# Geometrically irreducible $p$-adic local systems are de Rham up to a twist

Alexander Petrov  Harvard University, USA<EMAIL_ADDRESS>
###### Abstract.

We prove that any geometrically irreducible $\overline{\mathbb{Q}}_{p}$-local
system on a smooth algebraic variety over a $p$-adic field $K$ becomes de Rham
after a twist by a character of the Galois group of $K$. In particular, for
any geometrically irreducible $\overline{\mathbb{Q}}_{p}$-local system on a
smooth variety over a number field the associated projective representation of
the fundamental group automatically satisfies the assumptions of the relative
Fontaine-Mazur conjecture. The proof uses $p$-adic Simpson and Riemann-Hilbert
correspondences of Diao-Lan-Liu-Zhu and the Sen operator on the decompletions
of those developed by Shimizu. Along the way, we observe that a $p$-adic local
system on a smooth geometrically connected algebraic variety over $K$ is
Hodge-Tate if its stalk at one closed point is a Hodge-Tate Galois
representation. Moreover, we prove a version of the main theorem for local
systems with arbitrary geometric monodromy, which allows us to conclude that
the Galois action on the pro-algebraic completion of $\pi_{1}^{\text{et}}$ is
de Rham.

##  1\. Introduction

Let $X$ be a smooth algebraic variety over a complete discretely valued field
$K$ of characteristic $0$ with a perfect residue field of characteristic
$p>0$. Inside the category of ${\mathbb{Q}}_{p}$-etale local systems on $X$
there is the subcategory of de Rham local systems [Sch13], [LZ17]. To any de
Rham local system one can canonically attach a vector bundle of the same rank
with integrable connection and filtration satisfying Griffiths transversality.
The primary examples of de Rham local systems are given by relative cohomology
$R^{n}\pi_{*}{\mathbb{Q}}_{p}$ of smooth proper morphisms $\pi:Y\to X$. Our
main result is the following

######  Theorem 1 (Corollary 5.3).

Suppose that ${\mathbb{L}}$ is a $\overline{\mathbb{Q}}_{p}$-local system on
$X$ such that the restriction of ${\mathbb{L}}$ to $X_{\overline{K}}$ has no
non-scalar endomorphisms:
$\operatorname{{End}}_{X_{\overline{K}}}({\mathbb{L}})=\overline{\mathbb{Q}}_{p}\cdot\mathrm{Id}$.
Then there exists a character of the Galois group
$\chi:G_{K}\to\overline{\mathbb{Q}}_{p}^{\times}$ such that
${\mathbb{L}}\otimes\chi$ is de Rham .

This fact has been conjectured by Daniel Litt. More generally, our proof works
for $X$ a smooth rigid analytic variety that can be embedded into a smooth
proper $\overline{X}$ such that $\overline{X}\setminus X$ is a normal
crossings divisor, and assuming additionally that ${\mathbb{L}}$ descends to
an ${\mathcal{O}}_{E}$-local system for a finite extension
$E\supset{\mathbb{Q}}_{p}$. Theorem 1 has the following implications on our
expectations of which conditions should be sufficient to guarantee that a
local system comes from geometry. We say that a
$\overline{\mathbb{Q}}_{p}$-etale local system on a scheme $X$ comes from
geometry if there exists a dense open subscheme $U\subset X$ and a morphism
$\pi:Y\to U$ such that ${\mathbb{L}}|_{U}$ is a subquotient of
$R^{i}\pi_{*}\overline{\mathbb{Q}}_{p}(j)$ for some $i,j$. Recall the relative
form of the Fontaine-Mazur conjecture [FM95], [LZ17]:

######  Conjecture 1.

Suppose that an irreducible $\overline{\mathbb{Q}}_{p}$-local system
${\mathbb{L}}$ on a smooth variety $X$ over a number field $F$ satisfies the
conditions

1)There is a finite set $S$ of places of $F$ and a smooth model
${\mathfrak{X}}$ of $X$ over ${\mathcal{O}}_{F,S}$ such that ${\mathbb{L}}$
extends to a local system on ${\mathfrak{X}}$.

2)For every place of $F$ above $p$ the restriction of ${\mathbb{L}}$ to
$X_{F_{v}}$ is a de Rham local system.

Then ${\mathbb{L}}$ comes from geometry.

The above theorem implies that for a geometrically irreducible local system
the second condition is automatically satisfied for the corresponding
projective representation of the arithmetic fundamental group. First condition
is also easily seen to be automatic (as proven in [Lit21], we recall the proof
in Proposition 6.1) yielding:

######  Theorem 2.

Let ${\mathbb{L}}$ be a $\overline{\mathbb{Q}}_{p}$-local system on a smooth
variety $X$ over a number field $F$ with the corresponding representation
$\rho:\pi_{1}^{\text{et}}(X,\overline{x})\to
GL_{d}(\overline{\mathbb{Q}}_{p})$. If
$\rho|_{\pi_{1}^{\text{et}}(X_{\overline{F}},\overline{x})}$ is irreducible
then

1)$\rho$ extends to $\pi_{1}^{\text{et}}({\mathfrak{X}},\overline{x})$ where
${\mathfrak{X}}$ is a smooth scheme over ${\mathcal{O}}_{F,S}$ for some finite
set of places $S$ with ${\mathfrak{X}}\times_{{\mathcal{O}}_{F,S}}F=X$.

2)For every place $v$ of $F$ above $p$ the projective representation
$\rho_{\mathrm{proj}}|_{F_{v}}:\pi_{1}^{\text{et}}(X_{F_{v}},\overline{x})\to
GL_{d}(\overline{\mathbb{Q}}_{p})\to PGL_{d}(\overline{\mathbb{Q}}_{p})$ is de
Rham.

The conclusion (2) of Theorem 2 is weaker than saying that ${\mathbb{L}}$ can
be made de Rham by twisting it by a character of $G_{F}$ as [Con, Example 6.8]
shows. This issue does not occur when working with local systems over local
fields by a result of Patrikis [Pat19, Corollary 3.2.13]. This allows us to
prove Theorem 1 in the stated form. Nevertheless, Moritz Kerz observed (Lemma
6.2) that Theorem 2 yields the following corollary of the relative Fontaine-
Mazur conjecture:

######  Conjecture 1 bis.

Let ${\mathbb{L}}$ be a $\overline{\mathbb{Q}}_{p}$-local system on a smooth
variety $X$ over a number field $F$ such that the local system
${\mathbb{L}}|_{X_{\overline{F}}}$ on $X_{\overline{F}}$ is irreducible. Then
${\mathbb{L}}|_{X_{\overline{F}}}$ comes from geometry.

Alexander Beilinson observed that the above results can be generalized to the
case of not necessarily irreducible local systems. Using the same techniques
as in the proof of Theorem 1, we show

######  Theorem 3 (Theorem 8.1).

For any $\overline{\mathbb{Q}}_{p}$-local system ${\mathbb{L}}$ on a smooth
variety $X$ over $K$ there exists a de Rham local system ${\mathbb{M}}$ such
that there is an inclusion
${\mathbb{L}}|_{X_{\overline{K}}}\subset{\mathbb{M}}|_{X_{\overline{K}}}$ of
local systems on $X_{\overline{K}}$.

This result can be restated in terms of the Galois action on the pro-algebraic
completion of the fundamental group of $X_{\overline{K}}$, generalizing the
results of [Vol03], [Ols11] regarding the pro-unipotent completion.

######  Corollary 1.1 (Proposition 8.5).

Any finite-dimensional subrepresentation
$V\subset\overline{\mathbb{Q}}_{p}[\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X_{\overline{K}},\overline{x})]$
of the Galois group $G_{K}$ is de Rham.

We will now sketch the proof of Theorem 1 in the case where $X$ is proper
assuming also that ${\mathbb{L}}$ is a ${\mathbb{Q}}_{p}$-local system with
$\operatorname{{End}}_{X_{\overline{K}}}{\mathbb{L}}={\mathbb{Q}}_{p}\cdot\mathrm{Id}$
with the conclusion of the theorem weakened to
${\mathbb{L}}\otimes_{{\mathbb{Q}}_{p}}\chi$ being Hodge-Tate. The main tool
used in the proof is a decompleted version of the $p$-adic Simpson
correspondence as developed by [LZ17] and [Shi18], summarized in the
following. Put $K_{\infty}=K(\mu_{p^{n}}|n\in{\mathbb{N}})$.

######  Proposition 1.2.

For a smooth proper algebraic variety $X$ over $K$ to any
${\mathbb{Q}}_{p}$-local system ${\mathbb{L}}$ on $X$ one can associate a
Higgs bundle $(\mathrm{H}({\mathbb{L}}),\theta)$ on $X_{K_{\infty}}$ equipped
with an ${\mathcal{O}}_{X_{K_{\infty}}}$-linear endomorphism
$\phi:\mathrm{H}({\mathbb{L}})\to\mathrm{H}({\mathbb{L}})$ such that
$\theta\circ\phi=(\phi-1)\circ\theta$. This construction satisfies the
following properties:

$(\mathrm{i})$ $\mathrm{H}$ is a tensor functor compatible with duality.

$(\mathrm{ii})$ We have
$\dim_{{\mathbb{Q}}_{p}}H^{i}_{\text{et}}(X_{\overline{K}},{\mathbb{L}})=\dim_{K_{\infty}}H^{i}_{\mathrm{Higgs}}(X_{K_{\infty}},\mathrm{H}({\mathbb{L}}))$.

$(\mathrm{iii})$ The characteristic polynomial
$\det(T\cdot\mathrm{Id}_{\mathrm{H}({\mathbb{L}})}-\phi)$ of $\phi$ has
coefficients in $K$. A local system ${\mathbb{L}}$ is Hodge-Tate if and only
if $\phi$ is a semi-simple operator with eigenvalues in ${\mathbb{Z}}$.

The relation $\theta\circ\phi=(\phi-1)\circ\theta$ implies that generalized
eigenspace decomposition of $\mathrm{H}({\mathbb{L}})$ with respect to the
endomorphism $\phi$ gives $(\mathrm{H}({\mathbb{L}}),\theta)$ a structure of a
system of Hodge bundles in the sense of [Sim92, Section 4]. By Proposition
1.2, the condition
$\operatorname{{End}}_{X_{\overline{K}}}{\mathbb{L}}={\mathbb{Q}}_{p}\cdot\mathrm{Id}$
implies that all endomorphisms of the Higgs bundle
$(\mathrm{H}({\mathbb{L}}),\theta)$ are scalars. In particular, it is
indecomposable and therefore eigenvalues of $\phi$ must lie in a subset of the
form $a+{\mathbb{Z}}\subset\overline{K}$ for some $a\in\overline{K}$. Twisting
by an appopriate character we may assume that $a=0$ so $\phi$ has integral
eigenvalues and, if $\phi$ was not semi-simple, the Higgs bundle would have a
non-zero nilpotent endomorphism. Therefore, the twsited local system is Hodge-
Tate.

We then deduce (Theorem 5.2) that a twist of ${\mathbb{L}}$ is not just Hodge-
Tate but is also de Rham by employing the $p$-adic Riemann-Hilbert
correspondence of [LZ17], [DLLZ19]. Building on the work of [Shi18], we obtain
(Proposition 3.5) a decompleted version $\mathrm{RH}({\mathbb{L}})$ of the
correspondence. The main observation then (Proposition 5.1) is that for any
local system ${\mathbb{L}}$ with integral Hodge-Tate weights one can strip off
the associated $K_{\infty}((t))$-linear filtered vector bundle with connection
$\mathrm{RH}({\mathbb{L}})$ to get a $K_{\infty}$-linear filtered vector
bundle with connection whose associated graded is $\mathrm{H}({\mathbb{L}})$.
One then utilizes the same idea as above: if ${\mathbb{L}}$ was not de Rham,
the Sen endomorphism would provide a non-scalar endomorphism of the Higgs
bundle $\mathrm{H}({\mathbb{L}})$.

To generalize this proof from ${\mathbb{Q}}_{p}$ local systems to
$\overline{\mathbb{Q}}_{p}$-local systems one simply needs to redo the above
argument with ${\mathbb{L}}$ equipped with an action of a finite extension $E$
of ${\mathbb{Q}}_{p}$. To cover the case of a non-proper $X$ we use the
logarithmic versions of $p$-adic Simpson and Riemann-Hilbert correspondences
developed in [DLLZ19].

As a byproduct of the above method, we show the following rigidity statement
for the Hodge-Tate property of fibers of a local system on an algebraic
variety

######  Proposition 1.3 (Proposition 7.2 & Corollary 7.3).

If ${\mathbb{L}}$ is a ${\mathbb{Q}}_{p}$-local system on a smooth
geometrically connected algebraic variety $X$ over $K$, then the Higgs sheaf
$D_{\mathrm{HT}}({\mathbb{L}})$ is locally free. In particular,
$\dim_{k(x)}D_{\mathrm{HT}}({\mathbb{L}}_{\overline{x}})$ does not depend on
the choice of a closed point $x\in X$ and, if the representation of the Galois
group $\operatorname{{Gal}}{\overline{k(x)}/k(x)}$ is Hodge-Tate for some
closed point $x\in X$ then the local system ${\mathbb{L}}$ is Hodge-Tate.

The proof rests on the observation that the Higgs bundles attached to local
systems on connected algebraic varieties via $p$-adic Simpson correspondence
satisfy a special property (Lemma 7.1): if a section annihilated by the Higgs
field vanishes at one point then it is identically zero. This forces the Sen
operator to be semi-simple if it is known that it is semi-simple at one point.
The statement of this result is similar to Liu-Zhu’s rigidity theorem for the
de Rham condition [LZ17, Theorem 1.3] and Shimizu’s constancy of Hodge-Tate
weights that was already used above. The proofs, however, have different
flavors and are exploiting entirely different effects: the rigidity theorems
of [LZ17] and [Shi18] are local statements that are proven for arbitrary
smooth rigid-analytic bases while our result is global: it uses crucially the
comparison between the global sections of a local system and the associated
Higgs bundle. It might well be the case that the rigidity is false for local
systems on non-partially proper rigid analytic varieties.

Acknowledgements. I am grateful to Alexander Beilinson, Hélène Esnault, Mark
Kisin, Moritz Kerz, Dmitry Kubrak, Daniel Litt, Koji Shimizu, and Bogdan
Zavyalov for several interesting and useful discussions. I am especially
thankful to Alexander Beilinson for his generalization of the main result to
the non-irreducible case and for the idea of looking at these results from the
vantage point of Galois action on the fundamental group. I also want to thank
Hélène Esnault, Mark Kisin, Shizhang Li, and Koji Shimizu for the comments on
the draft of this text.

##  2\. Preliminaries on $p$-adic Riemann-Hilbert correspondence

Generalizing Fontaine’s notions of Hodge-Tate and de Rham representations of
the Galois group of $K$, Brinon [Bri08], Scholze [Sch13] and in this particuar
form Liu and Zhu [LZ17] have constructed the following functors for any smooth
rigid-analytic variety $X$ over $K$. First, there is a functor

$D_{\mathrm{dR}}:\\{{\mathbb{Q}}_{p}-\text{local systems on
}X\\}\rightarrow\\{\parbox{150.00023pt}{\centering Filtered vector bundles on
$X$ with a Griffiths transverse connection\@add@centering}\\}$

A local system ${\mathbb{L}}$ is called de Rham if the output of this functor
$D_{\mathrm{dR}}({\mathbb{L}})$ has rank equal to
$\operatorname{{rk}}_{{\mathbb{Q}}_{p}}{\mathbb{L}}$. Likewise, there is a
functor

$D_{\mathrm{HT}}:\\{{\mathbb{Q}}_{p}-\text{local systems on
}X\\}\rightarrow\\{\parbox{90.00014pt}{\centering Coherent sheaves on $X$ with
a Higgs field\@add@centering}\\}$

A Hodge-Tate local system is one for which $D_{\mathrm{HT}}({\mathbb{L}})$ is
a locally free sheaf of rank
$\operatorname{{rk}}_{{\mathbb{Q}}_{p}}{\mathbb{L}}$. For convenience of
notation let us pick a geometric point $\overline{x}$ of $X$. Let
$\pi_{1}^{\text{et}}(X,\overline{x})$ denote the fundamental group of $X$ in
the sense of de Jong [dJ95].

######  Definition 2.1.

An etale $\overline{\mathbb{Q}}_{p}$-local system of rank $d$ on a rigid-
analytic variety $X$ is a continuous homomorphism
$\rho:\pi_{1}^{\text{et}}(X,\overline{x})\to
GL_{d}(\overline{\mathbb{Q}}_{p})$ that factors through $GL_{d}(E)$ for some
finite extension $E\supset{\mathbb{Q}}_{p}$. We say that an
$\overline{\mathbb{Q}}_{p}$-local system admits a
$\overline{{\mathbb{Z}}}_{p}$-lattice if $\rho$ is, moreover, conjugate to a
homomorphism into $GL_{d}({\mathcal{O}}_{E})\subset
GL_{d}(\overline{\mathbb{Q}}_{p})$. This condition is equivalent to $\rho$
factoring through the profinite completion of
$\pi_{1}^{\text{et}}(X,\overline{x})$.

If $X$ appears as the analytification of an algebraic variety then any
$\overline{\mathbb{Q}}_{p}$-local system on this algebraic variety gives rise
to a $\overline{\mathbb{Q}}_{p}$-local system in the above sense and it
automatically admits a $\overline{{\mathbb{Z}}}_{p}$-lattice. The proofs of
all our results will require the existence of
$\overline{{\mathbb{Z}}}_{p}$-lattices.

