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\title{Some remarks on the companions conjecture for normal varieties}

\author{\firstname{Marco} \lastname{D'Addezio}}
\address{Institut de Mathématiques de Jussieu-Paris Rive Gauche, SU - 4 place Jussieu, Case 247, 75005 Paris}
\email{daddezio@imj-prg.fr}

\subjclass{14G15}
\keywords{$\ell$-adic representation, independence of $\ell$, étale fundamental group}

\begin{abstract}
Drinfeld in 2010 proved the companions conjecture for smooth varieties over a finite field, generalizing L.~Lafforgue's result for smooth curves. We study the obstruction to prove the conjecture for arbitrary normal varieties. To do this, we introduce a new property of morphisms. We verify this property in some cases, showing thereby the companions conjecture for some singular normal varieties.
\end{abstract}


\dateposted{2024-02-02}
\begin{document}
\maketitle
%\tableofcontents

\section{Introduction}

\subsection{The companions conjecture}

Let $\Fq$ be a finite field of characteristic $p$ and let $X_0$ be a connected normal variety over $\Fq$. For a prime $\ell$ different from $p$, the category of Weil lisse $\Qlbar$-sheaves over $X_0$ carries much information on the arithmetic and the geometry of $X_0$. An invariant that is associated to any Weil lisse $\Qlbar$-sheaf is the trace field $E\subseteq\Qlbar$ generated by the coefficients of the Frobenius polynomials at closed points. Deligne proved in~\cite{Del} that if $\calV_0$ is an irreducible Weil lisse $\Qlbar$-sheaf over $X_0$ with finite order determinant, then $E$ is a finite extension of $\Q$. He showed this finiteness by reduction to the case of curves, where it was proven by L.~Lafforgue in~\cite[Thm.~VII.6]{Laf} as a consequence of the Langlands correspondence. This property of the trace field was one of the conjectures proposed by Deligne in~\cite[Conj.~1.2.10]{Weil2}. In the same list, he also formulated the following conjecture.

\begin{conj}[Companions conjecture]\label{intro-companions-c}
After possibly replacing $E$ with a finite field extension, for every finite place $\lambda$ not dividing $p$, there exists a Weil lisse $E_{\lambda}$-sheaf $E$-compatible\footnote{Cf.~\cite[Def.~3.1.15]{Dad}.} with $\calV_0$.
\end{conj}

When the dimension of $X_0$ is $1$, the conjecture is again a consequence of the Langlands correspondence. For higher dimensional varieties, Drinfeld in~\cite{Dri} proved Conjecture~\ref{intro-companions-c} when $X_0$ is smooth. As noticed in \cite[\S 6]{Dri}, his method cannot be applied directly to prove the full conjecture.

\subsection{The obstruction}\label{obst:ss}

Suppose for simplicity that the singular locus of $X_0$ consists of one closed point and that we can solve that singularity. In other words, suppose that there exists a smooth variety $Y_0$ and a proper morphism $h_0:Y_0\to X_0$ which sends a closed subscheme $i_0:Z_0\hookrightarrow Y_0$ to a closed point $x_0\in |X_0|$ and such that $h_0$ is an isomorphism outside $Z_0$. Write $\F$ for an algebraic closure of $\Fq$ and suppose that $Z:=Z_0\otimes_{\Fq}\F$ is connected.

\begin{lemm}[{\cite[Cor.~IX.6.11]{SGA1}}]\label{i-contraction-fundamental-group-lemma}
For every geometric point $z$ of $Z_0$ there exists an exact sequence
\[
\pi_1^{\et}(Z,z)\xrightarrow{i_*} \pi_1^{\et}(Y_0,z)\xrightarrow{h_{0*}} \pi_1^{\et}(X_0,h(z))\to 1
\]
in the sense that the smallest normal closed subgroup containing the image of $i_*$ is equal to the kernel of $h_{0*}$.
\end{lemm}

By the lemma, every étale Weil lisse sheaf $\calV_0$ over $Y_0$ which is trivial over $Z$ is the inverse image of an étale Weil lisse sheaf defined over $X_0$. Since we know the companions conjecture for $Y_0$, in order to deduce it for $X_0$ we have to verify the following property.
\begin{itemize}
\item[$\mathcal{P}(Z_0)$ :] For every pair $(\calV_0,\calW_0)$ of compatible absolutely irreducible Weil lisse sheaves with finite order determinant over $Y_0$, the sheaf $\calV_0$ is trivial over $Z$ if and only if the same is true for~$\calW_0$.
\end{itemize}
If $Z_0\subseteq Y_0$ satisfies $\calP(Z_0)$ we say that $Z_0$ is a \emph{$\lambda$-uniform subvariety}. Thanks to Lemma~\ref{i-contraction-fundamental-group-lemma} and the companions conjecture for smooth varieties, we have the following result.

