}$-crystal.

If $(V,\nabla)$ is further assumed to be cohomologically rigid, the above
integer $f$ may be taken to be $|\text{CohRig}_{d}^{n}(X,\mathbb{C})|!$, i.e.,
the size of the permutation group of the set
$\text{CohRig}_{d}^{n}(X,\mathbb{C})$.

######  Remark 6.1.3.

We note that the assumption that $\mathbb{F}_{p^{f}}\subset k$ is in fact
unnecessary, if we assume that our de Rham bundle $(V,\nabla)$ is
cohomologically rigid (as opposed to merely rigid). Indeed, this follows from
the alternative argument due to Esnault-de Jong, see [Esn22, Theorem 8.4]. The
point here is that Frobenius pullback permutes the finite collection of
isomorphism classes of formal flat connections (a.k.a. crystals on the
nilpotent crystalline site $NCRYS(\mathfrak{X}_{s}/\mathbb{Z}_{p})$) that are
cohomologically rigid.

######  Remark 6.1.4.

We emphasize that, in the context of Theorem 6.1.2, the choice of the $F^{f}$
structure is _not_ canonical: for instance, we can scale it by any number in
$\mathbb{Z}_{p}^{\times}$. As a result, there is _no reason_ to suppose that
the output $F^{f}$-isocrystal is “algebraic” (in the sense of having algebraic
determinant, or having traces of Frobenius elements being algebraic numbers).

The following fact relating $F^{f}$-isocrystals to $F$-isocrystals with
coefficients is well-known.

######  Fact 6.1.5.

Let $Y/\mathbb{F}_{q}$ be a smooth variety and let $f\geq 1$. If
$\mathbb{F}_{p^{f}}\subset\mathbb{F}_{q}$, then there is an natural
equivalence of categories between $F^{f}$-isocrystals on $Y$ and
$F$-isocrystals on Y with coefficients in $\mathbb{Q}_{p^{f}}$:

$\textbf{F}^{f}\textbf{-Isoc}(Y)\rightarrow\textbf{F-Isoc}(Y)_{\mathbb{Q}_{p^{f}}}$

######  Remark 6.1.6.

This fact is false without the hypothesis that
$\mathbb{F}_{p^{f}}\subset\mathbb{F}_{q}$. For instance, it is easy to find
counterexamples setting $Y={\rm Spec}(\mathbb{F}_{p})$.

To control the ramification of the field of Frobenius traces at
$\text{char}(s)$, we need a few basic results on $F$-isocrystals with
unramified coefficients. For simplicity, these are tailored to our
application. Before stating these, we need a definition.

######  Definition 6.1.7.

Let $Y/k$ be a smooth variety over a finite field of characteristic $p$, let
$r\geq 1$, and let $K$ be a $p$-adic local field. Let
$\mathcal{E}\in\textbf{F-Isoc}(Y)_{K}$. We set:

  * •

$k_{r}/k$ to be the smallest field extension that contains
$\mathbb{F}_{p^{r}}$;

  * •

$Y_{r}=Y\times_{{\rm Spec}(k)}{\rm Spec}(k_{r})$ and

  * •

$\mathcal{E}_{r}$ to be the _restriction_ of $\mathcal{E}$ to $Y_{r}$.

######  Proposition 6.1.8.

Let $k=\mathbb{F}_{p^{a}}$, let
$\psi\in\textbf{F-Isoc}(k)_{\mathbb{Q}_{p^{f}}}$ be a rank 1 object, and let
$m\geq 1$ be an integer. Then there are exactly $m$ isomorphism classes of
rank 1 objects $\chi_{i}\in\textbf{F-Isoc}(k)_{\overline{\mathbb{Q}}_{p}}$
with the property that $\chi_{i}^{\otimes m}\cong\psi$.

Suppose further that $f|a$ and that $1\leq m<p$ and set $M:=fm$. Then for each
$1\leq i\leq m$, there exists an extension $K_{i}/\mathbb{Q}_{p^{a}}$ of
degree at most $m$, such that the object $\chi_{i}$ satisfies the following
properties:

  * •

$\chi_{i}\in\textbf{F-Isoc}(k)_{K_{i}}$; and

  * •

the restriction $(\chi_{i})_{M}$ of $\chi_{i}$ to $k_{M}$ has Frobenius trace
in an unramified extension of $\mathbb{Q}_{p}$.