######  Definition 2.2.

For a reductive group $G$ over ${\mathbb{Q}}_{p}$ a continuous representation
$\pi_{1}^{\text{et}}(X,\overline{x})\to G({\mathbb{Q}}_{p})$ is de Rham (resp.
Hodge-Tate) if for some (equivalently, every) embedding $G\hookrightarrow
GL_{n,{\mathbb{Q}}_{p}}$ the rank $n$ ${\mathbb{Q}}_{p}$-local system
corresponding to $\pi_{1}^{\text{et}}(X,\overline{x})\to
G({\mathbb{Q}}_{p})\to GL_{n}({\mathbb{Q}}_{p})$ is de Rham (resp. Hodge-
Tate). For any finite extension $E$ of ${\mathbb{Q}}_{p}$ an $E$-local system
${\mathbb{L}}$ on $X$ is called de Rham (resp. Hodge-Tate) if the
corresponding representation into
$(\operatorname{{Res}}^{E}_{{\mathbb{Q}}_{p}}GL_{n})({\mathbb{Q}}_{p})$ is de
Rham (resp. Hodge-Tate). We call a $\overline{\mathbb{Q}}_{p}$-local system de
Rham (resp. Hodge-Tate) if any (equivalently, every) of its descents to an
$E$-local system for some finite extension $E/{\mathbb{Q}}_{p}$ is de Rham
(resp. Hodge-Tate).

The property of being de Rham or Hodge-Tate indeed does not depend on the
choice of a faithful representation $G\hookrightarrow GL_{n,{\mathbb{Q}}_{p}}$
because any other representation of the reductive group $G$ can be embedded
into a representation of the form $V^{\otimes a}\otimes V^{\vee\otimes b}$
where $V$ denotes the tautological representation of
$GL_{n,{\mathbb{Q}}_{p}}$.

######  Example 2.3.

According to this definition, for a $\overline{\mathbb{Q}}_{p}$-local system
${\mathbb{L}}$ on $X$ the corresponding projective representation
$\rho:\pi_{1}^{\text{et}}(X,\overline{x})\to
PGL_{n}(\overline{\mathbb{Q}}_{p})$ is de Rham (resp. Hodge-Tate) if and only
if its composition with the adjoint representation
$PGL_{n,\overline{\mathbb{Q}}_{p}}\to GL_{n^{2},\overline{\mathbb{Q}}_{p}}$
has the corresponding property. In terms of the local system ${\mathbb{L}}$
this is equivalent to saying that the local system
${\mathbb{L}}\otimes_{\overline{\mathbb{Q}}_{p}}{\mathbb{L}}^{\vee}$ is de
Rham (resp. Hodge-Tate).

We summarize in this section the setup and the properties of $p$-adic Simpson
and Riemann-Hilbert correspondences of [LZ17] and [DLLZ19]. From now on
suppose that $X$ can be embedded into a smooth proper rigid-analytic variety
$\overline{X}$ such that the complement $D=\overline{X}\setminus X$ is a
normal crossings divisor.

Let $K_{n}=K(\mu_{p^{n}})$ for $n\geq 0$ denote the $p$-cyclotomic extensions
of $K$ and let $K_{\infty}=\bigcup\limits_{n\geq 1}K_{n}$ be their union with
$\widehat{K}_{\infty}$ denoting its completion. The Galois group
$\operatorname{{Gal}}(K_{\infty}/K)$ is an open subgroup of
${\mathbb{Z}}_{p}^{\times}$ which we denote by $\Gamma$. We will denote by
${\mathbb{Q}}_{p}(1)=(\lim\limits_{n}\mu_{p^{n}}(\overline{K}))[\frac{1}{p}]$
the cyclotomic character of $G_{K}$, sometimes viewed as a character of
$\Gamma$. Denote also by $H_{K}$ the subgroup
$\operatorname{{Gal}}(\overline{K}/K_{\infty})\subset G_{K}$. Evaluating
Fontaine’s construction of $B_{\mathrm{dR}}$ on the perfectoid field
$\widehat{K}_{\infty}$ we get rings
$L_{\mathrm{dR}}=B_{\mathrm{dR}}(\widehat{K}_{\infty}),L_{\mathrm{dR}}^{+}=B_{\mathrm{dR}}^{+}(\widehat{K}_{\infty})$
equipped with the filtrations such that $L_{\mathrm{dR}}^{+}$ is complete. For
the convenience of notation we fix a compatible system of roots of unity
$\zeta_{p},\zeta_{p^{2}},\ldots\in\widehat{K}_{\infty}$ and denote the
corresponding element $\log[(1,\zeta_{p},\zeta_{p^{2}},\ldots)]\in
L_{\mathrm{dR}}^{+}$ by $t$. Denote also by $C$ the completed algebraic
closure $\widehat{\overline{K}}$ of $K$.

Consider the ringed space $\overline{{\mathcal{X}}}$ defined as
$(\overline{X}_{\widehat{K}_{\infty}},({\mathcal{O}}_{\overline{X}}\widehat{\otimes_{K}}L_{\mathrm{dR}}^{+})[t^{-1}])$.
It is equipped with the sheaf of logarithmic differentials
$\Omega^{1,\log}_{\overline{{\mathcal{X}}}/L_{\mathrm{dR}}}=\Omega^{1,\log}_{\overline{X}/K}\widehat{\otimes}_{K}L_{\mathrm{dR}}$
where $\Omega^{1,\log}_{\overline{X}/K}$ is the sheaf of $1$-forms that are
regular on $X$ and have at worst logarithmic poles along $D$. See [LZ17,
Definitions 3.5, 3.6] and [DLLZ19, Subsection 3.1] for complete definitions of
these objects and for the notion of a filtered vector bundle with connection
on $\overline{{\mathcal{X}}}$.

Here is the excerpt from [DLLZ19] that we will need:

######  Theorem 2.4.

There exists an exact functor
${\mathbb{L}}\mapsto{\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}})$ from
${\mathbb{Q}}_{p}$-local systems on $X$ to $\Gamma$-equivariant filtered
vector bundles on $\overline{{\mathcal{X}}}$ with a logarithmic integrable
connection that satisfies Griffiths transversality such that

(i) Denoting by ${\mathcal{H}}^{\log}({\mathbb{L}})$ the 0th graded piece
$\operatorname{{gr}}^{0}{\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}})$ we
have canonical $\Gamma$-equivariant isomorphisms
$\operatorname{{gr}}^{i}{\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}})\simeq{\mathcal{H}}^{\log}({\mathbb{L}})(i)$.
The connection on ${\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}})$ induces a
$\Gamma$-equivariant Higgs field

$\theta:{\mathcal{H}}^{\log}({\mathbb{L}})\to{\mathcal{H}}^{\log}({\mathbb{L}})(-1)\otimes_{{\mathcal{O}}_{\overline{X}_{\widehat{K}_{\infty}}}}\Omega^{1,\log}_{\overline{X}_{\widehat{K}_{\infty}}/\widehat{K}_{\infty}}$

(ii) The restrictions
${\mathbb{L}}\mapsto{\mathcal{H}}^{\log}({\mathbb{L}})|_{X_{\widehat{K}_{\infty}}}$
and
${\mathbb{L}}\mapsto{\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}})|_{{\mathcal{X}}}$
are monoidal functors compatible with duality.

(iii) If ${\mathbb{L}}$ admits a ${\mathbb{Z}}_{p}$-lattice then there is a
$H_{K}$-equivariant isomorphism (with respect to the trivial action on
$\operatorname{{End}}_{\overline{X}_{\widehat{K}_{\infty}},\mathrm{Higgs}}{\mathcal{H}}^{\log}({\mathbb{L}})$)
of $C$-algebras

$\operatorname{{End}}_{\overline{X}_{\widehat{K}_{\infty}},\mathrm{Higgs}}{\mathcal{H}}^{\log}({\mathbb{L}})\otimes_{\widehat{K}_{\infty}}C\simeq\operatorname{{End}}_{X_{\overline{K}}}{\mathbb{L}}|_{X_{\overline{K}}}\otimes_{{\mathbb{Q}}_{p}}C.$

(iv) We have

${\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}})^{\Gamma}|_{X}\simeq
D_{\mathrm{dR}}({\mathbb{L}})\qquad(\bigoplus\limits_{i\in{\mathbb{Z}}}{\mathcal{H}}^{\log}({\mathbb{L}})(i))^{\Gamma}|_{X}\simeq
D_{\mathrm{HT}}({\mathbb{L}})$

(v) For any classical point $x\in X$, choosing a geometric point
$\overline{x}:\operatorname{{Spa}}(\overline{K},{\mathcal{O}}_{\overline{K}})\to
X$ above it, we get a
$\operatorname{{Gal}}(\overline{k(x)}/k(x)(\mu_{p^{\infty}}))$-equivariant
isomorphism (with respect to trivial action on
${\mathcal{H}}^{\log}({\mathbb{L}})_{x}$)

${\mathcal{H}}^{\log}({\mathbb{L}})_{x}\otimes_{\widehat{k(x)(\mu_{p^{\infty}})}}C\simeq{\mathbb{L}}_{\overline{x}}\otimes_{{\mathbb{Q}}_{p}}C$

###### Proof.

Part (i) is [DLLZ19, Theorems 3.2.3 and 3.2.4], part (ii) is [LZ17, Theorem
2.1 (iv) and Theorem 3.8 (i)], and part (v) is [LZ17, Theorem 2.1 (ii)]. Part
(iv) is the definition of functors $D_{\mathrm{dR}}$ [LZ17, Section 3.2] and
$D_{\mathrm{HT}}({\mathbb{L}})$ [Shi18, Section 5] together with the fact that
${\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}})|_{X}\simeq{\mathcal{R}}{\mathcal{H}}({\mathbb{L}})$
and
${\mathcal{H}}^{\log}({\mathbb{L}})|_{X}\simeq{\mathcal{H}}({\mathbb{L}})$.

The statement (iii) would be immediate from the Hodge-Tate comparison [DLLZ19,
Theorem 3.2.4(ii)] applied to
${\mathbb{L}}\otimes_{{\mathbb{Q}}_{p}}{\mathbb{L}}^{\vee}$, if
${\mathcal{H}}^{\log}$ was monoidal. Unfortunately, in general only a weaker
statement hold:

######  Proposition 2.5.

The functor ${\mathbb{L}}\mapsto{\mathcal{H}}^{\log}({\mathbb{L}})$ is lax
monoidal: for any ${\mathbb{L}}_{1},{\mathbb{L}}_{2}$ there exists a canonical
map
${\mathcal{H}}^{\log}({\mathbb{L}}_{1})\otimes_{{\mathcal{O}}_{\overline{X}_{\widehat{K}_{\infty}}}}{\mathcal{H}}^{\log}({\mathbb{L}}_{2})\to{\mathcal{H}}^{\log}({\mathbb{L}}_{1}\otimes_{{\mathbb{Q}}_{p}}{\mathbb{L}}_{2})$.
This map is an isomorphism if both ${\mathbb{L}}_{1}$ and ${\mathbb{L}}_{2}$
have unipotent (as opposed to merely quasi-unipotent) monodromy along $D$.

###### Proof.

The lax monoidality is immediate from the definition of ${\mathcal{H}}^{\log}$
because the period sheaf ${\mathcal{O}}{\mathbb{C}}_{\log,\overline{X}}$ is a
sheaf of rings. The second statement is [DLLZ19, Theorem 3.2.12(1)] ∎

This issue is not specific to the $p$-adic situation and comes down to the
fact that, when defining ${\mathcal{R}}{\mathcal{H}}^{\log}$ and
${\mathcal{H}}^{\log}$ via Deligne’s canoncial extension, we choose the
residues of the extension to lie in $[0,1)$, but the residues of the tensor
product
${\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}}_{1})\otimes_{{\mathcal{O}}_{\overline{{\mathcal{X}}}}}{\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}}_{2})$
need not belong to $[0,1)$.

Nevertheless, we will prove that for any ${\mathbb{Z}}_{p}$-local system
${\mathbb{L}}$ on $X$ there is a canonical isomorphism

$H^{0}_{\mathrm{Higgs}}(\overline{X}_{\widehat{K}_{\infty}},{\mathcal{H}}^{\log}({\mathbb{L}}\otimes_{{\mathbb{Z}}_{p}}{\mathbb{L}}^{\vee}))\simeq
H^{0}_{\mathrm{Higgs}}(\overline{X}_{\widehat{K}_{\infty}},{\mathcal{H}}^{\log}({\mathbb{L}})\otimes_{\overline{X}_{\widehat{K}_{\infty}}}{\mathcal{H}}^{\log}({\mathbb{L}})^{\vee})$

which will yield the result by [DLLZ19, Theorem 3.2.4(ii)]. We have a
canonical map of Higgs bundles
${\mathcal{H}}^{\log}({\mathbb{L}}\otimes_{{\mathbb{Z}}_{p}}{\mathbb{L}}^{\vee})\otimes_{{\mathcal{O}}_{\overline{X}_{\widehat{K}_{\infty}}}}{\mathcal{H}}^{\log}({\mathbb{L}})\to{\mathcal{H}}^{\log}({\mathbb{L}})$
that induces the map

(2.1)
$\alpha_{{\mathbb{L}}}:{\mathcal{H}}^{\log}({\mathbb{L}}\otimes_{{\mathbb{Z}}_{p}}{\mathbb{L}}^{\vee})\to{\mathcal{H}}^{\log}({\mathbb{L}})\otimes_{{\mathcal{O}}_{X_{\widehat{K}_{\infty}}}}{\mathcal{H}}^{\log}({\mathbb{L}})^{\vee}$

which is an isomorphism when restricted to
$X_{\widehat{K}_{\infty}}\subset\overline{X}_{\widehat{K}_{\infty}}$ and, in
particular, is injective. We will prove that $\alpha_{{\mathbb{L}}}$ induces
an isomorphism on
$H^{0}_{\mathrm{Higgs}}(\overline{X}_{\widehat{K}_{\infty}},-)$ by passing to
a finite etale cover to make the monodromy of ${\mathbb{L}}$ around $D$
unipotent. Denote the rank of ${\mathbb{L}}$ by $d$ and let $\pi:Y\to X$ be
the finite $G:=GL_{d}({\mathbb{Z}}/p^{d})$-Galois cover of $X$ that
trivializes ${\mathbb{L}}/p^{d}$.

Let $\pi^{\prime}:Y^{\prime}\to\overline{X}$ be the normalization of
$\overline{X}$ in $Y$. The action of $G$ extends on $Y^{\prime}$ and
$\pi^{\prime}$ is a $G$-equivariant finite morphism. Applying functorial
resolution of singularities [Tem18, Theorem 1.1.13(iii)] we can find a diagram

${Y}$${\overline{Y}}$${X}$${\overline{X}}$$\scriptstyle{\pi}$$\scriptstyle{\overline{\pi}}$

such that $\overline{Y}$ is a proper smooth rigid-analytic variety, the
complement $D^{\prime}=\overline{Y}\setminus Y$ is a normal crossings divisor,
and $G$-action extends from $Y$ to $\overline{Y}$ and $\overline{\pi}$ is
$G$-equivariant. Since every quasi-unipotent element of
$1+p^{d}\mathrm{Mat}_{d}({\mathbb{Z}}_{p})$ is unipotent,
$\pi^{*}{\mathbb{L}}$ has unipotent monodromy around $D^{\prime}$. The maps
$\alpha_{{\mathbb{L}}}$ and $\alpha_{\pi^{*}{\mathbb{L}}}$ induce the diagram

(2.2)
${\operatorname{{End}}_{X_{\overline{K}}}({\mathbb{L}})\otimes_{{\mathbb{Z}}_{p}}C}$${H^{0}_{\mathrm{Higgs}}(\overline{X}_{\widehat{K}_{\infty}},{\mathcal{H}}^{\log}({\mathbb{L}}\otimes_{{\mathbb{Z}}_{p}}{\mathbb{L}}^{\vee}))\otimes_{\widehat{K}_{\infty}}C}$${\operatorname{{End}}_{\overline{X}_{\widehat{K}_{\infty}},\mathrm{Higgs}}{\mathcal{H}}^{\log}({\mathbb{L}})\otimes_{\widehat{K}_{\infty}}C}$${\operatorname{{End}}_{Y_{\overline{K}}}(\pi^{*}{\mathbb{L}})^{G}\otimes_{{\mathbb{Z}}_{p}}C}$${H^{0}_{\mathrm{Higgs}}(\overline{Y}_{\widehat{K}_{\infty}},{\mathcal{H}}^{\log}(\pi^{*}({\mathbb{L}}\otimes_{{\mathbb{Z}}_{p}}{\mathbb{L}}^{\vee})))^{G}\otimes_{K_{\widehat{K}_{\infty}}}C}$${\operatorname{{End}}_{\overline{Y}_{\widehat{K}_{\infty}},\mathrm{Higgs}}{\mathcal{H}}^{\log}(\pi^{*}{\mathbb{L}})^{G}\otimes_{\widehat{K}_{\infty}}C}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\alpha_{{\mathbb{L}}}}$$\scriptstyle{\sim}$$\scriptstyle{\alpha_{{\mathbb{L}}}}$$\scriptstyle{\sim}$

The first arrows in both rows are isomorphisms by [DLLZ19, Theorem 3.2.4(2)],
the second one in the second row is by Theorem 3.2.12(1) of loc. cit. and the
leftmost vertical one is an isomorphism because $\pi:Y_{\overline{K}}\to
X_{\overline{K}}$ is an etale $G$-Galois cover. The second arrow in the first
row and the rightmost vertical one are both injections, so the composition of
the first row is an isomorphism, as desired. ∎

######  Remark 2.6.

In part (iii) the condition on the existence of a ${\mathbb{Z}}_{p}$-lattice
is necessary. Consider a Tate elliptic curve
$X=\overline{X}={\mathbb{G}}_{m}/\langle\gamma\rangle$ defined by quotienting
${\mathbb{G}}_{m,K}=(\operatorname{{Spec}}K[x,x^{-1}])^{\mathrm{an}}$ by the
automorphism $\gamma$ defined as $\gamma^{*}(x)=px$. Define a rank $1$ local
system ${\mathbb{M}}$ on $X$ by taking the constant local system
$\underline{{\mathbb{Q}}_{p}}$ on ${\mathbb{G}}_{m}$ and equipping it with the
descent datum
$\gamma^{*}\underline{{\mathbb{Q}}_{p}}\xrightarrow{p}\underline{{\mathbb{Q}}_{p}}$.
Then $H^{0}(X_{\overline{K}},{\mathbb{M}})=0$ while
$H^{0}(X_{\widehat{K}_{\infty}},{\mathcal{H}}({\mathbb{M}}))^{\theta=0}=H^{0}_{\mathrm{proet}}(X_{\widehat{K}_{\infty}},{\mathbb{M}}\otimes_{{\mathbb{Q}}_{p}}\widehat{{\mathcal{O}}})=H^{0}_{\mathrm{proet}}({\mathbb{G}}_{m,\widehat{K}_{\infty}},\widehat{{\mathcal{O}}})^{\gamma=p^{-1}}=\widehat{K}_{\infty}\cdot
x^{-1}$. In particular, the comparison between etale and Higgs cohomology
fails already for cohomology in degree $0$ and (iii) is false e.g. for
${\mathbb{L}}=\underline{{\mathbb{Q}}_{p}}\oplus{\mathbb{M}}$.