\begin{prop}\label{lamd-unif-comp:p}
If $Z_0\subseteq Y_0$ is $\lambda$-uniform, $X_0$ satisfies the companions conjecture.
\end{prop}
\subsection{Main results}

The aim of this text is to shed some new lights on the companions conjecture for normal varieties. For this scope, we focus on $\lambda$-uniformity. We extend the notion of $\lambda$-uniform subvarieties in Section~\ref{obst:ss} to the one of \emph{$\lambda$-uniform morphisms} of varieties (Definition~\ref{uniform-morphisms:d}) and we investigate the following conjecture.
\begin{conj}[Conjecture~\ref{unif:c}]\label{i-unif:c}
Let $Y_0$ and $Z_0$ be varieties over $\Fq$. If $Y_0$ is normal, every morphism $f_0:Z_0\to Y_0$ is $\lambda$-uniform (cf. Definition~\ref{uniform-morphisms:d}).
\end{conj}

We shall verify Conjecture~\ref{i-unif:c} in some particular cases.

\begin{theo}[Theorem~\ref{tree:t},
Theorem~\ref{finite:t}]\label{i-main:t}
Let $f_0: Z_0 \to Y_0$ be a morphism of geometrically connected varieties over $\Fq$ with $Y_0$ normal and let $z$ be a geometric point of $Z_0$. The morphism $f_0$ is $\lambda$-uniform in the following cases.
\begin{enumerate}\romanenumi
\item \label{5i} If $Z_0$ has a simply connected dual complex and normal irreducible components.
\item \label{5ii} If the smallest closed normal subgroup of $\pi_1^{\et}(Y_0,f(z))$ containing the image of $\pi_1^{\et}(Z,z)$ is open inside $\pi_1^{\et}(Y,f(z))$.
\item \label{5iII} If $\pi_1^{\et}(Y,f(z))$ contains an open solvable profinite subgroup.
\end{enumerate}
\end{theo}

Combining the previous results we get the following consequence.

\begin{coro}
Let $Y_0$ be a smooth geometrically connected variety over $\Fq$. If $X_0$ is a normal variety that can be written as a contraction of a geometrically connected subvariety $Z_0\subseteq Y_0$ satisfying one of the conditions of Theorem~\ref{i-main:t}, then $X_0$ verifies Conjecture~\ref{intro-companions-c}.
\end{coro}

An independent property we prove in this text is a property of invariance of $\lambda$-uniformity ``under deformations'' (Theorem~\ref{homotopy:t}). This might be useful for further developments in the direction of Conjecture~\ref{i-unif:c}. Moreover, we present in Section~\ref{fina-comm:ss} some variants of Conjecture~\ref{i-unif:c} and we present a concrete example, proposed by de Jong, where these conjectures are not known.


\subsection{Notation and conventions}
%\subsubsection{}
For us, a \emph{variety} over a field $k$ is a separated scheme of finite type over $k$. We write $X_0,Y_0,Z_0,\dots$ for varieties over $\F_q$ and $X,Y,Z,\dots$ for the base change to $\F$. Further, we put a subscript $_0$ to indicate objects and morphisms defined over $\F_q$ and the suppression of this subscript shall mean that we are extending the scalars to $\F$. If $E$ is a number field we write $|E|_{\neq p}$ for the set of finite places of $E$ which do not divide $p$. For every $\lambda\in |E|_{\neq p}$, we denote by $E_{\lambda}$ the completion of $E$ with respect to~$\lambda$.

%\subsubsection{}
We use the notation for Weil lisse sheaves as in~\cite[\S2.2]{Dad}. We say that a Weil lisse $E_\lambda$-sheaf $\calV_0$ is \emph{split untwisted} if every irreducible subquotient of $\calV_0$ is absolutely irreducible and has finite order determinant. We say instead that $\calV_0$ is \emph{untwisted} if it is split untwisted after possibly extending $E_\lambda$. Recall that if $Y_0$ is a normal variety over $\Fq$, every untwisted Weil lisse sheaf over $Y_0$ is pure of weight $0$ and geometrically semi-simple by~\cite[Thm.~3.4.1]{Weil2} and~\cite[Thm.~1.6]{Del}. In addition, by~\cite[Prop.~1.3.14]{Weil2} and~\cite[Prop.~3.1.16]{Dad}, every untwisted Weil lisse sheaf is étale.