In other words, for each $m^{\text{th}}$-root $\chi$ of $\psi$, if
$k\supset\mathbb{F}_{p^{fm}}$, then the Frobenius trace of $\chi$ is
unramified over $\mathbb{Q}_{p}$. It may be helpful to have in mind the
example of
$\overline{\mathbb{Q}}_{p}(-1/2)\in\textbf{F-Isoc}(\mathbb{F}_{p})_{\overline{\mathbb{Q}}_{p}}$;
when restricted to $\mathbb{F}_{p^{a}}$, where $a$ is even, the Frobenius
trace is in $\mathbb{Q}_{p}$.

###### Proof.

First of all, $\psi$ corresponds to a pair $(M,F)$, where $M$ is a free
$\mathbb{Q}_{p^{a}}\otimes\mathbb{Q}_{p^{f}}$-module of rank 1 and $F$ is a
$\sigma\otimes\text{id}$-linear map by [Kri22, Proposition 5.2]. Then it
follows from [Kri22, Proposition 5.8] and the surrounding discussion that
$\psi$ is determined by the element
$\lambda:=\text{Tr}(F^{a})\in\mathbb{Q}_{p^{f}}^{\times}$. (In other words,
$F^{a}$ is a _linear_ map $M\rightarrow M$ between free
$\mathbb{Q}_{p^{a}}\otimes\mathbb{Q}_{p^{f}}$-modules of rank 1, so is given
by multiplication by an element of
$\mathbb{Q}_{p^{a}}\otimes\mathbb{Q}_{p^{f}}$. One proves this element in fact
lives in the subring
$\mathbb{Q}_{p}\otimes\mathbb{Q}_{p^{f}}\cong\mathbb{Q}_{p^{f}}$.)

Fix an $m$-th root $\lambda^{1/m}\in\overline{\mathbb{Q}}_{p}$ of $\lambda$.
Consider the cyclic algebra map
$\overline{\mathbb{Q}}_{p}\rightarrow\mathbb{Q}_{p^{a}}\otimes\overline{\mathbb{Q}}_{p}$
(with cyclic structure given by $\sigma\otimes\text{id}$). The resulting norm
map is clearly surjective:
$\mathbb{Q}_{p^{a}}\otimes\overline{\mathbb{Q}}_{p}\cong\prod\overline{\mathbb{Q}}_{p}$,
where the cyclic action on the right hand side is given by shifting. Therefore
$\lambda^{1/m}$ is in the image of the norm map, and hence there exists a rank
1 object $\chi\in\textbf{F-Isoc}(k)_{\overline{\mathbb{Q}}_{p}}$ with $F^{a}$
being multiplication by $\lambda^{1/m}$, by the discussion in between
Proposition 5.7 and Proposition 5.8 of [Kri22]; therefore $\chi^{\otimes
m}\cong\psi$. Moreover, any $m^{\text{th}}$-root of $\psi$ in
$\textbf{F-Isoc}(k)_{\overline{\mathbb{Q}}_{p}}$ is given by this
construction.

Note that $\lambda$ has the following special property with respect to the
cyclic algebra map
$\mathbb{Q}_{p^{f}}\rightarrow\mathbb{Q}_{p^{a}}\otimes\mathbb{Q}_{p^{f}}$
(with cyclic structure given by $\sigma\otimes\text{id}$): $\lambda$ lies in
the image of the resulting norm map
$(\mathbb{Q}_{p^{a}}\otimes\mathbb{Q}_{p^{f}})^{\times}\rightarrow\mathbb{Q}_{p^{f}}^{\times}$.

Now, let us assume that $f|a$. Since
$\mathbb{Q}_{p^{a}}\otimes\mathbb{Q}_{p^{f}}\simeq\prod\mathbb{Q}_{p^{a}}$ and
the norm map from the latter to $\mathbb{Q}_{p^{f}}$ is the product of the
individual norm maps, we have that $\lambda$ lies in the image of
$\text{Nm}:\mathbb{Q}_{p^{a}}\rightarrow\mathbb{Q}_{p^{f}}$. Therefore
$\frac{a}{f}\mid v_{p}(\lambda)$, by the explicit description of the image of
the norm map of the extension $\mathbb{Q}_{p^{a}}/\mathbb{Q}_{p^{f}}$.