This construction translates the issue of proving that a local system
${\mathbb{L}}$ is de Rham into a question about the action of $\Gamma$ on
${\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}})$. We will analyze this action
using Fontaine-Sen’s technique of decompletion.

##  3\. Decompleted Simpson and Riemann-Hilbert correspondences

The decompleted versions of ${\mathcal{H}}({\mathbb{L}})$ and
${\mathcal{R}}{\mathcal{H}}({\mathbb{L}})$ we need have essentially been
constructed by Koji Shimizu. Sections 3.2 and 4.1 of [Shi18] construct those
over affinoid bases where the bundle ${\mathcal{H}}({\mathbb{L}})$ is trivial.
We check in this section that these constructions can be globalized to work
over arbitrary smooth quasi-compact rigid-analytic varieties, the main result
of this section is Proposition 3.5.

We will start with a general discussion of decompletions in relative setting
that will later be specialized to the cases of
${\mathcal{H}}^{\log}({\mathbb{L}})$ and
${\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}})$. For the duration of this
section let $Y$ be a smooth quasi-compact rigid-analytic variety over $K$
which we will view as an adic space over
$\operatorname{{Spa}}(K,{\mathcal{O}}_{K})$. Consider the sheaves
${\mathcal{O}}_{{Y}_{K_{\infty}}}:={\mathcal{O}}_{Y}\otimes_{K}K_{\infty}$,
${\mathcal{O}}_{{Y}_{K_{\infty}}}[[t]]:=\lim\limits_{\leftarrow}{\mathcal{O}}_{Y}\otimes_{K}K_{\infty}[t]/t^{n}$,
and
${\mathcal{O}}_{{Y}_{K_{\infty}}}{((t))}:={\mathcal{O}}_{{Y}_{K_{\infty}}}{[[t]]}\otimes_{K_{\infty}[[t]]}K_{\infty}((t))$.
Denote by ${Y}_{K_{\infty}}$, $Y_{K_{\infty}[[t]]}$, and
${Y}_{K_{\infty}((t))}$ the ringed spaces
$({Y},{\mathcal{O}}_{{Y}_{K_{\infty}}})$,
$(Y,{\mathcal{O}}_{Y_{K_{\infty}}}{[[t]]})$, and
$({Y},{\mathcal{O}}_{{Y}_{K_{\infty}}}{((t))})$ respectively. We will call
locally free sheaves of finite rank on these ringed spaces simply ’vector
bundles’. Recall that ${Y}_{\widehat{K}_{\infty}}$ is the base change
${Y}\times_{\operatorname{{Spa}}(K,{\mathcal{O}}_{K})}\operatorname{{Spa}}(\widehat{K}_{\infty},{\mathcal{O}}_{\widehat{K}_{\infty}})$
and ${\mathcal{Y}}^{+}$, ${\mathcal{Y}}$ are the ringed spaces
$({Y}_{\widehat{K}_{\infty}},{\mathcal{O}}_{Y}\widehat{\otimes}_{K}L^{+}_{\mathrm{dR}})$
and
$({Y}_{\widehat{K}_{\infty}},{\mathcal{O}}_{Y}\widehat{\otimes}_{K}L_{\mathrm{dR}})$
respectively.

Note that it is not immediate that a vector bundle on some $Y_{K_{n}}$ gives
rise to a vector bundle on $Y_{K_{\infty}}$ because the underlying topological
spaces $|Y_{K_{n}}|$ and $|Y|$ are different. The next lemma, however, shows
that one can always find a cover of $Y$ that trivializes a vector bundle on
$Y_{K_{n}}$.

######  Lemma 3.1.

(i) For any quasi-compact open $U\subset Y_{\widehat{K}_{\infty}}$ there
exists $n\in{\mathbb{N}}$ and an open $U_{n}\subset Y_{K_{n}}$ such that
$U=U_{n}\times_{K_{n}}\widehat{K}_{\infty}$.

(ii) If $E$ is a vector bundle on $Y_{K_{n}}$ for some $n\in{\mathbb{N}}$ then
there exists a finite cover of $Y=\bigcup\limits_{i}U_{i}$ such that the cover
$\\{U_{i,K_{n}}\\}_{i\in I}$ of $Y_{K_{n}}$ trivializes $E$.

###### Proof.

(i) Since it is enough to prove the statement of each member of a finite cover
$U=\bigcup\limits_{i\in I}V_{i}$ we may assume that $Y$ is affinoid and $U$ is
a rational subset. The claim is then given by [Ber07, Lemma 2.1.3(ii)].

(ii) As $Y$ is assumed to be quasi-compact, we may assume that it is moreover
affinoid $Y=\operatorname{{Spa}}(A,A^{+})$ so $E$ is associated to a finite
projective module $M$ over $A_{K_{n}}$. We will prove the statement by
induction on the maximum of ranks of restrictions of $E$ to connected
components of $Y_{K_{n}}$ (we cannot assume that $Y_{K_{n}}$ is connected even
if it is given that $Y$ is connected). It is enough to find a cover
$Y=\bigcup\limits_{i}U_{i}$ by affinoids
$U_{i}=\operatorname{{Spa}}(A_{i},A_{i}^{+})$ such that on each $U_{i,K_{n}}$
there is a section $s:{\mathcal{O}}_{U_{i,K_{n}}}\to E|_{U_{i,K_{n}}}$ that
does not vanish at the points of connected components of $U_{i,K_{n}}$ where
$E$ is not identically zero. The exact sequences
$0\to{\mathcal{O}}_{U_{i,K_{n}}}\xrightarrow{s}E|_{U_{i,K_{n}}}\to\operatorname{coker}(s)\to
0$ split automatically by Tate’s acyclicity and we can find a trivializing
cover for $\operatorname{coker}(s)$ by induction assumption. For a point
$x\in|Y|$ consider the finite set
$\\{x_{1},\dots,x_{r}\\}=\pi_{n}^{-1}(x)\subset|Y_{K_{n}}|$ where
$\pi_{n}:Y_{K_{n}}\to Y$ is the canonical projection. For each $x_{i}$ such
that $E$ is not zero on the connected component containing $x_{i}$ we can find
a section $s_{i}:A_{K_{n}}\to M$ that does not vanish on some $x_{i}\in
V_{i}\subset Y_{K_{n}}$ and a function $f_{i}\in A$ that does not vanish in
the residue field $k(x_{i})$ but vanishes in each $k(x_{j})$ for $j\neq i$. If
$E$ vanishes on the connected component containing $x_{i}$ we declare
$s_{i}=0$. Let $\widetilde{V}\subset Y_{K_{n}}$ be union of the open subset
where the section $\sum\limits_{i=1}^{r}f_{i}s_{i}:A_{K_{n}}\to M$ does not
vanish and all the connected components of $Y_{K_{n}}$ where $E$ is zero. Take
$V\subset Y_{K_{n}}$ to be the intersection of its
$\operatorname{{Gal}}(K_{n}/K)$-orbit. Then $V=\pi_{n}^{-1}(U)$ for some open
$U\subset Y$ and $E|_{U_{i,K_{n}}}$ admits a section with desired property. ∎

######  Proposition 3.2.

Given a $\Gamma$-equivariant vector bundle $E$ on $Y_{\widehat{K}_{\infty}}$,
there is a $\Gamma$-equivariant vector bundle $E^{\mathrm{fin}}$ on
$Y_{K_{\infty}}$ with an isomorphism
$E^{\mathrm{fin}}_{\widehat{K}_{\infty}}\simeq E$. For an affinoid open
$\operatorname{{Spa}}(A,A^{+})=U\subset Y$ the sections of $E^{\mathrm{fin}}$
are given by:

(3.1) $E^{\mathrm{fin}}(U)=\\{s\in E(U_{\widehat{K}_{\infty}})\text{ such that
the }A\text{-span of the }\Gamma\text{-orbit of }s\text{ is finitely generated
over }A\\}$

Moreover, there is an endomorphism $\phi:E^{\mathrm{fin}}\to E^{\mathrm{fin}}$
such that for any section $s\in E^{\mathrm{fin}}(U_{K_{\infty}})$ we have

(3.2) $\gamma(s)=\exp(\phi\cdot\log\chi_{\mathrm{cycl}}(\gamma))s$

for all $\gamma$ in some open subgroup $\Gamma_{s}$ of $\Gamma$. The
association $E\mapsto(E^{\mathrm{fin}},\phi)$ is functorial with respect to
morphisms of equivariant vector bundles on $Y_{\widehat{K}_{\infty}}$ and is
compatible with tensor products.

###### Proof.

Denote by $E^{\mathrm{fin}}_{0}$ the presheaf of
${\mathcal{O}}_{Y}\otimes_{K}K_{\infty}$-modules defined on affinoid opens in
$Y$ by (3.1).

First, let us prove that $E^{\mathrm{fin}}_{0}$ satisfies the sheaf condition:
if $U=\bigcup\limits_{i\in I}U_{i}$ is an open covering of an affinoid by
affinoids, we need to check that a section $s\in E(U_{\widehat{K}_{\infty}})$
lies in $E^{\mathrm{fin}}_{0}(U)$ if all its restrictions to
$U_{i,{K_{\infty}}}$ lie in the corresponding submodules. We may assume that
the covering has finitely many opens as $U$ is quasi-compact. For each open
$U_{i}=\operatorname{{Spa}}(A_{i},A_{i}^{+})$ there is a finite subset
$\Gamma_{i}\subset\Gamma$ such that the finite set $\Gamma_{i}\cdot
s|_{U_{i}}$ generates the module $\langle\Gamma\cdot s|_{U_{i}}\rangle$ over
$A_{i}$. Taking the union of these $\Gamma_{i}$ for all $i$ we obtain a finite
set of elements $\gamma_{1},\dots,\gamma_{N}\in\Gamma$ such that for all $i$
the inclusion
$\langle\gamma_{1}(s)|_{U_{i}},\dots,\gamma_{N}(s)|_{U_{i}}\rangle_{A_{i}}\subset\langle\Gamma\cdot
s|_{U_{i}}\rangle_{A_{i}}$ is equality. Since $E$ is, in particular, a
coherent sheaf on $Y_{\widehat{K}_{\infty}}$, we have
$E(U_{i})=E(U)\otimes_{A_{\widehat{K}_{\infty}}}A_{i,\widehat{K}_{\infty}}=E(U)\otimes_{A}A_{i}$.
The map $A\to A_{i}$ is flat so for any set $I\subset E(U)$ we have $\langle
x|x\in I\rangle_{A}\otimes_{A}A_{i}\simeq\langle x|_{U_{i}}|x\in
I\rangle_{A_{i}}$ under the canonical map. Therefore
$\langle\gamma_{1}(s),\dots,\gamma_{N}(s)\rangle_{A}\subset\langle\Gamma\cdot
s\rangle_{A}$ becomes an equality after taking the tensor product with any
$A_{i}$ over $A$. Since the map $A\to\prod\limits_{i\in I}A_{i}$ is faithfully
flat, this inclusion is in fact equality, as desired.

By [Shi18, Theorem 2.9] for any affinoid open $U\subset Y$ such that
$E|_{U_{\widehat{K}_{\infty}}}$ is trivial as a vector bundle we have
$E^{\mathrm{fin}}_{0}(U)\otimes_{{\mathcal{O}}(U_{K_{\infty}})}{{\mathcal{O}}(U_{\widehat{K}_{\infty}})}\simeq
E(U_{\widehat{K}_{\infty}})$. By Lemma 3.1 (i) there exists $n$ and a finite
cover of $Y_{K_{n}}$ such that its base change to $Y_{\widehat{K}_{\infty}}$
trivializes $E$ so $E^{\mathrm{fin}}_{0}$ defines a locally free sheaf of
${\mathcal{O}}_{Y_{K_{\infty}}}$-modules on ${Y}_{K_{n}}$. Possibly after
increasing $n$, there is a vector bundle $E_{n}$ on ${Y}_{K_{n}}$ such that
$E_{n}\otimes_{K_{n}}K_{\infty}\simeq E_{0}^{\mathrm{fin}}$. By Lemma 3.1
(ii), the sheaf $E_{n}$ and therefore $E^{\mathrm{fin}}_{0}$ is locally free
on $Y$, as desired.

The local construction of the operator $\phi$ from [Shi18, Proposition 2.20]
then immediately globalizes because $\phi$ is uniquely characterized by (3.2).
The last assertion about functoriality and tensor compatibility follows from
the base change functor from vector bundles on $Y_{K_{\infty}}$ to
$Y_{\widehat{K}_{\infty}}$ being faithful and conservative. ∎

######  Definition 3.3.

A filtration on a vector bundle $E$ on $Y_{K_{\infty}((t))}$ is a collection
$\\{F^{i}E\\}_{i\in{\mathbb{Z}}}$ of locally free
${\mathcal{O}}_{Y_{K_{\infty}}}[[t]]$-subsheaves $F^{i}E\subset E$ such that
$F^{i+1}E\subset F^{i}E$ and $t\cdot F^{i}E=F^{i+1}E$ for all $i$. A filtered
vector bundle with integrable connection on $Y_{K_{\infty}((t))}$ is a
filtered vector bundles $(E,\\{F^{i}E\\}_{i\in{\mathbb{Z}}})$ as above
together with an integrable $K_{\infty}((t))$-linear connection $\nabla:E\to
E\otimes_{{\mathcal{O}}_{Y_{K_{\infty}}}((t))}\Omega^{1}_{Y_{K_{\infty}}/K_{\infty}}((t))$
such that $\nabla(F^{i}E)\subset
F^{i-1}E\otimes_{{\mathcal{O}}_{Y_{K_{\infty}}}[[t]]}\Omega^{1}_{Y_{K_{\infty}}/K_{\infty}}[[t]]$.

Note that given a vector bundle $E$ on $Y_{K_{\infty}[[t]]}$ we can attach to
it a vector bundle on ${\mathcal{Y}}^{+}$ as follows. Suppose that
$Y=\bigcup\limits_{i\in I}U_{i}$ is a trivializing cover for $E$ and define a
vector bundle $E_{L_{\mathrm{dR}}^{+}}$ on ${\mathcal{Y}}^{+}$ trivialized by
the cover $\bigcup\limits_{i\in
I}|U_{i,\widehat{K}_{\infty}}|=|{\mathcal{Y}}^{+}|$ using the gluing data
obtained from that for $E$ applying the maps
${\mathcal{O}}_{U_{ij}}[[t]]\to{\mathcal{O}}_{U_{ij}}\widehat{\otimes}_{K}L_{\mathrm{dR}}^{+}$.

######  Proposition 3.4.

Suppose that $E$ is a $\Gamma$-equivariant vector bundle on
${\mathcal{Y}}^{+}$. Then there is a $\Gamma$-equivariant vector bundle
$E^{\mathrm{fin}}$ on $Y_{K_{\infty}[[t]]}$ together with an isomorphism
$E^{\mathrm{fin}}_{L_{\mathrm{dR}}^{+}}\simeq E$. For every $n$ the sections
of $E^{\mathrm{fin}}/t^{n}$ on an affinoid open
$\operatorname{{Spa}}(A,A^{+})=U\subset Y$ are given by:

(3.3) $(E^{\mathrm{fin}}/t^{n})(U)=\\{s\in
E({\mathcal{U}}^{+}_{L_{\mathrm{dR}}^{+}/t^{n}})\text{ such that the
}A\text{-span of the }\Gamma\text{-orbit of }s\text{ is finitely generated
over }A\\}$

Moreover, there is an endomorphism $\phi:E^{\mathrm{fin}}\to E^{\mathrm{fin}}$
that is ${\mathcal{O}}_{Y_{K_{\infty}}}$-linear and satisfies
$\phi(ts)=t\phi(s)+ts$ for any local section $s\in E^{\mathrm{fin}}(U)$. For
any $n$ and $s\in E^{\mathrm{fin}}(U)$

(3.4) $\gamma(s)=\exp(\phi\cdot\log\chi_{\mathrm{cycl}}(\gamma))s$

for all $\gamma$ in some open subgroup $\Gamma_{n,s}$ of $\Gamma$. The
association $E\mapsto(E^{\mathrm{fin}},\phi)$ is functorial with respect to
morphisms of equivariant vector bundles on ${\mathcal{Y}}^{+}$ and is
compatible with tensor products.

###### Proof.

Define the presheaf of $K_{\infty}[t]/t^{n}$-module on affinoid open subspaces
of $Y$ by

$E^{\mathrm{fin}}_{n}(U)=\\{s\in
E({\mathcal{U}}^{+}_{L_{\mathrm{dR}}^{+}/t^{n}})\text{ such that the
}A\text{-span of }\Gamma\text{-orbit of }s\text{ is finitely generated over
}A\\}.$

There is a left exact sequence of presheaves $0\to
E_{n-1}^{\mathrm{fin}}\xrightarrow{t}E_{n}^{\mathrm{fin}}\to(E/t)^{\mathrm{fin}}$.
We will prove by induction on $n$ that $E_{n}^{\mathrm{fin}}$ is a sheaf, the
base case $n=1$ being covered by Proposition 3.2. Suppose that
$U=\bigcup\limits_{i\in I}U_{i}$ is a finite cover of an affinoid by affinoids
and a section $s\in E({\mathcal{U}}^{+}_{L_{\mathrm{dR}}^{+}/t^{n}})$ is such
that every restriction $s|_{U_{i}}$ lies in $E_{n}^{\mathrm{fin}}(U_{i})$. We
want to conclude that $s$ lies in $E^{\mathrm{fin}}_{n}(U)$. Denote by
$\overline{s}$ the image of $s$ in $(E/t^{n-1})(U_{\widehat{K}_{\infty}})$. By
the inductive assumption, $E_{n-1}^{\mathrm{fin}}$ is a sheaf so there is a
finite set of elements $\gamma_{1},\dots,\gamma_{N}\in\Gamma$ such that
$\gamma_{1}\cdot\overline{s},\dots\gamma_{N}\cdot\overline{s}$ span
$\langle\Gamma\cdot\overline{s}\rangle_{A}$ inside
$E({\mathcal{U}}_{L_{\mathrm{dR}}^{+}/t^{n-1}})$. This automatically implies
that $\gamma_{1}\cdot s,\dots\gamma_{N}\cdot s,t\gamma_{1}(s),\dots
t\gamma_{N}(s)$ span $\langle\Gamma\cdot s\rangle_{A}$ inside
$E({\mathcal{U}}^{+}_{L_{\mathrm{dR}}^{+}/t^{n}})$. Therefore,
$E^{\mathrm{fin}}_{n}$ is a sheaf.

Next, by [Shi18, Proposition 2.11] for any affinoid $U\subset X$ such that
$E({\mathcal{U}}^{+}_{L_{\mathrm{dR}}^{+}/t^{n}})$ is a free
$A\widehat{\otimes}_{K}L_{\mathrm{dR}}^{+}/t^{n}$-module the subspace
$E_{n}^{\mathrm{fin}}(U)$ is a free $K_{\infty}[t]/t^{n}$-module of the same
rank and the canonical map
$E_{n}^{\mathrm{fin}}(U)\widehat{\otimes}_{K_{\infty}[t]/t^{n}}L_{\mathrm{dR}}^{+}/t^{n}\simeq
E({\mathcal{U}}^{+}_{L_{\mathrm{dR}}^{+}/t^{n}})$ is an isomorphism.

Finally, there is a cover of $Y_{\widehat{K}_{\infty}}$ by affinoids that
trivializes $E$, it descends to some $K_{m}$ by Lemma 3.1(i) so
$E_{n}^{\mathrm{fin}}$ defines a locally free
${\mathcal{O}}_{Y_{K_{\infty}}}{[t]/t^{n}}$-module on $Y_{K_{m}}$. After
increasing $m$ this module may be non-canonically descended to a locally free
sheaf of ${\mathcal{O}}_{Y_{K_{m}}}[t]/t^{n}$-modules on $Y_{K_{m}}$ and Lemma
3.1(ii) provides a cover of $Y$ whose base change to $K_{m}$ trivializes this
locally free sheaf. This cover therefore also trivializes the
${\mathcal{O}}_{Y_{K_{\infty}}}[t]/t^{n}$-module $E_{n}^{\mathrm{fin}}$ and we
define the desired $E^{\mathrm{fin}}$ as
$\lim\limits_{n}E_{n}^{\mathrm{fin}}$. [Shi18, Proposition 2.24] gives us the
desired Sen endomorphism $\phi$ over the trivializing cover and it glues to a
global endomorphism because of the characterizing property (3.4). ∎

Now let us put ourselves in the setting of Section 2: $\overline{X}$ is a
geometrically connected smooth proper rigid-analytic variety over $K$ and
$X\subset\overline{X}$ is an open subvariety such that
$D=\overline{X}\setminus X$ is a normal crossings divisor. Applying these
decompletion results to ${\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}})$ for
a local system ${\mathbb{L}}$ on $X$ gives

######  Proposition 3.5.

To every ${\mathbb{Q}}_{p}$-local system ${\mathbb{L}}$ on $X$ we can assign a
filtered vector bundles with integrable connection
$\mathrm{RH}^{\log}({\mathbb{L}})$ on $\overline{X}_{K_{\infty}((t))}$ that
comes equipped with a $K_{\infty}$-linear endomorphism $\phi$ preserving the
filtration and commuting with the connection.

(i) Denote the graded piece of the filtration
$\operatorname{{gr}}^{0}\mathrm{RH}^{\log}({\mathbb{L}})$ by
$\mathrm{H}^{\log}({\mathbb{L}})$. All the other graded pieces
$\operatorname{{gr}}^{i}\mathrm{RH}^{\log}({\mathbb{L}})$ are then canonically
identified with $\mathrm{H}^{\log}({\mathbb{L}})$ as well.
$\mathrm{H}^{\log}({\mathbb{L}})$ is a vector bundle of rank
$\operatorname{{rk}}_{{\mathbb{Q}}_{p}}{\mathbb{L}}$ on
$\overline{X}_{K_{\infty}}$ and connection induces a Higgs field
$\theta:\mathrm{H}^{\log}({\mathbb{L}})\to\mathrm{H}^{\log}({\mathbb{L}})\otimes\Omega^{1}_{\overline{X}_{K_{\infty}}/K_{\infty}}$.
The endomorphism $\phi$ defines an endomorphism $\phi$ of the vector bundle
$\mathrm{H}^{\log}({\mathbb{L}})$ such that
$\theta\circ\phi=(\phi-1)\circ\theta$.

(ii) The characteristic polynomial
$\det(T\cdot\mathrm{Id}_{\mathrm{H}^{\log}({\mathbb{L}})}-\phi)$ has
coefficients in $K$.

(iii) If ${\mathbb{L}}$ admits a ${\mathbb{Z}}_{p}$-lattice then there is an
$H_{K}$-equivariant isomorphism of $C$-algebras

$\operatorname{{End}}_{X_{\overline{K}}}({{\mathbb{L}}})\otimes_{{\mathbb{Q}}_{p}}C\simeq\operatorname{{End}}_{\overline{X}_{K_{\infty}},\mathrm{Higgs}}(\mathrm{H}^{\log}({\mathbb{L}}))\otimes_{K_{\infty}}C.$

###### Proof.

Start with ${\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}})$ provided by
Theorem 2.4 and apply Proposition 3.4 to
$E=\mathrm{Fil}^{0}{\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}})$ with
$Y=\overline{X}$. Define $\mathrm{RH}^{\log}({\mathbb{L}})$ as
$E^{\mathrm{fin}}[1/t]$. By construction, its
${\mathcal{O}}_{X}\otimes_{K}L_{\mathrm{dR}}$-rank is equal to
$\operatorname{{rk}}_{{\mathbb{Q}}_{p}}{\mathbb{L}}$ and is equipped with an
endomorphism $\phi$ related to the action of $\Gamma$ on
${\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}})$ via (3.3).

For part (i), the identification
$\operatorname{{gr}}^{i}{\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}})\simeq{\mathcal{H}}^{\log}({\mathbb{L}})(i)$
induces
$\operatorname{{gr}}^{i}\mathrm{RH}^{\log}({\mathbb{L}})\simeq\mathrm{H}^{\log}({\mathbb{L}})$
where we use our choice of $t$ to identify
${\mathcal{H}}^{\log}({\mathbb{L}})(i)$ with
${\mathcal{H}}^{\log}({\mathbb{L}})$. The Sen endomorphism on
$\operatorname{{gr}}^{-1}\mathrm{RH}^{\log}({\mathbb{L}})$ is given by
$\phi-1$ in terms of this identification where $\phi$ is the Sen endomophism
on $\mathrm{H}^{\log}({\mathbb{L}})$. This gives the equality
$\theta\circ\phi=(\phi-1)\circ\theta$ of maps from
$\mathrm{H}^{\log}({\mathbb{L}})$ to
$\mathrm{H}^{\log}({\mathbb{L}})\otimes_{{\mathcal{O}}_{\overline{X}_{K_{\infty}}}}\Omega^{1}_{\overline{X}_{K_{\infty}}/K_{\infty}}$.

In part (ii), the fact the coefficients of the characteristic polynomial of
$\phi$ are constant (that is, lie in $\overline{K}$) across $\overline{X}$ is
[Shi18, Theorem 4.8]. It is, in fact, a purely local statement and is a
consequence of the fact that the Higgs bundle
$\mathrm{H}^{\log}({\mathbb{L}})$ together with the endomorphism $\phi$
appears as a graded piece of a vector bundle with an integrable connection,
but in our setting of a proper $\overline{X}$ it also follows immediately from
all global functions being constant
$H^{0}(\overline{X}_{K_{\infty}},{\mathcal{O}}_{X_{K_{\infty}}})=K_{\infty}$
(recall that $X$ is assumed to be geometrically connected, otherwise the
statement (ii) is false as the example of
$X=\operatorname{{Spa}}(K^{\prime},{\mathcal{O}}_{K^{\prime}})$ for a finite
extension $K^{\prime}\supset K$ shows).

It remains to prove that the coefficients of
$\det(T\cdot\mathrm{Id}_{\mathrm{H}^{\log}({\mathbb{L}})}-\phi)$ in fact lie
in $K$. If $X$ has a $K$-point, this follows from [Sen81, Theorem 5] applied
to the stalk of ${\mathbb{L}}$ at that point. To cover the case of $X$ not
admitting a rational point, we can repeat the proof of Sen’s result in our
relative setting: the Sen endomorphism $\phi$ induces an
$\widehat{K}_{\infty}$-linear endomorphism of
${\mathcal{H}}^{\log}({\mathbb{L}})$ that commutes with the semi-linear
$\Gamma$-action on that vector bundle. Therefore the coefficients of the
characteristic polynomial of $\phi$ lie in
$H^{0}(\overline{X}_{\widehat{K}_{\infty}},{\mathcal{O}}_{\overline{X}_{\widehat{K}_{\infty}}})^{\Gamma}=K$.

Part (iii) follows from Theorem 2.4 (iii) together with the observation that
$H^{0}(\overline{X}_{K_{\infty}},E^{\mathrm{fin}})=H^{0}(\overline{X}_{\widehat{K}_{\infty}},E)^{\mathrm{fin}}$
for any $\Gamma$-equivariant vector bundle $E$ on $\overline{X}$. ∎

We will call the roots of the characteristic polynomial
$\det(T\cdot\mathrm{Id}_{\mathrm{H}^{\log}({\mathbb{L}})}-\phi)$ viewed as
elemens of $\overline{K}$ the Hodge-Tate weights of ${\mathbb{L}}$.

######  Lemma 3.6.

(i) A local system ${\mathbb{L}}$ is de Rham if and only if
$\mathrm{RH}^{\log}({\mathbb{L}})^{\phi=0}$ is a vector bundle of rank
$\operatorname{{rk}}_{{\mathbb{Q}}_{p}}{\mathbb{L}}$ on
$\overline{X}_{K_{\infty}}$.

(ii) A local system ${\mathbb{L}}$ is Hodge-Tate if and only if $\phi$ is a
semi-simple endomorphism of $\mathrm{H}^{\log}({\mathbb{L}})$ and Hodge-Tate
weights of ${\mathbb{L}}$ lie in ${\mathbb{Z}}$.

###### Proof.

(i) By definition, a local system is de Rham if and only if the invariants
$D_{\mathrm{dR},\log}({\mathbb{L}}):={\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}})^{\Gamma}$
form a vector bundle on $X$ of rank
$\operatorname{{rk}}_{{\mathbb{Q}}_{p}}{\mathbb{L}}$. Suppose that
$\mathrm{RH}^{\log}({\mathbb{L}})^{\phi=0}$ is a vector bundle of rank
$\operatorname{{rk}}_{{\mathbb{Q}}_{p}}{\mathbb{L}}$ on $X_{K_{\infty}}$.
Therefore there is a finite cover $\overline{X}=\bigcup\limits_{i}U_{i}$ such
that for each $U_{i}$ the kernel
$\mathrm{RH}^{\log}({\mathbb{L}})(U_{i})^{\phi=0}$ is spanned by
$\operatorname{{rk}}_{{\mathbb{Q}}_{p}}{\mathbb{L}}$ elements
$s_{i1},\dots,s_{id}$ with $\phi(s_{ij})=0$ for all $j$. By (3.3) each
$s_{ij}$ is stabilized by an open subgroup of $\Gamma$. Therefore for large
enough $n$ they all are stabilized by
$\Gamma_{n}:=\operatorname{{Gal}}(K_{\infty}/K_{n})$ so
$\operatorname{{rk}}_{{\mathcal{O}}_{X_{K_{n}}}}D_{\mathrm{dR}}({\mathbb{L}}|_{X_{K_{n}}})\geq\operatorname{{rk}}_{{\mathbb{Q}}_{p}}{\mathbb{L}}$.
These ranks are hence equal and ${\mathbb{L}}|_{X_{K_{n}}}$ is de Rham which
implies that ${\mathbb{L}}$ itself is de Rham.

Conversely, suppose that ${\mathbb{L}}$ is de Rham. By [DLLZ19, Theorem
3.2.7(2)] the sheaf
$D_{\mathrm{dR},\log}({\mathbb{L}})={\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}})^{\Gamma}$
is a locally free sheaf of rank
$\operatorname{{rk}}_{{\mathbb{Z}}_{p}}{\mathbb{L}}$. The canonical injective
map
$D_{\mathrm{dR},\log}({\mathbb{L}})\otimes_{K}K_{\infty}\to\mathrm{RH}^{\log}({\mathbb{L}})^{\phi=0}$
is an isomorphism because the map
$D_{\mathrm{dR},\log}({\mathbb{L}})\widehat{\otimes}_{K}{L_{\mathrm{dR}}}\to{\mathcal{R}}{\mathcal{H}}^{\log}({\mathbb{L}})$
is. In particular, $\mathrm{RH}^{\log}({\mathbb{L}})^{\phi=0}$ is a vector
bundle of rank $\operatorname{{rk}}_{{\mathbb{Q}}_{p}}{\mathbb{L}}$ on
$\overline{X}_{K_{\infty}}$.

(ii) The operator $\phi$ on $\mathrm{H}^{\log}({\mathbb{L}})$ is semi-simple
with integer eigenvalues if and only if its restriction
$\mathrm{H}^{\log}({\mathbb{L}})|_{X}$ is. The latter condition is a
restatement of what it means for ${\mathbb{L}}$ to be Hodge-Tate by [Shi18,
Theorem 5.5]. ∎

##  4\. Geometrically irreducible local systems are Hodge-Tate up to a twist

As before, $X$ is a geometrically connected smooth rigid-analytic variety over
$K$ embedded as a Zariski open subvariety into a smooth proper rigid-analytic
variety $\overline{X}$ such that $\overline{X}\setminus X$ is a normal
crossings divisor.

######  Theorem 4.1.

Suppose that ${\mathbb{L}}$ is a $\overline{\mathbb{Q}}_{p}$-local system on
$X$ that admits a $\overline{{\mathbb{Z}}}_{p}$-lattice. If
${\mathbb{L}}|_{X_{\overline{K}}}$ is irreducible then there exists a
character $\chi:G_{K}\to\overline{\mathbb{Q}}_{p}^{\times}$ such that
${\mathbb{L}}\otimes\chi$ is Hodge-Tate.

The next lemma says that proving the theorem amounts to proving that the
projective representation of the fundamental group corresponding to
${\mathbb{L}}$ is Hodge-Tate. In the case where $K$ is a finite extension of
${\mathbb{Q}}_{p}$ this statement follows from [Pat19, Corollary 3.2.12].

Let us introduce some notation for working with the Sen endomorphism
associated to a local system ${\mathbb{L}}$ of $E$-vector spaces, for a finite
extension $E\supset{\mathbb{Q}}_{p}$. Viewing ${\mathbb{L}}$ as a
${\mathbb{Q}}_{p}$-local system (of rank
$\operatorname{{rk}}_{E}{\mathbb{L}}\cdot[E:{\mathbb{Q}}_{p}]$) we may
consider the associated Higgs bundle $\mathrm{H}^{\log}({\mathbb{L}})$ on
$\overline{X}_{K_{\infty}}$ that has a structure of an $E$-module, by
functoriality. In particular, $\mathrm{H}^{\log}({\mathbb{L}})$ carries an
action of the algebra $E\otimes_{{\mathbb{Q}}_{p}}K$. This algebra can be
decomposed as
$E\otimes_{{\mathbb{Q}}_{p}}K=\bigoplus\limits_{\tau:E\to\overline{K}}K_{\tau}$
where $\tau$ runs over all ${\mathbb{Q}}_{p}$-linear embeddings of $E$ into
$\overline{K}$ and $K_{\tau}$ is the subfield of $\overline{K}$ generated by
$K$ and $\tau(E)$. This decomposition induces a decomposition of every
$E\otimes_{{\mathbb{Q}}_{p}}K$-module. In particular, we get a decomposition
$\mathrm{H}^{\log}({\mathbb{L}})=\bigoplus\limits_{\tau}\mathrm{H}^{\log}({\mathbb{L}})_{\tau}$
where $\mathrm{H}^{\log}({\mathbb{L}})_{\tau}$ is defined as
$\mathrm{H}^{\log}({\mathbb{L}})\otimes_{E\otimes_{{\mathbb{Q}}_{p}}K}K_{\tau}$.
This decomposition is respected by the Sen endomorphism $\phi$ and we denote
by $\phi_{\tau}$ its component on the summand
$\mathrm{H}^{\log}({\mathbb{L}})_{\tau}$. For the duration of the proofs of
Lemmas 4.2 and 4.3 we will enhance the notation $\phi_{\tau}$ to
$\phi_{{\mathbb{L}},\tau}$ as the discussion will involve Sen endomorphisms of
various local systems. We will refer to eigenvalues of
$\phi_{{\mathbb{L}},\tau}$ as $\tau$-Hodge-Tate weights of ${\mathbb{L}}$.