Let $K:=\mathbb{Q}_{p^{a}}(\lambda^{1/m})\subset\overline{\mathbb{Q}}_{p}$. We
now claim that $\lambda^{1/m}\in K^{*}$ is a norm with respect to the cyclic
algebra map $K\rightarrow\mathbb{Q}_{p^{a}}\otimes K$, where the cyclic
structure is again given by $\sigma\otimes\text{id}$.

Indeed,

$\mathbb{Q}_{p^{a}}\otimes K\cong\prod K,$

and the norm map on the left hand side transports to the product on the right
hand side. Therefore, the above norm map is surjective. It follows from
[Kri22, Corollary 5.9] that $\chi$ descends to an object of
$\textbf{F-Isoc}(k)_{K}$.

Let $M=fm$. We now claim that given our choice of
$\lambda^{1/m}\in\overline{\mathbb{Q}}_{p}$ and the induced $\chi$, the
Frobenius trace field of $\chi_{M}$ (the restriction of $\chi$ to
$\textbf{F-Isoc}(k_{M})_{K}$) has Frobenius trace in an unramified extension
of $\mathbb{Q}_{p}$.

To prove this, note that the construction of $\chi$ as an $m^{\text{th}}$-root
of $\psi$ may as just as well have been done on $k_{M}$. Therefore, to prove
the above, we may as well assume that $M|a$; since $\chi$ has coefficients in
$K$, it suffices to show that $K$ is unramified over $\mathbb{Q}_{p}$. In this
case, note that $m|v_{p}(\lambda)$ (normalized with $v_{p}(p)=1$): indeed,
$\lambda$ is in the image of the norm map
$\text{Nm}\colon\mathbb{Q}_{p^{a}}^{\times}\rightarrow\mathbb{Q}_{p^{f}}^{\times}$,
with $mf|a$. As $m<p$, it follows that $K/\mathbb{Q}_{p^{a}}$ is an unramified
extension, as required. ∎

The following lemma uses Proposition 6.1.8 to deal with higher rank
$F$-isocrystals . We recommend first parsing the following lemma when $d=1$
and $g=0$ to understand what it is saying.

######  Lemma 6.1.9.

Let $d,f\geq 1$, let $\mathbb{F}_{p^{f}}\subset k$ be a finite field, let
$Y/k$ be a smooth variety and let

$\mathcal{E}\in\textbf{F-Isoc}(Y)_{\mathbb{Q}_{p^{f}}}$

be an object of rank $n$. Suppose that the underlying (convergent) isocrystal
of $\mathcal{E}$ has determinant of order dividing $d$, and furthermore that
$nd<p$. Let $g\in\mathbb{Z}$ and set

$S_{g,d}:=\\{\mathcal{E}^{\prime}\cong\mathcal{E}\otimes\chi\in\textbf{F-Isoc}(Y)_{\overline{\mathbb{Q}}_{p}}|\
\chi\in\textbf{F-Isoc}(k)_{\overline{\mathbb{Q}}_{p}}^{\text{rank 1}},\
\det(\mathcal{E}^{\prime})^{d}\cong\overline{\mathbb{Q}}_{p}(dg)\\},$

i.e., $S_{g,d}$ is the set of isomorphism classes of twists of $\mathcal{E}$
whose determinant is isomorphic to $\overline{\mathbb{Q}}_{p}(g)$ tensor a
rank 1 object of order $d$.

Then $|S_{g,d}|=dn$, and for every $\mathcal{E}^{\prime}\in S_{g,d}$, the
restriction $(\mathcal{E}^{\prime})_{fdn}$ has Frobenius traces in an
unramified extension of $\mathbb{Q}_{p}$.