######  Lemma 4.2.

Let $\overline{x}$ be arbitrary geometric point of $X$ and
$E\supset{\mathbb{Q}}_{p}$ be a finite extension. If
$\rho:\pi_{1}^{\text{et}}(X,\overline{x})\to GL_{d}({\mathcal{O}}_{E})$ is a
continuous representation such that the corresponding projective
representation $\rho_{\mathrm{proj}}:\pi_{1}^{\text{et}}(X,\overline{x})\to
PGL_{d}({\mathcal{O}}_{E})$ is Hodge-Tate then there exists a character
$\chi:G_{K}\to\overline{\mathbb{Q}}_{p}^{\times}$ such that $\rho\otimes\chi$
is Hodge-Tate.

###### Proof.

Denote by ${\mathbb{L}}$ the $E$-local system corresponding to $\rho$. By
definition, $\rho_{\mathrm{proj}}$ being Hodge-Tate means that the adjoint
local system ${\mathbb{L}}\otimes_{E}{\mathbb{L}}^{\vee}$ is Hodge-Tate, cf.
Example 2.3.

The assumption that ${\mathbb{L}}\otimes_{E}{\mathbb{L}}^{\vee}$ is Hodge-Tate
translates into the fact that the endomorphism $\phi_{{\mathbb{L}}}\otimes
1-1\otimes\phi_{{\mathbb{L}}}^{\vee}$ of
$\mathrm{H}^{\log}({\mathbb{L}})\otimes_{E}\mathrm{H}^{\log}({\mathbb{L}})^{\vee}$
is semi-simple with integer eigenvalues. Here we have used that the functor
$\mathrm{H}^{\log}|_{X_{K_{\infty}}}$ is monoidal together with the
observation that semi-simplicity and integrality of eigenvalues can be checked
after restricting to a dense open in $\overline{X}_{K_{\infty}}$. In the
notation of the paragraph preceeding the statement of the lemma, this is
equivalent to saying that for every $\tau$ the Sen endomorphism
$\phi_{{\mathbb{L}},\tau}\otimes 1-1\otimes\phi_{{\mathbb{L}},\tau}^{\vee}$ on
$\mathrm{H}^{\log}({\mathbb{L}})_{\tau}\otimes_{{\mathcal{O}}_{\overline{X}_{K_{\infty}^{\tau}}}}\mathrm{H}^{\log}({\mathbb{L}})^{\vee}_{\tau}$
is semi-simple with eigenvalues in ${\mathbb{Z}}$. This means that for every
$\tau$ the operator $\phi_{{\mathbb{L}},\tau}$ is semi-simple and there exists
an element $a_{\tau}\in\overline{K}$ such that the eigenvalues of
$\phi_{{\mathbb{L}},\tau}$ lie in $a_{\tau}+{\mathbb{Z}}$.

Our goal now is to find a character $\chi:G_{K}\to E^{\prime\times}$ for a
finite extension $E^{\prime}\supset E$ such that the Sen endomorphism
corresponding to the $E^{\prime}$-local system ${\mathbb{L}}\otimes_{E}\chi$
has integral eigenvalues. This would be immediate if $K$ was contained in
$\overline{\mathbb{Q}}_{p}$ as then [DM13, Lemma 2.1.3] can be used to provide
a character $\chi$ with the eigenvalue of $\phi_{\chi,\tau}$ on
$\mathrm{H}(\chi)_{\tau}$ equal to $-a_{\tau}$ for every $\tau$. We don’t know
if the analog of [DM13, Lemma 2.1.3] is true for an arbitrary $K$ but we can
nevertheless construct $\chi$ as follows.

Suppose that eigenvalues of $\phi_{{\mathbb{L}},{\tau}}$ on
$\mathrm{H}^{\log}({\mathbb{L}})_{\tau}$ are
$a_{\tau}+n_{\tau,1},\dots,a_{\tau}+n_{\tau,d}$ for some
$n_{\tau,j}\in{\mathbb{Z}}$. Then the eigenvalue of
$\phi_{\det{\mathbb{L}},\tau}$ on $\mathrm{H}^{\log}(\det{\mathbb{L}})_{\tau}$
is equal to $d\cdot a_{\tau}+\sum\limits_{j=1}^{d}n_{\tau,j}$. To get started
on the construction of $\chi$, we would like to have a character
$\chi_{0}:G_{K}\to(E^{\prime})^{\times}$ with values in a finite extension
$E^{\prime}\supset E$, such that the eigenvalue of $\phi_{\chi_{0},\tau}$ on
$\mathrm{H}(\chi_{0})_{\tau}$ is equal to that of
$\phi_{\det{\mathbb{L}},\tau}$ on
$\mathrm{H}^{\log}(\det{\mathbb{L}})_{\tau}$. If $X$ has a $K$-point $x$ then
such $\chi_{0}$ is provided simply by $(\det{\mathbb{L}})_{x}$. In general, if
$X$ acquires a rational point $x\in X(K^{\prime})$ after a finite extension
$K^{\prime}\supset K$ we define $\chi_{0}$ as the output of Lemma 4.3 (ii)
applied to $\eta^{\prime}=(\det{\mathbb{L}})_{x}$, the assumption of this
lemma being satisfied by Proposition 3.5 (ii).

Next, we will construct a character $\chi_{1}:G_{K}\to E^{\times}$ such that
the $\tau$-Hodge-Tate weight of $\chi_{1,\tau}$ is the integer
$\sum\limits_{j=1}^{d}n_{\tau,j}$, for every $\tau$. Let $L$ be the field
$\overline{\mathbb{Q}}_{p}\cap K\subset\overline{K}$. The inclusion
$\overline{L}=\overline{\mathbb{Q}}_{p}\subset\overline{K}$ defines a map of
Galois groups $\alpha:G_{K}\to G_{L}$ and, if $\eta:G_{L}\to E^{\times}$ is a
character, the $\tau$-weight of $\eta$ coincides with the $\tau$ weight of
$\eta\circ\alpha$, for every embedding $\tau:E\to\overline{K}$ (all embeddings
necessarily factor through $\overline{L}$). The character $\chi_{1}$ can be
now obtained by precomposing with $\alpha$ the character provided by [DM13,
Lemma 2.1.3] applied to the field $L$ and the set of weights
$(\sum\limits_{j=1}^{d}n_{\tau,j})_{\tau}$.

The desired character $\chi$ can now be defined as the ”$d$-th root” of
$\chi_{1}\otimes\chi_{0}^{-1}$, provided by Lemma 4.3(i). ∎

######  Lemma 4.3.

  1. (i)

For any character $\eta:G_{K}\to(E^{\prime})^{\times}$ and an integer $d$
there exists a character $\eta^{\prime}:G_{K}\to E^{\prime\times}$ with values
in a finite extension $E^{\prime}\supset E$ such that $(\eta^{\prime})^{d}$ is
equal to $\eta$ when restricted to an open subgroup of $G_{K}$.

  2. (ii)

Suppose that $K^{\prime}\supset K$ is a finite extension and
$\eta^{\prime}:G_{K^{\prime}}\to E^{\times}$ is a character such that for
every $\tau:E\to\overline{K}=\overline{K}^{\prime}$ the $\tau$-Hodge-Tate
weight of $\eta^{\prime}$ lies in $K_{\tau}$ (rather than in a larger field
$K^{\prime}_{\tau}$). Then there exists a character
$\eta:G_{K}\to(E^{\prime})^{\times}$ with values in a finite extension
$E^{\prime}\supset E$ such that the $\tau$-Hodge-Tate weight of $\eta$ is
equal to the $\tau|_{E}$-Hodge-Tate weight of $\eta^{\prime}$, for every
$\tau:E^{\prime}\to\overline{K}$.

###### Proof.

(i) A very similar argument to the one we are going to give appears in the
proof of [DM13, Lemma 2.1.3]. We will prove that there exists a continuous
homomorphism
$r_{d}:{\mathcal{O}}_{E}^{\times}\to\overline{\mathbb{Q}}_{p}^{\times}$ that
is given by the formula
$r_{d}(x)=(1+x)^{1/d}:=\sum\limits_{i=0}^{\infty}\binom{1/d}{i}x^{i}$ for $x$
from an open subgroup of ${\mathcal{O}}_{E}^{\times}$. Indeed, the topological
group ${\mathcal{O}}_{E}^{\times}$ can be decomposed as a product
${\mathcal{O}}_{E}^{\times}=({\mathcal{O}}_{E}^{\times})_{\mathrm{tors}}\times
U_{E}$ where $U_{E}$ is an open subgroup isomorphic to
${\mathbb{Z}}_{p}^{[E:{\mathbb{Q}}_{p}]}$. We can define a homomorphism
$r^{\prime}_{d}:U_{E}\to\overline{\mathbb{Q}}_{p}^{\times}$ by the formula
$r^{\prime}_{d}(x)=(1+x)^{1/d}$ on an open subgroup of $U_{E}$ where the
series $(1+x)^{1/d}$ converges and then extend it arbitrarily to all of
$U_{E}$ using that $\overline{\mathbb{Q}}_{p}^{\times}$ is a divisible group.
The desired $r_{d}$ is then obtained by precomposing $r^{\prime}_{d}$ with the
projection ${\mathcal{O}}_{E}^{\times}\to U_{E}$.

The ”$d$-th root” of any $\eta^{\prime}$ can now be defined as
$r_{d}\circ\eta^{\prime}$ using that the image of $\eta^{\prime}$ is contained
in ${\mathcal{O}}_{E}^{\times}\subset E^{\times}$.

(ii) Extending $K^{\prime}$, we may assume that it is Galois over $K$. The
finite Galois group $\operatorname{{Gal}}(K^{\prime}/K)$ then acts on
$G_{K^{\prime}}$ by conjugation and, for an element
$\sigma\in\operatorname{{Gal}}(K^{\prime}/K)$ we denote by
$\eta^{\prime\sigma}$ the precomposition of the character $\eta^{\prime}$ with
the automorphism induced by $\sigma$. By our assumption, the Hodge-Tate
weights of $\eta^{\prime}$ and $\eta^{\prime\sigma}$ coincide for every
$\sigma$. Hence the $\tau$-Hodge-Tate weight of the character
$\varepsilon^{\prime}:=\prod\limits_{\sigma\in\operatorname{{Gal}}(K^{\prime}/K)}\eta^{\prime\sigma}$
is equal to $[K^{\prime}:K]$ times the $\tau$-weight of $\eta^{\prime}$, for
every $\tau$. The character $\varepsilon^{\prime}$ visibly extends to the
character
$\varepsilon:=\det\operatorname{{Ind}}^{G_{K^{\prime}}}_{G_{K}}\eta^{\prime}$
of $G_{K}$ so the desired $\eta$ can be obtained as the ”$[K^{\prime}:K]$th
root” of $\varepsilon$. ∎

Let us recall the generalized eigenspace decomposition of vector bundles with
respect to an endomorphism. Suppose that we are given a locally free finite
rank sheaf $M$ on a ringed space $(Z,{\mathcal{O}}_{Z})$ equipped with an
endomorphism $f:M\to M$ such that the characteristic polynomial
$\chi_{f}(T):=\det(T\cdot\mathrm{Id}_{M}-f)\in H^{0}(Z,{\mathcal{O}}_{Z})[T]$
has coefficients and all of its roots in a field $K\subset
H^{0}(Z,{\mathcal{O}}_{Z})$. If $\chi_{f}(T)$ decomposes as
$\prod(T-\lambda_{i})^{m_{i}}$ with $\lambda_{1},\dots\lambda_{r}$ being
pairwise distinct elements of $K$ then the algebra
$K[T]/\chi_{f}(T)\simeq\bigoplus\limits_{i}K[T]/(T-\lambda_{i})^{m_{i}}$ acts
on $M$ and induces a decomposition $M=\bigoplus\limits_{i}M_{\lambda_{i}}$
where each $M_{\lambda_{i}}:=\ker(f-\lambda_{i}\cdot\mathrm{Id}_{M})^{m_{i}}$
is a locally free sheaf on $(Z,{\mathcal{O}}_{Z})$.

###### Proof of Theorem 4.1.

Let $E$ be a finite extension of ${\mathbb{Q}}_{p}$ such that ${\mathbb{L}}$
is a scalar extension of a local system with coefficients in $E$, we will
denote this local system by the same symbol ${\mathbb{L}}$. By Lemma 4.2 it is
enough to prove that the projective representation of
$\pi_{1}^{\text{et}}(X,\overline{x})$ corresponding to ${\mathbb{L}}$ is
Hodge-Tate. That can be checked after the base change of $X$ and
${\mathbb{L}}$ to a finite extension of $K$, we may therefore assume that any
embedding $E\to\overline{K}$ factors through $K$, and all the eigenvalues of
the Sen endomorphism belong to $K$.

We have
$\operatorname{{End}}_{E}{\mathbb{L}}|_{X_{\overline{K}}}=E\cdot\mathrm{Id}$.
Viewing ${\mathbb{L}}$ as a ${\mathbb{Q}}_{p}$-local system, we can form a
Higgs bundle $\mathrm{H}^{\log}({\mathbb{L}})$ on $\overline{X}_{K_{\infty}}$
of rank $[E:{\mathbb{Q}}_{p}]\cdot\operatorname{{rk}}_{E}{\mathbb{L}}$
equipped with an action of $E$. This action commutes with the Sen endomorphism
$\phi$ of the bundle $\mathrm{H}^{\log}({\mathbb{L}})$. By Proposition 3.5
(iii) we have
$(\operatorname{{End}}_{\overline{X}_{K_{\infty}},\mathrm{Higgs}}\mathrm{H}^{\log}({\mathbb{L}}))\otimes_{K_{\infty}}\widehat{K}_{\infty}=(\operatorname{{End}}_{X_{\overline{K}}}{\mathbb{L}}\otimes_{{\mathbb{Q}}_{p}}C)^{H_{K}}=E\otimes_{{\mathbb{Q}}_{p}}C^{H_{K}}=E\otimes_{{\mathbb{Q}}_{p}}\widehat{K}_{\infty}$
where the second equality follows from the fact that all endomorphisms of
${\mathbb{L}}|_{X_{\overline{K}}}$ extend to endomorphisms of ${\mathbb{L}}$
over $K$, so the Galois action on the endomorphisms of
${\mathbb{L}}|_{X_{\overline{K}}}$ is trivial. Therefore, the map
$E\to\operatorname{{End}}_{\overline{X}_{K_{\infty}},\mathrm{Higgs}}\mathrm{H}^{\log}({\mathbb{L}})$
induces an isomorphism
$E\otimes_{{\mathbb{Q}}_{p}}K_{\infty}\simeq\operatorname{{End}}_{\overline{X}_{K_{\infty}},\mathrm{Higgs}}\mathrm{H}^{\log}({\mathbb{L}})$.

As we have discussed at the beginning of this section, we get a decomposition
$\mathrm{H}^{\log}({\mathbb{L}})=\bigoplus\limits_{\tau}\mathrm{H}^{\log}({\mathbb{L}})_{\tau}$.
By the description of endomorphisms, for each $\tau$ we have that all
endomorphisms of the Higgs bundle $\mathrm{H}^{\log}({\mathbb{L}})_{\tau}$ are
scalars from $K_{\infty}$. For a given $\tau$, suppose that
$\prod\limits_{i}(t-\lambda_{i})^{m_{i}}$ with $\lambda_{i}\in\overline{K}$ is
the minimal polynomial of $\phi_{\tau}$ on
$\mathrm{H}^{\log}({\mathbb{L}})_{\tau}$. Enlarging $K$, we may assume that
all $\lambda_{i}$s lie in $K$. Consider the generalized eigenspace
decomposition
$\mathrm{H}^{\log}({\mathbb{L}})_{\tau}=\bigoplus\limits_{i}\mathrm{H}^{\log}({\mathbb{L}})_{\tau,\lambda_{i}}$
with respect to $\phi_{\tau}$. Since $\theta\circ\phi=(\phi-1)\circ\theta$,
for a local section
$s\in\mathrm{H}^{\log}({\mathbb{L}})_{\tau,\lambda_{i}}(U)$ we have

$\phi_{\tau}(\theta(s))=\theta(\phi_{\tau}(s)+s)=\theta(\lambda_{i}s+s)=(\lambda_{i}+1)\cdot\theta(s)$

In other words, the Higgs field $\theta$ takes the summand
$\mathrm{H}^{\log}({\mathbb{L}})_{\tau,\lambda}$ to
$\mathrm{H}^{\log}({\mathbb{L}})_{\tau,\lambda+1}\otimes_{{\mathcal{O}}_{K_{\infty}}}\Omega^{1}_{X_{K_{\infty}}/K_{\infty}}$,
for every $\lambda$.

If all $\lambda_{i}$s do not belong to a single coset of ${\mathbb{Z}}$ in
$\overline{K}$ then $\mathrm{H}^{\log}({\mathbb{L}})_{\tau}$ is decomposable
into a direct sum of two non-zero Higgs bundles, which contradicts the fact
that all endomorphisms are scalars. It remains to show that $\phi_{\tau}$ is
semi-simple on $\mathrm{H}^{\log}({\mathbb{L}})_{\tau}$. Define an
endomorphism $f$ of
$\mathrm{H}^{\log}({\mathbb{L}})_{\tau}=\bigoplus\limits_{i}\mathrm{H}^{\log}({\mathbb{L}})_{\tau,\lambda_{i}}$
by $\phi_{\tau}-\lambda_{i}$ on the summand
$\mathrm{H}^{\log}({\mathbb{L}})_{\tau,\lambda_{i}}$. By construction, it
commutes with the Higgs field and is nilpotent. Therefore, it is equal to zero
so $\prod\limits_{i}(\phi_{\tau}-\lambda_{i})=0$ on
$\mathrm{H}^{\log}({\mathbb{L}})_{\tau}$. In other words, $\phi_{\tau}$ is
semi-simple. We infer that for each $\tau$ there is an element
$a_{\tau}\in\overline{K}$ such that $\phi_{\tau}$ acts on
$\mathrm{H}^{\log}({\mathbb{L}})_{\tau}$ semi-simply with eigenvalues from
$a_{\tau}+{\mathbb{Z}}$. This means the the projective representation
corresponding to ${\mathbb{L}}$ is Hodge-Tate. ∎

##  5\. A criterion of de Rhamness for local systems with integral weights

We stay in the same setup: $X$ is a smooth rigid-analytic variety over $K$
embedded into a smooth proper $\overline{X}$ such that $\overline{X}\setminus
X$ is a normal crossings divisor.

######  Proposition 5.1.

Suppose that ${\mathbb{L}}$ is a ${\mathbb{Q}}_{p}$-local system of rank $d$
on $X$ with Hodge-Tate weights in ${\mathbb{Z}}$. Then the sheaf

(5.1)
${\mathcal{A}}({\mathbb{L}}):=\mathrm{RH}^{\log}({\mathbb{L}})^{\phi^{d}=0}$

is a locally free sheaf of rank $d$ of
${\mathcal{O}}_{\overline{X}_{K_{\infty}}}$-modules on
$\overline{X}_{K_{\infty}}$ equipped with an integrable connection and
Griffiths transverse filtration, induced from those on
$\mathrm{RH}^{\log}({\mathbb{L}})$. Moreover, the Higgs bundle
$(\bigoplus\limits_{i\in{\mathbb{Z}}}\operatorname{{gr}}^{i}{\mathcal{A}}({\mathbb{L}}),\operatorname{{gr}}(\nabla))$
is isomorphic to $\mathrm{H}^{\log}({\mathbb{L}})$ (the isomorphism is not
compatible with the Sen endomorphisms).

###### Proof.

First, note that ${\mathcal{A}}({\mathbb{L}})$ is indeed a subsheaf of
${\mathcal{O}}_{\overline{X}_{K_{\infty}}}$-modules inside
$\mathrm{RH}^{\log}({\mathbb{L}})$ and it is preserved by the connection
$\nabla$. Suppose that the weights of ${\mathbb{L}}$ are contained in the set
$\\{i,i-1,\dots,i-n\\}$ for $i\in{\mathbb{Z}},n\in{\mathbb{N}}$. Then
${\mathcal{A}}({\mathbb{L}})$ is contained in
$F^{-i}\mathrm{RH}^{\log}({\mathbb{L}})$ because for $j<-i$ we have
$(\operatorname{{gr}}^{j}\mathrm{RH}^{\log}({\mathbb{L}}))^{\phi^{d}=0}=(\mathrm{H}^{\log}({\mathbb{L}})(j))^{\phi^{d}=0}=0$
as all eigenvalues of $\phi$ on $\mathrm{H}^{\log}({\mathbb{L}})(j)$ are in
the range $\\{i+j,i+j-1,\dots i+j-n\\}\subset{\mathbb{Z}}_{<0}$. Similarly,
the map ${\mathcal{A}}({\mathbb{L}})\to
F^{[-i,-i+n]}\mathrm{RH}^{\log}({\mathbb{L}}):=F^{-i}\mathrm{RH}^{\log}({\mathbb{L}})/F^{-i+n+1}\mathrm{RH}^{\log}({\mathbb{L}})$
is injective.

The subquotient $F^{[-i,-i+n]}\mathrm{RH}^{\log}({\mathbb{L}})$ is a locally
free sheaf on $\overline{X}_{K_{\infty}}$ (of rank $(n+1)\cdot d$) equipped
with an ${\mathcal{O}}_{\overline{X}_{K_{\infty}}}$-linear endomorphism $\phi$
and ${\mathcal{A}}({\mathbb{L}})$ is its generalized eigenbundle with
eigenvalue $0$. In particular, it is a direct summand of
$F^{[-i,-i+n]}\mathrm{RH}^{\log}({\mathbb{L}})$ and is therefore locally free.

The generalized eigenspace of eigenvalue $0$ on the graded pieces
$\operatorname{{gr}}^{j}\mathrm{RH}^{\log}({\mathbb{L}})$ for $j\in[-i,-i+n]$
is given by $\mathrm{H}^{\log}({\mathbb{L}})_{-j}$. Therefore we have an
isomorphism of vector bundles

(5.2)
$\operatorname{{gr}}{\mathcal{A}}({\mathbb{L}})\simeq\bigoplus\limits_{j=-i}^{-i+n}\mathrm{H}^{\log}({\mathbb{L}})_{-j}=\mathrm{H}^{\log}({\mathbb{L}})$

The Higgs field
$\operatorname{{gr}}(\nabla):\operatorname{{gr}}^{j}{\mathcal{A}}({\mathbb{L}})\to\operatorname{{gr}}^{j-1}{\mathcal{A}}({\mathbb{L}})\otimes\Omega^{1}_{X_{K_{\infty}}}$
is restricted from
$\operatorname{{gr}}^{j}\mathrm{RH}^{\log}({\mathbb{L}})\to\operatorname{{gr}}^{j-1}\mathrm{RH}^{\log}({\mathbb{L}})\otimes\Omega^{1}_{X_{K_{\infty}}}$
and is thus given by
$\theta_{\mathrm{H}^{\log}({\mathbb{L}})}:\mathrm{H}^{\log}({\mathbb{L}})_{-j}\to\mathrm{H}^{\log}({\mathbb{L}})_{-j+1}\otimes\Omega^{1}_{X_{K_{\infty}}}$
so (5.2) is compatible with Higgs fields. ∎

######  Theorem 5.2.

Suppose that ${\mathbb{L}}$ is a ${\mathbb{Q}}_{p}$-local system admitting a
${\mathbb{Z}}_{p}$-lattice on $X$ with Hodge-Tate weights in ${\mathbb{Z}}$
such that
$\operatorname{{End}}_{X_{\overline{K}}}({\mathbb{L}})\otimes_{{\mathbb{Q}}_{p}}C$
has no nilpotent elements. Then ${\mathbb{L}}$ is de Rham.

###### Proof.

We are going to use the criterion for de Rhamness given by Lemma 3.6(i) which
says that ${\mathbb{L}}$ is de Rham if the nilpotent endomorphism $\phi$ of
${\mathcal{A}}({\mathbb{L}})$ is in fact zero. Suppose that it is not zero.
Let $i$ be the maximal integer such that
$\phi(F^{m}{\mathcal{A}}({\mathbb{L}}))\subset
F^{m+i}{\mathcal{A}}({\mathbb{L}})$ for all $m\in{\mathbb{Z}}$. We will view
$\phi$ as a morphism
${\mathcal{A}}({\mathbb{L}})\to{\mathcal{A}}({\mathbb{L}})(i)$ of filtered
vector bundles with connection where the twist denotes the shift of the
filtration:
$F^{m}({\mathcal{A}}({\mathbb{L}})(i))=F^{m+i}{\mathcal{A}}({\mathbb{L}})$.
For at least one $m$ the induced map $\phi:F^{m}/F^{m+1}\to F^{m+i}/F^{m+i+1}$
is non-zero so $\phi$ induces a non-zero nilpotent endomorphism of the Higgs
bundle
$\mathrm{H}^{\log}({\mathbb{L}})=\bigoplus\limits_{m\in{\mathbb{Z}}}\operatorname{{gr}}^{m}{\mathcal{A}}({\mathbb{L}})$.

By Proposition 3.5(iii) the algebra
$\operatorname{{End}}_{\overline{X}_{K_{\infty}},\mathrm{Higgs}}\mathrm{H}^{\log}({\mathbb{L}})\otimes_{K_{\infty}}C$
has no non-zero nilpotents under the assumption of the theorem. The same is
then true for the algebra
$\operatorname{{End}}_{\overline{X}_{K_{\infty}},\mathrm{Higgs}}\mathrm{H}^{\log}({\mathbb{L}})$.
This gives us a contradiction. ∎

######  Corollary 5.3.

Let $X$ be a smooth algebraic variety over $K$ and ${\mathbb{L}}$ be an etale
$\overline{\mathbb{Q}}_{p}$-etale local system such that
$\operatorname{{End}}_{X_{\overline{K}}}{\mathbb{L}}=\overline{\mathbb{Q}}_{p}\cdot\mathrm{Id}$.
Then there exists a character
$\chi:G_{K}\to\overline{\mathbb{Q}}_{p}^{\times}$ such that
$\chi\otimes{\mathbb{L}}$ is de Rham.

###### Proof.

Using resolution of singularities, choose a smooth proper algebraic variety
$\overline{X}\supset X$ containing $X$ as a dense open such that
$\overline{X}\setminus X$ is a normal crossing divisor and apply the
discussion of sections 2-5 to the analytifications of $X$, $\overline{X}$ and
${\mathbb{L}}$. We are in a position to do so because ${\mathbb{L}}$, being a
$\overline{\mathbb{Q}}_{p}$-local system on an algebraic variety, admits a
$\overline{{\mathbb{Z}}}_{p}$-lattice. Theorem 4.1 applied to ${\mathbb{L}}$
provides the character $\chi$ and Theorem 5.2 applied to a descent of
${\mathbb{L}}\otimes\chi$ to a finite extension $E$ of ${\mathbb{Q}}_{p}$
viewed as a ${\mathbb{Q}}_{p}$-local system gives the result. ∎

##  6\. Assumptions in the relative Fontaine-Mazur conjecture

For the duration of this section let $F$ be a number field and
${\mathfrak{X}}$ be a smooth scheme over ${\mathcal{O}}_{F,S}$ where $S$ is a
finite set of places of $F$. Denote by $X$ the generic fiber
${\mathfrak{X}}_{F}$. We show here that the first condition of the relative
Fontaine-Mazur conjecture is automatic for geometrically irreducible local
systems. Together with Corollary 5.3 this completes the proof of Theorem 2. I
have learned about this fact from Daniel Litt. The proof, in the case of
curves, appears in Step 2 of the proof of [Lit21, Theorem 1.1.3]. We include
here (an essentially equivalent) proof for the sake of completeness. A very
similar argument also appears in [LZ17, Proposition 4.1].

######  Proposition 6.1 (D. Litt).

For any $\overline{\mathbb{Q}}_{p}$-local system ${\mathbb{L}}$ on $X$ such
that
$\operatorname{{End}}_{X_{\overline{F}}}{\mathbb{L}}|_{X_{\overline{F}}}=\overline{\mathbb{Q}}_{p}\cdot\mathrm{Id}$
there exists a finite set $S^{\prime}\supset S$ such that ${\mathbb{L}}$
extends to a $\overline{\mathbb{Q}}_{p}$-local system on
${\mathfrak{X}}_{{\mathcal{O}}_{F,S^{\prime}}}$.

###### Proof.

Any $\overline{\mathbb{Q}}_{p}$-local system descends to an $E$-local system
for some finite extension $E$ of ${\mathbb{Q}}_{p}$ and we can choose an
${\mathcal{O}}_{E}$-lattice in this descent. We may therefore work with an
${\mathcal{O}}_{E}$-local system ${\mathbb{L}}$ such that
$\operatorname{{End}}_{X_{\overline{F}}}{\mathbb{L}}|_{X_{\overline{F}}}[\frac{1}{p}]=E\cdot\mathrm{Id}$.
Note that it is enough to prove the claim for $X$ and $F$ replaced with
$X_{F^{\prime}}$ and $F^{\prime}$ for some finite field extension
$F^{\prime}\supset F$. Hence, we may assume that $X$ admits an $F$-point
$x:\operatorname{{Spec}}F\to X$ and we will pick a geometric point
$\overline{x}$ above $x$. Denote the representation corresponding to
${\mathbb{L}}$ by $\rho:\pi_{1}^{\text{et}}(X,\overline{x})\to
GL_{d}({\mathcal{O}}_{E})$. Restriction $x^{*}{\mathbb{L}}$ can be viewed as a
Galois representation $\rho_{x}:G_{F}\to GL_{d}({\mathcal{O}}_{E})$. Replace
again $F$ by a finite extension to make the residual representation
$\overline{\rho}_{x}:G_{F}\to GL_{d}({\mathcal{O}}_{E}/p^{2})$ trivial. Also,
pick a smooth proper compactification $\overline{X}\supset X$ such that
$D=\overline{X}\setminus X$ is a normal crossings divisor. After enlarging $S$
we may assume that $\overline{X}$ has a smooth proper model
$\overline{{\mathfrak{X}}}$ over ${\mathcal{O}}_{F,S}$ such that $D$ extends
to a horizontal normal crossings divisor
${\mathfrak{D}}\subset\overline{{\mathfrak{X}}}$ with
${\mathfrak{X}}=\overline{{\mathfrak{X}}}\setminus{\mathfrak{D}}$ and,
moreover, $x$ extends to a point in ${\mathfrak{X}}({\mathcal{O}}_{F,S})$.

Let now $S^{\prime}$ be the union of $S$ with the set of places of $F$ such
that (i) residue characteristic of $v$ divides the order of the finite group
$GL_{d}({\mathcal{O}}_{E}/p^{2})$ or (ii) the character $\det\rho_{x}$ of the
Galois group $G_{F}$ is ramified at $v$. The set of places satisfying the
second condition is finite by [Ser68, Corollary III-2.2] so $S^{\prime}$ is
finite. We will now prove that the local system ${\mathbb{L}}$ extends to a
local system on ${\mathfrak{X}}_{{\mathcal{O}}_{F,S^{\prime}}}$.

First, we will show that the Galois representation $\rho_{x}$ is unramified at
all places $v$ outside of $S^{\prime}$. Let $v$ be any such place and $l$ be
its residue characteristics. The prime $l$ does not divide the order of any of
the groups $GL_{d}({\mathcal{O}}_{E}/p^{n})$ for $n\in{\mathbb{N}}$ by
assumption so the representation $\rho:\pi_{1}^{\text{et}}(X,\overline{x})\to
GL_{d}({\mathcal{O}}_{E})$ factors through the prime-to-$l$ quotient of
$\pi_{1}^{\text{et}}(X,\overline{x})$ which we will denote by
$\pi_{1}^{\text{et}}(X,\overline{x})^{(l^{\prime})}$. The action of $G_{F}$ on
$\pi_{1}(X_{\overline{F}},\overline{x})^{(l^{\prime})}$ is unramified at $v$
by [LO10, Corollary A.12], thanks to the fact that we assumed the existence of
good compactification ${\mathfrak{X}}$ for ${\mathfrak{X}}$. In other words,
the image of the inertia subgroup $I_{v}\subset G_{F_{v}}$ inside
$\pi_{1}^{\text{et}}(X,\overline{x})^{(l^{\prime})}$ commutes with the image
of
$\pi_{1}^{\text{et}}(X_{\overline{F}},\overline{x})^{(l^{\prime})}\to\pi_{1}^{\text{et}}(X,\overline{x})^{(l^{\prime})}$.
It means that $\rho_{x}(I_{F_{v}})$ commutes with every element of
$\rho(\pi_{1}^{\text{et}}(X_{\overline{F}},\overline{x}))$. By our assumption
$\operatorname{{End}}_{X_{\overline{F}}}{\mathbb{L}}|_{X_{\overline{F}}}[\frac{1}{p}]=E\cdot\mathrm{Id}$
every element of $GL_{d}({\mathcal{O}}_{E})$ commuting with
$\rho(\pi_{1}^{\text{et}}(X_{\overline{F}},\overline{x}))$ has to be a scalar
multiple of identity so for every $g\in I_{F_{v}}$ we have
$\rho_{x}(g)=\lambda\cdot\mathrm{Id}$ for some
$\lambda\in{\mathcal{O}}_{E}^{\times}$. By our preparatory assumption
$\lambda\equiv 1$ modulo $p^{2}$ and by the definition of $S^{\prime}$ the
power $\lambda^{d}$ is equal to $1$. This implies that $\lambda$ is in fact
equal to $1$ so $\rho_{x}(I_{F_{v}})=\\{\mathrm{Id}\\}$ and $\rho_{x}$ is
unramified at all places outside of $S^{\prime}$.

This now forces ${\mathbb{L}}$ to extend to
${\mathfrak{X}}_{{\mathcal{O}}_{F,S^{\prime}}}$: let $F_{n}$ be the finite
extension of $F$ trivializing $x^{*}({\mathbb{L}}/p^{n})$, by the above it is
unramified at all $v\not\in S^{\prime}$ and to show that ${\mathbb{L}}/p^{n}$
extends from $X$ to ${\mathfrak{X}}_{{\mathcal{O}}_{F,S^{\prime}}}$ it is
enough to do so after the base change to ${\mathcal{O}}_{F_{n},S^{\prime}}$.
This extension exists by [ST, Lemma 57.31.7]. ∎

Despite the fact that it is not clear whether for a geometrically irreducible
${\mathbb{L}}$ on $X$ there always exists a character
$\chi:G_{F}\to\overline{\mathbb{Q}}_{p}^{\times}$ of the global Galois group
such that ${\mathbb{L}}\otimes\chi$ is de Rham at all places above $p$, Moritz
Kerz observed that one can still conclude from the relative Fontaine-Mazur
conjecture that the underlying geometric local system comes from geometry.

######  Lemma 6.2 (M. Kerz).

Assume that Conjecture 1 holds for local systems on the variety $X\times X$.
Then for any $\overline{\mathbb{Q}}_{p}$-local system ${\mathbb{L}}$ on $X$
such that ${\mathbb{L}}|_{X_{\overline{F}}}$ is irreducible the local system
${\mathbb{L}}|_{X_{\overline{F}}}$ comes from geometry.