###### Proof.

First of all, we claim that $\det(\mathcal{E})^{\otimes
d}\in\textbf{F-Isoc}(Y)_{\mathbb{Q}_{p^{f}}}$ is the pull back of an object of
$\textbf{F-Isoc}(k)_{\mathbb{Q}_{p^{f}}}$. This follows from the fact that the
convergent isocrystal underlying $\det(\mathcal{E})^{\otimes d}$ is trivial,
and the fact that an $F$-isocrystal with trivial underlying isocrystal is
pulled back from the base.

We will first prove the lemma for $g=0$. Let us first construct $nd$ such rank
1 objects $\chi$. Set $\psi:=\det(\mathcal{E})^{-d}$ and $m=dn$. Let $\chi$ be
one of the $dn$ outputs of Proposition 6.1.8. Note that
$(\mathcal{E}\otimes\chi)\in\textbf{F-Isoc}(Y)_{K}$ has determinant of order
$d$; indeed,
$\det(\mathcal{E}\otimes\chi)^{d}=\det(\mathcal{E})\otimes\chi^{dn}$. Then
Proposition 6.1.8 implies that $(\mathcal{E}\otimes\chi)_{fdn}$ has Frobenius
traces in a field unramified over $\mathbb{Q}_{p}$.

On the other hand, if
$\mathcal{F}=\mathcal{E}\otimes\chi\in\textbf{F-Isoc}(Y)_{\overline{\mathbb{Q}}_{p}}$
is a twist with order $d$ determinant, then
$\chi^{-dn}=(\det{\mathcal{E}})^{d}$. Hence Proposition 6.1.8 implies that the
above construction exhausts $S_{g,d}$.

To prove the lemma for arbitrary $g\in\mathbb{Z}$, pick any
$p^{-1/n}\in\overline{\mathbb{Q}}_{p}$ and consider the following rank 1
object $(M,F)$ in
$\textbf{F-Isoc}(\mathbb{F}_{p})_{\overline{\mathbb{Q}}_{p}}$, which we call
$\overline{\mathbb{Q}}_{p}(1/n)$: $M$ is a free rank 1 vector space over
$\overline{\mathbb{Q}}_{p}$ and $F$ is given by multiplication by $p^{-1/n}$.
We may define
$\overline{\mathbb{Q}}_{p}(g/n):=\overline{\mathbb{Q}}_{p}(1/n)^{\otimes g}$.
Then there is a bijection:

$S_{0,d}\xrightarrow[]{\otimes\overline{\mathbb{Q}}_{p}(g/n)}S_{g,d}.$

By the same argument as in Proposition 6.1.8, it follows that the Frobenius
trace of the restriction
$(\overline{\mathbb{Q}}_{p}(g/n))_{n}\in\textbf{F-Isoc}(\mathbb{F}_{p^{n}})_{\overline{\mathbb{Q}}_{p}}$
is in $\mathbb{Q}_{p}$ (in fact, it is exactly $1/p^{g}$), and the lemma
follows. ∎

Putting the above together, we have the following theorem, which provides a
$p$-adic bound on the field of Frobenius traces in the theorem of Esnault-
Groechenig. Loosely, it says that for any algebraic $F$-isocrystal that is an
output of their theorem, after restricting to a slightly bigger base field,
the number field generated by Frobenius traces is unramified over $p$.

######  Theorem 6.1.10.

Notation as in Esnault-Groechenig’s Theorem 6.1.2. Let $(V,\nabla)$ be a
cohomologically rigid de Rham bundle, with canonical spreading out
$(\mathcal{V},\nabla)$. Let $p>dn$.

Then for any point $\tilde{s}\colon W(\mathbb{F}_{q})\rightarrow S$, where
$\mathbb{F}_{q}\supset\mathbb{F}_{p^{f}}$, and for any
$\mathcal{E}\in\textbf{F}^{f}\textbf{-Isoc}(\mathfrak{X}_{s})\cong\textbf{F-Isoc}(\mathfrak{X}_{s})_{\mathbb{Q}_{p^{f}}}$
such that the underlying convergent isocrystal of $\mathcal{E}$ admits
$(\mathcal{V},\nabla)$ as a lattice, and for any twist
$\mathcal{E}^{\prime}\in\textbf{F-Isoc}(\mathfrak{X}_{s})_{\overline{\mathbb{Q}}_{p}}$
of $\mathcal{E}$ where $\det(\mathcal{E}^{\prime})^{\otimes
d}\cong\overline{\mathbb{Q}}_{p}(g)$ for some $g\in\mathbb{Z}$, the number
field generated by Frobenius traces of $\mathcal{E}^{\prime}_{fdn}$ is
unramified over $p$.

Note that Theorem 6.1.2 implies that the theorem is not vacuous.

###### Proof.

It follows from Theorem 6.1.2 and Lemma 6.1.9 that for any such
$\mathcal{E}^{\prime}$, the field generated by Frobenius traces of
$\mathcal{E}^{\prime}_{fdn}$, as a subfield of $\overline{\mathbb{Q}}_{p}$, is
contained in an unramified extension of $\mathbb{Q}_{p}$. This in particular
means that the _number field_ generated by Frobenius traces has at least one
$p$-adic place $\lambda$ that is unramified.

The obstruction to proving that the number field generated by Frobenius traces
is unramified over $p$ is the following: we do not a priori know that the
$p$-adic companions of $\mathcal{E}^{\prime}$ all come from the same procedure
that produced $\mathcal{E}$. This follows from [EG20, Theorem 7.3(1)](together
with Lemma 6.1.9), but we spell out the argument.

From the explicit conditions on $\mathfrak{X}\rightarrow S$ from [EG20,
Proposition 4.10] that any stable cohomologically rigid flat connection with
determinant $d$-torsion on $\mathfrak{X}_{s}$ globalizes to a (stable,
cohomologically rigid) flat connection with determinant $d$-torsion on
$\mathfrak{X}\rightarrow S$. (See in particular the conditions in Proposition
3.3 of _loc. cit._) Moreover, any companion of $\mathcal{E}^{\prime}$ is
cohomologically rigid (with the same order of determinant) using the standard
$L$-function argument.777Note that cohomological rigidity does not depend on
the Frobenius structure. Therefore any $p$-adic companion of
$\mathcal{E}^{\prime}$ arises _from the same procedure that produced
$\mathcal{E}^{\prime}$_, starting with a possibly different cohomologically
rigid stable flat connection on $\mathfrak{X}/S$. Hence the number field
generated by Frobenius traces of $\mathcal{E}^{\prime}_{fdn}$ is unramified
over $p$. ∎

######  Corollary 6.1.11.

Let $\mathfrak{X}\rightarrow S$ be a spreading out as in Setup 6.1.1. Let
$p>nd$.

Then there exists an $f\geq 1$ with the following property. Let $\mathbb{L}$
be any such $\overline{\mathbb{Q}}_{\ell}$-local system. Let $q=p^{a}$, with
$fdn|a$. Then for any point

$s\colon{\rm Spec}(\mathbb{F}_{q})\rightarrow S,$

the Frobenius trace field of $\mathbb{L}|_{\mathfrak{X}_{s}}$ is unramified
over $p$. In fact, $f$ make be taken to be
$|\text{CohRig}_{d}^{n}(X,\mathbb{C})|!$.

###  6.2. Proof of Theorem 1.1.7

###### Proof.

Parts (1) and (2) of Theorem 1.1.7 were proven in Sections 4 and 5
respectively, so it remains to prove part (3).

Let $(\mathfrak{X},\mathcal{L},S,\iota)$ be as in the statement of Theorem
1.1.7.

By Corollary 6.1.11, there exists888more precisely, we can take a spreading
out of $X$ to a $S^{\prime}$ which lies over the spreading out to $S$ as well
as the one in Setup 6.1.1 a finite type scheme $S^{\prime}$, equipped with a
map $S^{\prime}\rightarrow S$ such that
$(\mathfrak{X}_{S^{\prime}},\mathcal{L}_{S^{\prime}})$ satisfies the
conclusion of points (1), (2) of Theorem 1.1.7, as well as the conclusion of
Corollary 6.1.11.