###### Proof.

Denote by $p_{1},p_{2}:X\times X\to X$ the two projections and consider the
local system
${\mathbb{L}}\boxtimes{\mathbb{L}}^{\vee}:=p_{1}^{*}{\mathbb{L}}\otimes
p_{2}^{*}{\mathbb{L}}^{\vee}$. By Corollary 5.3 the local system
$({\mathbb{L}}\boxtimes{\mathbb{L}}^{\vee})|_{X_{F_{v}}}$ is de Rham for every
place $v$ above $p$ and ${\mathbb{L}}\boxtimes{\mathbb{L}}^{\vee}$ extends to
a local system on
$({\mathfrak{X}}\times{\mathfrak{X}})_{{\mathcal{O}}_{F,S^{\prime}}}$ for some
finite set of places $S^{\prime}$ by Proposition 6.1. The assumptions of
Conjecture 1 are therefore satisfied for
${\mathbb{L}}\boxtimes{\mathbb{L}}^{\vee}$ so there exists an open $U\subset
X$ such that $({\mathbb{L}}\boxtimes{\mathbb{L}}^{\vee})|_{U}$ appears as a
subquotient of the sheaf $R^{i}\pi_{*}\overline{\mathbb{Q}}_{p}(j)$ for some
morphism $\pi:Y\to U$.

Choose a point $x\in X(\overline{F})$ such that the intersection of $U$ with
$\\{x\\}\times X$ is non-empty. Then the restriction
$i_{x}^{*}({\mathbb{L}}\boxtimes{\mathbb{L}}^{\vee})$ of the local system
$({\mathbb{L}}\boxtimes{\mathbb{L}}^{\vee})_{(X\times X)_{\overline{F}}}$
along the inclusion $i_{x}:X_{\overline{F}}=\\{x\\}\times
X_{\overline{F}}\to(X\times X)_{\overline{F}}$ comes from geometry. However,
this restriction is isomorphic to
${\mathbb{L}}|_{X_{\overline{F}}}\otimes{\mathbb{L}}^{\vee}_{x}\simeq{\mathbb{L}}_{X_{\overline{F}}}^{\operatorname{{rk}}{\mathbb{L}}}$
so the local system ${\mathbb{L}}|_{X_{\overline{F}}}$ on $X_{\overline{F}}$
comes from geometry. ∎

##  7\. Rigidity for Hodge-Tate local systems on algebraic varieties

For this section we go back into the setup of Section 2: $X$ is a smooth
geometrically connected rigid-analytic variety over $K$ with a smooth proper
compactification $\overline{X}\supset X$ such that $\overline{X}\setminus X$
is a normal crossings divisor, the main example being provided by the
analytification of a smooth algebraic variety.

######  Lemma 7.1.

If ${\mathbb{L}}$ is a ${\mathbb{Q}}_{p}$-local system on $X$ that admits a
${\mathbb{Z}}_{p}$-lattice then any endomorphism
$f:\mathrm{H}^{\log}({\mathbb{L}})\to\mathrm{H}^{\log}({\mathbb{L}})$
commuting with the Higgs field that vanishes at one closed point $x\in
X_{K_{\infty}}$ has to be identically zero.

###### Proof.

By Theorem 2.4 (v) and Proposition 3.5 (ii) we have a commutative diagram

${\operatorname{{End}}_{\overline{X}_{K_{\infty}},\mathrm{Higgs}}\mathrm{H}^{\log}({\mathbb{L}})\otimes_{K_{\infty}}C}$${\operatorname{{End}}_{X_{\overline{K}}}{\mathbb{L}}|_{X_{\overline{K}}}\otimes_{{\mathbb{Q}}_{p}}C}$${\operatorname{{End}}_{K_{\infty}}\mathrm{H}^{\log}({\mathbb{L}})_{x}\otimes_{K_{\infty}}C}$${\operatorname{{End}}_{{\mathbb{Z}}_{p}}{\mathbb{L}}_{\overline{x}}\otimes_{{\mathbb{Q}}_{p}}C}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

As ${\mathbb{L}}\otimes_{{\mathbb{Z}}_{p}}{\mathbb{L}}^{\vee}$ is a local
system the right vertical arrow is injective, so the left one is injective as
well.

∎

######  Proposition 7.2.

For any ${\mathbb{Q}}_{p}$-local system on $X$ that admits a
${\mathbb{Z}}_{p}$-lattice the associated coherent Higgs sheaf
$D_{\mathrm{HT}}({\mathbb{L}})$ is locally free.

###### Proof.

By [Shi18, Proposition 5.3] this statement amounts to the fact that
$\bigoplus\limits_{i\in{\mathbb{Z}}}\mathrm{H}^{\log}({\mathbb{L}})^{\phi=i}|_{X_{K_{\infty}}}$
is locally free. Assume, by enlarging $K$, that all Hodge-Tate weights of
${\mathbb{L}}$ belong to $K$. Let
$\mathrm{H}^{\log}({\mathbb{L}})=\bigoplus\mathrm{H}^{\log}({\mathbb{L}})_{\lambda}$
be the generalized eigenbundle decomposition associated to the endomorphism
$\phi$. Denote by $f$ the endomorphism of $\mathrm{H}^{\log}({\mathbb{L}})$
defined by $\phi-\lambda$ on the component
$\mathrm{H}^{\log}({\mathbb{L}})_{\lambda}$. It commutes with the Higgs field
because of the identity $\theta\circ\phi=(\phi-1)\circ\theta$. To prove that
$\bigoplus\limits_{i\in{\mathbb{Z}}}\mathrm{H}({\mathbb{L}})^{\phi=i}|_{X_{K_{\infty}}}$
is locally free, it is enough to show that the rank of $f$ is constant across
$X_{K_{\infty}}$. Equivalently, we need to prove that, if for some
$m\in{\mathbb{N}}$ the exterior power
$\Lambda^{m}f\in\operatorname{{End}}_{\overline{X}_{K_{\infty}},\mathrm{Higgs}}(\Lambda^{m}\mathrm{H}^{\log}({\mathbb{L}}))$
vanishes at a point $x\in X_{K_{\infty}}$, then $\Lambda^{m}f$ is identically
zero. This follows from Lemma 7.1. ∎

######  Corollary 7.3.

Suppose that ${\mathbb{L}}$ is a ${\mathbb{Q}}_{p}$-local system on $X$
admiting a ${\mathbb{Z}}_{p}$-lattice. If for one classical point $x\in X$ the
restriction ${\mathbb{L}}_{\overline{x}}$ is a Hodge-Tate representation of
$\operatorname{{Gal}}(\overline{k(x)}/k(x))$ then ${\mathbb{L}}$ is Hodge-
Tate. In particular, for any other closed point $y\in X$ the Galois
representation ${\mathbb{L}}_{\overline{y}}$ is Hodge-Tate as well.

###### Proof.

By the assumption, all Hodge-Tate weights of ${\mathbb{L}}$ belong to
${\mathbb{Z}}$. The proof of Proposition 7.2 shows that for every
$i\in{\mathbb{Z}}$ the rank of $\phi-i$ is constant across $X$. Since the
restriction of $\phi-i$ to $x$ annihilates all of the generalized eigenspace
corresponding to the weight $i$, the operator $\phi-i$ does the same globally.
Hence $\phi$ is semi-simple with integral eigenvalues. ∎

##  8\. Reducible local systems and Galois action on the fundamental group

In this section, we extend Theorem 1 to local systems that are not necessarily
geometrically irreducible by proving the following result. I learned about
this generalization and its interpretation in terms of the Galois action on
$\pi_{1}^{\text{et}}(X_{\overline{K}},\overline{x})$ from Alexander Beilinson.
As before, $X$ is a smooth rigid-analytic variety that admits a smooth proper
compactification $\overline{X}\supset X$ over a complete discretely valued
field $K$ of characteristic zero with a perfect residue field of
characteristic $p$.

######  Theorem 8.1.

For any $\overline{\mathbb{Q}}_{p}$-local system ${\mathbb{L}}$ on $X$ there
exists a de Rham local system ${\mathbb{M}}$ on $X$ such that there is an
inclusion
${\mathbb{L}}|_{X_{\overline{K}}}\subset{\mathbb{M}}|_{X_{\overline{K}}}$ of
local systems on $X_{\overline{K}}$.

###### Proof.

Note that if the theorem is proven for the base change of $X$ to a finite
extension $K^{\prime}\supset K$ then it follows for the initial $X$ as well by
considering the pushforward of ${\mathbb{M}}$ along $X_{K^{\prime}}\to X$. We
may therefore assume that $X$ has a $K$-point $x$. Pick also a geometric point
$\overline{x}$ above $x$. We will view the stalk ${\mathbb{L}}_{\overline{x}}$
as a representation of $G_{K}$.

The local system ${\mathbb{M}}$ can now be defined constructively. Denote by
${\mathcal{F}}({\mathbb{L}})$ the minimal sub-local system of
${\mathbb{L}}^{\vee}\otimes_{\overline{\mathbb{Q}}_{p}}{\mathbb{L}}_{\overline{x}}$
whose stalk ${\mathcal{F}}({\mathbb{L}})_{\overline{x}}$ contains the identity
endomorphism
$\operatorname{{id}}_{{\mathbb{L}}_{\overline{x}}}\in{\mathbb{L}}^{\vee}_{\overline{x}}\otimes{\mathbb{L}}_{\overline{x}}=\operatorname{{End}}_{\overline{\mathbb{Q}}_{p}}({\mathbb{L}}_{\overline{x}})$.
In terms of representations of $\pi_{1}^{\text{et}}(X,\overline{x})$, the
stalk ${\mathcal{F}}({\mathbb{L}})_{\overline{x}}$ is the
$\overline{\mathbb{Q}}_{p}$-span of the image of the geometric fundamental
group $\pi_{1}^{\text{et}}(X_{\overline{K}},\overline{x})$ inside
$\operatorname{{End}}_{\overline{\mathbb{Q}}_{p}}({\mathbb{L}}_{\overline{x}})$.
Note that ${\mathcal{F}}({\mathbb{L}})|_{X_{\overline{K}}}$ is therefore
minimal among local systems on $X_{\overline{K}}$ contained in
$({\mathbb{L}}^{\vee}\otimes{\mathbb{L}}_{\overline{x}})|_{X_{\overline{K}}}$
and containing $\operatorname{{id}}_{{\mathbb{L}}_{\overline{x}}}$ in their
stalk at $\overline{x}$; this observation will play the main role in the proof
of Proposition 8.2 below.

Now define
${\mathbb{M}}=({\mathcal{F}}({\mathbb{L}})^{\vee})^{\oplus\operatorname{{rk}}{\mathbb{L}}}$.
By construction, ${\mathcal{F}}({\mathbb{L}})\otimes{\mathbb{L}}_{x}^{\vee}$
surjects onto ${\mathbb{L}}^{\vee}$, therefore
${\mathbb{L}}|_{X_{\overline{K}}}$ injects into
$({\mathcal{F}}({\mathbb{L}})^{\vee}\otimes{\mathbb{L}}_{x})|_{X_{\overline{K}}}={\mathbb{M}}|_{X_{\overline{K}}}$.
The proof of the theorem now amounts to the following proposition. ∎

######  Proposition 8.2.

For any $\overline{\mathbb{Q}}_{p}$-local system ${\mathbb{L}}$, the local
system ${\mathcal{F}}({\mathbb{L}})$ defined above is de Rham.

###### Proof.

Note that the subspace
${\mathcal{F}}({\mathbb{L}})_{\overline{x}}\subset\operatorname{{End}}_{\overline{\mathbb{Q}}_{p}}({\mathbb{L}}_{\overline{x}})$
only depends on ${\mathbb{L}}|_{X_{\overline{K}}}$ so, in particular, for a
finite extension $K^{\prime}\supset K$ we have
${\mathcal{F}}({\mathbb{L}}|_{X_{K^{\prime}}})={\mathcal{F}}({\mathbb{L}})|_{X_{K^{\prime}}}$.
We may thus freely replace $K$ by a finite extension. If ${\mathbb{K}}$ is a
local system of $E$-vector spaces on $X$ for a finite extension
$E\supset{\mathbb{Q}}_{p}$ we will denote by ${\mathcal{F}}({\mathbb{K}})$ the
sub local system of
${\mathbb{K}}^{\vee}\otimes_{E}{\mathbb{K}}_{\overline{x}}$ defined just as
above in the case of $\overline{\mathbb{Q}}_{p}$-local systems. If
${\mathbb{K}}_{{\mathbb{Q}}_{p}}$ denotes the same ${\mathbb{K}}$, considered
as a ${\mathbb{Q}}_{p}$-local system of rank
$\operatorname{{rk}}_{E}{\mathbb{K}}\cdot[E:{\mathbb{Q}}_{p}]$, then
${\mathcal{F}}({\mathbb{K}})$ can be recovered from
${\mathbb{K}}_{{\mathbb{Q}}_{p}}$ by the formula
${\mathcal{F}}({\mathbb{K}})={\mathcal{F}}({\mathbb{K}}_{{\mathbb{Q}}_{p}})\otimes_{E\otimes_{{\mathbb{Q}}_{p}}E,m}E$.
To prove the proposition, we may therefore work with ${\mathbb{L}}$ a
${\mathbb{Q}}_{p}$-local system to begin with.

As in the proof of Theorem 4.1, let us start by translating the defining
property of ${\mathcal{F}}({\mathbb{L}})$ into a statement about the
corresponding Higgs bundle $\mathrm{H}^{\log}({\mathcal{F}}({\mathbb{L}}))$.
By definition, ${\mathcal{F}}({\mathbb{L}})|_{X_{\overline{K}}}$ has no proper
sub local systems
${\mathbb{M}}^{\prime}\subset{\mathcal{F}}({\mathbb{L}})|_{X_{\overline{K}}}$
whose stalk ${{\mathbb{M}}}^{\prime}_{\overline{x}}$ contains
$\operatorname{{id}}_{{\mathbb{L}}_{\overline{x}}}\in{\mathcal{F}}({\mathbb{L}})_{\overline{x}}$.
In particular, any endomorphism
$f\in\operatorname{{End}}_{X_{\overline{K}}}({\mathcal{F}}({\mathbb{L}})|_{X_{\overline{K}}})$
satisfying
$f_{\overline{x}}(\operatorname{{id}}_{{\mathbb{L}}_{\overline{x}}})=0$ has to
be identically zero, as otherwise $\ker f$ would give an example of such
${\mathbb{M}}^{\prime}$. Under the comparison isomorphism
${\mathcal{F}}({\mathbb{L}})_{\overline{x}}\otimes_{{\mathbb{Q}}_{p}}C\simeq\mathrm{H}^{\log}({\mathcal{F}}({\mathbb{L}}))_{x}\otimes_{K_{\infty}}C$
the element $\operatorname{{id}}_{{\mathbb{L}}_{\overline{x}}}\otimes 1$ of
the left hand side gets carried to the element
$\operatorname{{id}}_{\mathrm{H}^{\log}({\mathbb{L}})_{x}}\otimes
1\in\mathrm{H}^{\log}({\mathcal{F}}({\mathbb{L}}))_{x}\otimes_{K_{\infty}}C\subset\operatorname{{End}}_{K_{\infty}}(\mathrm{H}^{\log}({\mathbb{L}})_{x})\otimes_{K_{\infty}}C$
because
$\operatorname{{id}}_{{\mathbb{L}}_{\overline{x}}}\in\operatorname{{End}}({\mathbb{L}}_{\overline{x}})$
is stable under the Galois action. By comparison isomorphism of Proposition
3.5 (iii), this implies that any endomorphism
$f\in\operatorname{{End}}_{\overline{X}_{K_{\infty}},\mathrm{Higgs}}\mathrm{H}^{\log}({\mathcal{F}}({\mathbb{L}}))$
satisfying $f(\operatorname{{id}}_{\mathrm{H}^{\log}({\mathbb{L}})_{x}})=0$
has to be zero.

We can now conclude that ${\mathcal{F}}({\mathbb{L}})$ has integral Hodge-Tate
weights. The direct sum
$\bigoplus\limits_{i\in{\mathbb{Z}}}\mathrm{H}^{\log}({\mathcal{F}}({\mathbb{L}}))_{i}\subset\mathrm{H}^{\log}({\mathcal{F}}({\mathbb{L}}))$
of generalized eigenbundles of $\phi$ with integral eigenvalues is a direct
summand of the Higgs bundle $\mathrm{H}^{\log}({\mathcal{F}}({\mathbb{L}}))$
or, in other words, there is an idempotent
$e\in\operatorname{{End}}_{X_{K_{\infty}},\mathrm{Higgs}}\mathrm{H}^{\log}({\mathcal{F}}({\mathbb{L}}))$
such that
$\ker(e)=\bigoplus\limits_{i\in{\mathbb{Z}}}\mathrm{H}^{\log}({\mathcal{F}}({\mathbb{L}}))_{i}$.
Since
$\operatorname{{id}}_{\mathrm{H}^{\log}({\mathbb{L}})_{x}}\in\operatorname{{End}}(\mathrm{H}^{\log}({\mathbb{L}})_{x})=\operatorname{{End}}(\mathrm{H}({\mathbb{L}}_{\overline{x}}))$
spans the result of applying $\mathrm{H}$ to the trivial Galois
representation,
$\phi(\operatorname{{id}}_{\mathrm{H}^{\log}({\mathbb{L}})_{x}})=0$ and hence
$e(\operatorname{{id}}_{\mathrm{H}^{\log}({\mathbb{L}})_{x}})=0$. This forces
$e$ to be zero, so every eigenvalue of $\phi$ on
$\mathrm{H}^{\log}({\mathcal{F}}({\mathbb{L}}))$ is an integer.