More precisely, there exists $\beta\in\mathbb{Z}_{\geq 1}$ such that for all
$s:{\rm Spec}(\mathbb{F}_{q})\rightarrow S^{\prime}$ such that
$\mathbb{F}_{p^{\alpha}}\subset\mathbb{F}_{q}$,
$\mathcal{L}_{\mathfrak{X}_{S^{\prime}}}|_{\mathfrak{X}_{S^{\prime}}\otimes
s}$ has trace field is

  * •

of degree $\leq N=N(X,n,d)$, and

  * •

unramified at any prime $m>\ell_{0}=\ell_{0}(\mathfrak{X},n,d)$.

Since the set of such number fields is finite [Neu13, Ch. II Theorem 2.13],
the local system $\mathcal{L}_{S^{\prime}}$ satisfies the conclusion of point
(3) of Theorem 1.1.7. Taking an étale map $T\rightarrow S$ such that the
induced morphism $S^{\prime}_{T}\vcentcolon=S^{\prime}\times_{S}T\rightarrow
T$ has a section, we may conclude. ∎

##  7\. The adjoint, stable trace fields, and a question

###  7.1. The adjoint representation

######  Proposition 7.1.1.

For $L\in\text{CohRig}_{d}^{n}(X,\overline{\mathbb{Q}}_{\ell})$, there exists
a spreading out $(\mathfrak{X},S,\mathcal{L},\iota)$ such that ${\rm
ad}(\mathcal{L})$ has bounded Frobenius traces.

To prove this, we require a couple of preliminaries.

######  Definition 7.1.2.

Let $Y/\mathbb{F}_{q}$ be a smooth, geometrically connected variety and let
$L$ be a $\overline{\mathbb{Q}}_{\ell}$-local system on $Y$. The _stable
Frobenius trace field_ of $L$ is the subfield of
$\overline{\mathbb{Q}}_{\ell}$ given by the intersection of the Frobenius
trace fields of

$L|_{Y\times_{\mathbb{F}_{q}}\mathbb{F}_{q^{\prime}}}$

as $q^{\prime}$ ranges over the positive integral powers of $q$.

The following follows directly from the definition, and perhaps provides some
intuition for the meaning of the stable Frobenius trace field.

######  Proposition 7.1.3.

Let $Y/\mathbb{F}_{q}$ be a smooth, geometrically connected variety and let
$L$ be a semi-simple $\overline{\mathbb{Q}}_{\ell}$-local system whose
irreducible constituents have finite order determinant. Then the following are
equivalent.

  1. (1)

The Frobenius trace field of $L$ is strictly bigger than the stable Frobenius
trace field of $L$.

  2. (2)

There exists a (possibly non-continuous)
$\tau\in\text{Aut}(\overline{\mathbb{Q}}_{\ell})$ such that ${}^{\tau}L$ (the
$\tau$-companion of $L$) and $L$ are not isomorphic but are isomorphic when
restricted to the geometric fundamental group
$\pi_{1}^{\text{\'{e}t}}(\bar{Y})$.

######  Lemma 7.1.4.

Let $Y/\mathbb{F}_{q}$ be a smooth projective variety and let $L$ be an
irreducible $\overline{\mathbb{Q}}_{\ell}$ local system with finite order
determinant on $Y$. Then the Frobenius trace field of ${\rm ad}(L)$ is a
subfield of the stable trace field of $L$.

###### Proof.

Set $E^{\prime}\subset E\subset\overline{\mathbb{Q}}_{\ell}$ to be the stable
trace field inside of the Frobenius trace field of $L$ and $K^{\prime}\subset
K\subset\overline{\mathbb{Q}}_{\ell}$ to be the stable trace field inside of
the Frobenius trace field of ${\rm ad}(L)$. Of course, $K\subset E$ and
$K^{\prime}\subset E^{\prime}$, and our goal is to prove that $K\subset
E^{\prime}$.

If $E=E^{\prime}$, then we are done, so we may assume that $E^{\prime}$
strictly contains $E$; we may suppose that $E^{\prime}$ is in fact the
Frobenius trace field of the restriction of $L$ to
$Y^{\prime}:=Y\times_{\mathbb{F}_{q}}\mathbb{F}_{q^{\prime}}$ for some power
$q^{\prime}$ of $q$. Furthermore, we may assume that there is $\lambda\in
K\subset E$ which is not in $E^{\prime}$.

Then there exists $\tau\in\text{Aut}(\overline{\mathbb{Q}}_{\ell})$ with the
following properties:

  1. (1)

${}^{\tau}L$ is not isomorphic to $L$ and ${}^{\tau}{\rm ad}(L)$ is not
isomorphic to ${\rm ad}(L)$

  2. (2)

${}^{\tau}L$ and $L$ are isomorphic on $Y^{\prime}$, and similarly with ${\rm
ad}(L)$.

Note that the operation of $\tau$-companions commutes with tensor product and
dual, so ${}^{\tau}{\rm ad}(L)\cong{\rm ad}(^{\tau}L)$.

On the other hand, as ${}^{\tau}L$ and $L$ are isomorphic over $Y^{\prime}$,
it follows that there exists a character $\psi$ such that ${}^{\tau}L\cong
L\otimes\psi$. It follows that ${\rm ad}(^{\tau}L)\cong{\rm ad}(L)$, which is
a contradiction. Therefore, such a $\lambda$ cannot exist and $K\subset
E^{\prime}$, as desired. ∎

###### Proof of Proposition 7.1.1.

By Theorem 1.1.7, there exists a spreading out
$(\mathfrak{X},S,\mathcal{L},\iota)$ and a number field $E$ such that the
stable trace field of $\mathcal{L}|_{\mathfrak{X}_{s}}$ is contained in $E$.
It follows from Lemma 7.1.4 that the Frobenius trace field of ${\rm
ad}(\mathcal{L})$ is contained in $E$, as desired. ∎

### 7.2.

The same arguments as in the proof of Theorem 1.1.7 gives the following, whose
proof we omit, though we make some comments regarding it in Remark 7.2.2. For
simplicity, We have opted to state the version with trivial determinant, but
as usual there is a version with determinant of order $d$.

######  Proposition 7.2.1.

Suppose $L$ is an irreducible $\overline{\mathbb{Q}}_{\ell}$-local system on
$X$ with trivial determinant, and of geometric origin of weight $w$, i.e. $L$
appears in cohomological degree $w$ of a smooth proper family. Then there
exists a spreading out $(\mathfrak{X},S,\mathcal{L},\iota)$. For any spreading
out such that $\mathcal{L}$ has weight $w$, for all closed points $s:{\rm
Spec}(k)\rightarrow S$, let $F_{s}$ denote the trace field of
$\mathcal{L}|_{\mathfrak{X}_{s}}$. Then

  1. (1)

There exists $N=N(X,L)$ such that, for all primes $m>N$ and different from the
characteristic of $s$, $F_{s}$ is unramified at $m$,

  2. (2)

For $f=\nu!$ where $\nu$ is the number of Galois conjugates of $L$, whenever
$\mathbb{F}_{p^{f}}\subset k$, $F_{s}$ is unramified at $p$.

######  Remark 7.2.2.

Note that part (2) of Proposition 7.2.1 is stronger than the analogous result
for (cohomologically) rigid local systems, where we must also assume that
$\mathbb{F}_{p^{nd}}\subset k$, and in the rigid case there is no assumption
on the weight of the spreading out $\mathcal{L}$. Recall that the proof
consists of comparing $F$-isocrystals obtained in two ways: one via companions
and one via Higgs-de Rham flows (or the Esnault-de Jong construction [Esn22,
Lecture 8]). The $\mathbb{F}_{p^{fdn}}$ assumption comes in because we have no
control over the weights of these isocrystals.

In the case of Proposition 7.2.1, by the assumption of being of geometric
origin, the Galois conjugates of $L$ all underlie $\mathbb{C}$-VHS of the same
weight. Hence choosing the spreading out $\mathcal{L}$ to have the same
weight, we have that the two $F$-isocrystals do have the same weights.

A positive answer to the following natural seeming question implies that we
can strengthen Theorem 1.1.7 part (3) in the same way; note that it is
certainly implied by Simpson’s motivicity conjecture.

######  Question 7.2.3.

Can one prove that Galois conjugates of a rigid local system underlie
$\mathbb{C}$-VHS of the same weight?

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