We can now finish the proof of the proposition analogously to the proof of
Theorem 5.2. Consider the sheaf
${\mathcal{A}}({\mathcal{F}}({\mathbb{L}}))=\mathrm{RH}^{\log}({\mathcal{F}}({\mathbb{L}}))^{\phi^{\operatorname{{rk}}({\mathcal{F}}({\mathbb{L}}))}=0}$
as in Proposition 5.1. If ${\mathcal{F}}({\mathbb{L}})$ is not de Rham, then
the Sen endomorphism on ${\mathcal{A}}({\mathcal{F}}({\mathbb{L}}))$ induces,
as described in the proof of Theorem 5.2, a non-zero nilpotent endomorphism of
the Higgs bundle $\mathrm{H}^{\log}({\mathcal{F}}({\mathbb{L}}))$. This
endomorphism, however, annihilates the element
$\operatorname{{id}}_{\mathrm{H}^{\log}({\mathbb{L}})_{x}}\in\mathrm{H}^{\log}({\mathcal{F}}({\mathbb{L}}))_{x}$,
so it has to be zero by the discussion above. Hence the local system
${\mathcal{F}}({\mathbb{L}})$ is de Rham. ∎

It is unclear if it is always possible to choose ${\mathbb{M}}$ such that
${\mathbb{L}}|_{X_{\overline{K}}}\simeq{\mathbb{M}}|_{X_{\overline{K}}}$ (note
that this is the case for a geometrically irreducible ${\mathbb{L}}$ by
Theorem 1). In particular, I do not know the answer to the following question

######  Question 8.3.

If ${\mathbb{L}}_{1},{\mathbb{L}}_{2}$ are de Rham local systems on $X$ and
$0\to{\mathbb{L}}_{1}\to{\mathbb{L}}\to{\mathbb{L}}_{2}\to 0$ is an extension
of local systems on $X$, does there exists an extension
$0\to{\mathbb{L}}_{1}\to{\mathbb{L}}^{\prime}\to{\mathbb{L}}_{2}\to 0$ such
that ${\mathbb{L}}^{\prime}$ is de Rham and there is an isomorphism
${\mathbb{L}}|_{X_{\overline{K}}}\simeq{\mathbb{L}}^{\prime}|_{X_{\overline{K}}}$
of extensions of ${\mathbb{L}}_{2}|_{X_{\overline{K}}}$ by
${\mathbb{L}}_{1}|_{X_{\overline{K}}}$?

From now on we assume that $x$ is either a $K$-point of $X$ or a tangential
base point of $X$ in the sense of [Del89, §15] and $\overline{x}$ is a
geometric base point above $x$. We can reformulate the above result in terms
of the Galois action on the pro-algebraic completion of the fundamental group
$\pi_{1}^{\text{et}}(X_{\overline{K}},\overline{x})$. We will now recall all
the necessary information about pro-algebraic completions of pro-finite groups
and refer the reader to Section 3.1 of [Dri18] for a more thorough
introduction.

######  Definition 8.4.

For a compact Hausdorff topological group $\Gamma$ the pro-algebraic
completion $\Gamma^{\mathrm{pro}-\mathrm{alg}}$ is the pro-algebraic group
over $\overline{\mathbb{Q}}_{p}$, equipped with a continuous map
$\alpha:\Gamma\to\Gamma^{\mathrm{pro}-\mathrm{alg}}(\overline{\mathbb{Q}}_{p})$
defined by the following universal property. For any continous homomorphism
$\rho:\Gamma\to GL_{n}(\overline{\mathbb{Q}}_{p})$ there exists a unique
algebraic morphism $\rho^{\mathrm{alg}}:\Gamma^{\mathrm{pro}-\mathrm{alg}}\to
GL_{n,\overline{\mathbb{Q}}_{p}}$ such that
$\alpha\circ\rho^{\mathrm{alg}}=\rho$.

Here, for a pro-algebraic group $G$ expressed as a limit $\lim\limits_{i\in
I}G_{i}$ of algebraic groups $G_{i}$ we endow
$G(\overline{\mathbb{Q}}_{p})=\lim\limits_{i\in
I}G_{i}(\overline{\mathbb{Q}}_{p})$ with a structure of a topological group
given by the limit of the topological groups
$G_{i}(\overline{\mathbb{Q}}_{p})$, each carrying the standard $p$-adic
topology. Note that, as $\Gamma$ is assumed to be compact and Hausdorff, any
continuous homomorphism $\Gamma\to GL_{n}(\overline{\mathbb{Q}}_{p})$ factors
through $GL_{n}(E)$ for some finite extension of $E$, by a Baire category
theorem argument, cf. [BM02, Lemma 2.2.1.1].

We will denote by
$\overline{\mathbb{Q}}_{p}[\Gamma^{\mathrm{pro}-\mathrm{alg}}]$ the ring of
regular functions on $\Gamma^{\mathrm{pro}-\mathrm{alg}}$. Our main objects of
interest are pro-algebraic compeltions of the profinite groups
$\pi_{1}^{\text{et}}(X_{\overline{K}},\overline{x})$ and
$\pi_{1}^{\text{et}}(X,\overline{x})$, which we denote by
$\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X_{\overline{K}},\overline{x})$ and
$\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X,\overline{x})$, respectively. The
continuous action of $G_{K}$ on
$\pi_{1}^{\text{et}}(X_{\overline{K}},\overline{x})$ induces an action on
$\overline{\mathbb{Q}}_{p}[\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X_{\overline{K}},\overline{x})]$.

######  Proposition 8.5.

Any finite-dimensional $\overline{\mathbb{Q}}_{p}$-representation $V$ of
$G_{K}$ that can be embedded into
$\overline{\mathbb{Q}}_{p}[\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X_{\overline{K}},\overline{x})]$
is de Rham.

###### Proof.

The span of the $G_{K}$-orbit of a function
$f\in\overline{\mathbb{Q}}_{p}[\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X_{\overline{K}},\overline{x})]$
is finite-dimensional if and only if $f$ extends to a regular function on the
pro-algebraic completion of the arithmetic fundamental group
$\pi_{1}^{\text{et}}(X,\overline{x})$. Therefore $V$ is contained in the image
of the restriction map
$\overline{\mathbb{Q}}_{p}[\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X,\overline{x})]\to\overline{\mathbb{Q}}_{p}[\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X_{\overline{K}},\overline{x})]$.

By Theorem 8.1, for any continuous homomorphism
$\rho:\pi_{1}^{\text{et}}(X,\overline{x})\to
GL_{n}(\overline{\mathbb{Q}}_{p})$ there exists another one
$\rho^{\prime}:\pi_{1}^{\text{et}}(X,\overline{x})\to
GL_{n+m}(\overline{\mathbb{Q}}_{p})$ such that the image of the restriction
map
$\rho^{*}:\overline{\mathbb{Q}}_{p}[GL_{n,\overline{\mathbb{Q}}_{p}}]\to\overline{\mathbb{Q}}_{p}[\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X_{\overline{K},\overline{x}})]$
is contained in the image of the restriction map $\rho^{\prime*}$, and
$\rho^{\prime}$ corresponds to a de Rham local system.

Denote by $W$ the Galois representation obtained by restricting
$\rho^{\prime}$ along
$x_{*}:G_{K}=\pi_{1}^{\text{et}}(\operatorname{{Spec}}K)\to\pi_{1}^{\text{et}}(X,\overline{x})$.
If $x\in X(K)$ is a classical base point then $W$ is evidently de Rham. If $x$
is a tangential base point, the fact that $W$ is de Rham was proven in
[DLLZ19, Corollary 4.3.4] as a consequence of their results on the
compatibility of the functor $D_{\mathrm{dR}}$ with the functor of nearby
cycles. The image
$\rho^{\prime*}(\overline{\mathbb{Q}}_{p}[GL_{n+m,\overline{\mathbb{Q}}_{p}}])\subset\overline{\mathbb{Q}}_{p}[\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X_{\overline{K}},\overline{x})]$
is a direct sum of Galois representations of the form $W^{\otimes a}\otimes
W^{\vee\otimes b}\otimes(\det W)^{c}$ for varying $a,b\in{\mathbb{Z}}_{\geq
0},c\in{\mathbb{Z}}$. Since $W$ is de Rham, the image of $\rho^{\prime*}$ is
thus a union of de Rham representations. We have therefore proven that the
image of
$\overline{\mathbb{Q}}_{p}[\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X,\overline{x})]\to\overline{\mathbb{Q}}_{p}[\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X_{\overline{K}},\overline{x})]$
is a union of finite dimensional de Rham representations, because any function
on the pro-algebraic completion
$\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X,\overline{x})$ factors through some
morphism of pro-algebraic groups
$\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X,\overline{x})\to
GL_{n,\overline{\mathbb{Q}}_{p}}$. ∎

######  Corollary 8.6.

Let $X$ be a smooth variety over a number field $F$, equipped with a base
point $x$ that is either a classical point $x\in X(F)$ or a tangential base
point. Any finite-dimensional representation of $G_{F}$ contained in
$\overline{\mathbb{Q}}_{p}[\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X_{\overline{F}},\overline{x})]$
is geometric in the sense of Fontaine-Mazur: it is de Rham at all places $v$
of $F$ above $p$ and it is almost everywhere unramified.

###### Proof.

Let
$V\subset\overline{\mathbb{Q}}_{p}[\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X_{\overline{F}},\overline{x})]$
be a finite-dimensional subspace stable under the action of $G_{F}$. The de
Rhamness has just been proven, so it remains to prove that $V$ is an almost
unramified representation. By definition, there exists a map of pro-algebraic
groups
$f:\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X_{\overline{F}},\overline{x})\to
GL_{n,\overline{\mathbb{Q}}_{p}}$ such that every function from $V$ factors
through it. Since the image of
$\pi_{1}^{\text{et}}(X_{\overline{F}},\overline{x})$ under $f$ is contained in
$GL_{n}({\mathcal{O}}_{E})\subset GL_{n}(\overline{\mathbb{Q}}_{p})$ for some
finite extension $E\supset{\mathbb{Q}}_{p}$, there exists a finite set $S$ of
rational primes such the restriction of $f$ to
$\pi_{1}^{\text{et}}(X_{\overline{F}},\overline{x})$ factors through the
pro-$S$ completion $\pi_{1}^{\text{et}}(X_{\overline{F}},\overline{x})^{(S)}$
of this group. Therefore, $V$ lies in the image of the canonical map
$\overline{\mathbb{Q}}_{p}[\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X_{\overline{F}},\overline{x})^{(S)}]\to\overline{\mathbb{Q}}_{p}[\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X_{\overline{F}},\overline{x})]$.
After enlarging $S$ we may assume that $X$ is the generic fiber of a smooth
${\mathcal{O}}_{F,S}$-scheme ${\mathfrak{X}}$ that can be established as the
complement to a horizontal normal crossings divisor
${\mathfrak{D}}\subset\overline{{\mathfrak{X}}}$ in a smooth proper
$\overline{{\mathfrak{X}}}$ and, moreover, $x$ extends to a point in
${\mathfrak{X}}({\mathcal{O}}_{F,S})$. By Lemma 8.7 below the action of
$G_{F}$ on
$\pi_{1}^{\mathrm{pro}-\mathrm{alg}}(X_{\overline{F}},\overline{x})^{(S)}$ is
unramified at all places not lying above $S$. Therefore $V$ is almost
everywhere unramified. ∎

######  Lemma 8.7.

Let $v$ be a finite place of $F$ of residue characteristics
$l=\mathrm{char}\,k(v)$. Suppose that ${\mathcal{Y}}$ is a smooth scheme over
${\mathcal{O}}_{F_{v}}$ that can be embedded into a smooth proper
${\mathcal{O}}_{F_{v}}$ scheme $\overline{{\mathcal{Y}}}$ such that
$\overline{{\mathcal{Y}}}\setminus{\mathcal{Y}}$ is a relative normal
crossings divisor. If $y$ is a classical or tangential base point of
${\mathcal{Y}}$ then there is an isomorphism

(8.1)
$\pi_{1}^{\text{et}}({\mathcal{Y}}_{\overline{F}_{v}},\overline{y}_{F_{v}})^{(l^{\prime})}\simeq\pi_{1}^{\text{et}}({\mathcal{Y}}_{\overline{k(v)}},\overline{y}_{k(v)})^{(l^{\prime})}$

compatible with the action of $G_{F_{v}}$ where the action on the right hand
side is given by precomposing the natural action of $G_{k(v)}$ with the
surjection $G_{F_{v}}\to G_{k(v)}$. Here $y_{F_{v}}$ (resp. $y_{k(v)}$)
denotes the base point of ${\mathcal{Y}}_{F_{v}}$ (resp.
${\mathcal{Y}}_{k(v)}$) induces by $y$.

###### Proof.

Let ${\mathcal{O}}^{nr}_{F_{v}}$ denote the completed ring of integers of the
maximal unramified extension of $F_{v}$. By [LO10, Theorem A.7] (or [SGA1,
Corollaire X.3.9] if ${\mathcal{Y}}=\overline{{\mathcal{Y}}}$) the categories
of Galois covers of degrees prime to $l$ of
${\mathcal{Y}}_{\overline{F}_{v}}$,
${\mathcal{Y}}_{{\mathcal{O}}^{nr}_{F_{v}}}$ and
${\mathcal{Y}}_{\overline{k(v)}}$ are equivalent via the natural functors. In
particular, these equivalences are compatible with the fiber functors induced
by $y$ which implies the result. ∎

## References

  * [Ber07] Vladimir G. Berkovich.  Integration of one-forms on $p$-adic analytic spaces, volume 162 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 2007.
  * [BM02] Christophe Breuil and Ariane Mézard.  Multiplicités modulaires et représentations de ${\rm GL}_{2}({\bf Z}_{p})$ et de ${\rm Gal}(\overline{\bf Q}_{p}/{\bf Q}_{p})$ en $l=p$.  Duke Math. J., 115(2):205–310, 2002.  With an appendix by Guy Henniart.
  * [Bri08] Olivier Brinon.  Représentations $p$-adiques cristallines et de de Rham dans le cas relatif.  Mém. Soc. Math. Fr. (N.S.), (112):vi+159, 2008.
  * [Con] Brian Conrad.  Lifting global representations with local properties.
  * [Del89] Pierre Deligne.  Le groupe fondamental de la droite projective moins trois points.  In Galois groups over ${\bf Q}$ (Berkeley, CA, 1987), volume 16 of Math. Sci. Res. Inst. Publ., pages 79–297. Springer, New York, 1989.
  * [dJ95] A. J. de Jong.  Étale fundamental groups of non-Archimedean analytic spaces.  volume 97, pages 89–118. 1995.  Special issue in honour of Frans Oort.
  * [DLLZ19] Hansheng Diao, Kai-Wen Lan, Ruochuan Liu, and Xinwen Zhu.  Logarithmic Riemann-Hilbert correspondences for rigid varieties, 2019\.
  * [DM13] Giovanni Di Matteo.  On admissible tensor products in $p$-adic Hodge theory.  Compos. Math., 149(3):417–429, 2013.
  * [Dri18] Vladimir Drinfeld.  On the pro-semisimple completion of the fundamental group of a smooth variety over a finite field.  Adv. Math., 327:708–788, 2018.
  * [FM95] Jean-Marc Fontaine and Barry Mazur.  Geometric Galois representations.  In Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), Ser. Number Theory, I, pages 41–78. Int. Press, Cambridge, MA, 1995.
  * [Lit21] Daniel Litt.  Arithmetic representations of fundamental groups, II: Finiteness.  Duke Math. J., 170(8):1851–1897, 2021.
  * [LO10] Max Lieblich and Martin Olsson.  Generators and relations for the étale fundamental group.  Pure Appl. Math. Q., 6(1, Special Issue: In honor of John Tate. Part 2):209–243, 2010.
  * [LZ17] Ruochuan Liu and Xinwen Zhu.  Rigidity and a Riemann-Hilbert correspondence for $p$-adic local systems.  Invent. Math., 207(1):291–343, 2017.
  * [Ols11] Martin C. Olsson.  Towards non-abelian $p$-adic Hodge theory in the good reduction case.  Mem. Amer. Math. Soc., 210(990):vi+157, 2011.
  * [ST] Stacks Project authors.  Stacks Project, 2019.
  * [Pat19] Stefan Patrikis.  Variations on a theorem of Tate.  Mem. Amer. Math. Soc., 258(1238):viii+156, 2019.
  * [Sch13] Peter Scholze.  $p$-adic Hodge theory for rigid-analytic varieties.  Forum Math. Pi, 1:e1, 77, 2013.
  * [Sen81] Shankar Sen.  Continuous cohomology and $p$-adic Galois representations.  Invent. Math., 62(1):89–116, 1980/81.
  * [Ser68] Jean-Pierre Serre.  Abelian $l$-adic representations and elliptic curves.  McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute. W. A. Benjamin, Inc., New York-Amsterdam, 1968.
  * [SGA1] Revêtements étales et groupe fondamental (SGA 1), volume 3 of Documents Mathématiques (Paris) [Mathematical Documents (Paris)].  Société Mathématique de France, Paris, 2003.  Séminaire de géométrie algébrique du Bois Marie 1960–61. [Algebraic Geometry Seminar of Bois Marie 1960-61], Directed by A. Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)].
  * [Shi18] Koji Shimizu.  Constancy of generalized Hodge-Tate weights of a local system.  Compos. Math., 154(12):2606–2642, 2018.
  * [Sim92] Carlos T. Simpson.  Higgs bundles and local systems.  Inst. Hautes Études Sci. Publ. Math., (75):5–95, 1992.
  * [Tem18] Michael Temkin.  Functorial desingularization over $\bf Q$: boundaries and the embedded case.  Israel J. Math., 224(1):455–504, 2018.
  * [Vol03] Vadim Vologodsky.  Hodge structure on the fundamental group and its application to $p$-adic integration.  Mosc. Math. J., 3(1):205–247, 260, 2003.