%\subsubsection{}
An \emph{$E$-compatible system} over $X_0$, denoted by $\underline{\calV_0}$, is a family $\{\calV_{\lambda,0}\}_{\lambda\in |E|_{\neq p}}$ where each $\calV_{\lambda,0}$ is an $E$-rational Weil lisse $E_{\lambda}$-sheaf and such that all sheaves are pairwise $E$-compatible. Each $\calV_{\lambda,0}$ is called the \emph{$\lambda$-component} of $\underline{\calV_0}$. We say that a compatible system is \emph{semi-simple, untwisted,} or \emph{split untwisted} if each $\lambda$-component has the respective property.


\section{\texorpdfstring{$\lambda$}{}-uniform morphisms}\label{the-conj:ss}

\subsection{General properties}

\begin{defi}\label{uniform-systems:d}
Let $Z_0$ be a connected variety. A compatible system $\underline{\calV_0}$ over $Z_0$ is \emph{$\lambda$-uniform} if one of the following disjoint conditions is verified.
\begin{enumerate}\romanenumi
\item For every $\lambda\nmid p$, the lisse sheaf\/ $\calV_{\lambda,0}$ is geometrically trivial.
\item For every $\lambda\nmid p$, the lisse sheaf\/ $\calV_{\lambda,0}$ is geometrically non-trivial.
\end{enumerate}
We say that $\underline{\calV_0}$ is \emph{strongly $\lambda$-uniform} if the dimension of $H^0(Z,\calV_{\lambda})$ does not depend on $\lambda$. Strongly $\lambda$-uniform compatible systems are clearly $\lambda$-uniform. If $Z_0$ is not connected we say that a compatible system is \emph{$\lambda$-uniform} (resp. \emph{strongly $\lambda$-uniform}) if it is $\lambda$-uniform (resp. strongly $\lambda$-uniform) over every connected component.
\end{defi}



\begin{prop}\label{normal-uniform:p}
Let $Z_0$ be a normal variety over $\F_q$. Every untwisted $E$-compatible system $\ucalVz$ over $Z_0$ is strongly $\lambda$-uniform.
\end{prop}

\begin{proof}
After extending the base field, we may assume that $Z_0$ is geometrically connected. Since $Z_0$ is a normal variety, each $\lambda$-component of $\ucalVz$ is pure of weight $0$ by~\cite[Thm.~1.6]{Del}. Also, if $U_0$ is the smooth locus of $Z_0$, by~\cite[Prop.~V.8.2]{SGA1}, the \'etale fundamental group of $U$ maps surjectively onto the \'etale fundamental group of $Z$. Therefore, we have a canonical isomorphism
\[
H^0(Z,\calV_{\lambda})=H^0(U,\calV_{\lambda}|_U)
\]
for every $\lambda$. By~\cite[Cor.~VI.3]{Laf}, we know that the dimension of $H^0(U,\calV_{\lambda}|_U)$ can be recovered from the $L$-function of $\calV_{\lambda,0}|_{U_0}$, thus we obtain the desired result.
\end{proof}

If we do not assume $Z_0$ normal, Proposition~\ref{normal-uniform:p} becomes false in general (Example~\ref{noda-curv:ex}). The issue is a different behaviour of weights for non-normal varieties. In what follows, we want to understand whether a weaker variant of Proposition~\ref{normal-uniform:p} is still true for singular varieties.

\begin{defi}\label{uniform-morphisms:d}
Let $f_0:Z_0\to Y_0$ be a morphism of varieties over $\Fq$. We say that $f_0$ is a \emph{$\lambda$-uniform morphism} if for every untwisted compatible system $\underline{\calV_0}$ over $Y_0$, the pullback $f_0^*\underline{\calV_0}$ is $\lambda$-uniform. If $f_0$ is a closed immersion we say that $Z_0$ is a \emph{$\lambda$-uniform subvariety} of $Y_0$.
\end{defi}

\begin{conj}\label{unif:c}
Let $Y_0$ and $Z_0$ be varieties over $\Fq$. If $Y_0$ is normal, every morphism $f_0:Z_0\to Y_0$ is $\lambda$-uniform.
\end{conj}

\subsection{Homotopic invariance}

Let us look more closely at $\lambda$-uniform morphisms by analysing the relation with the induced morphism on fundamental groups.

%\subsubsection{}

Let $f_0:Z_0\to Y_0$ be a morphism of geometrically connected varieties over $\Fq$ with $Y_0$ normal. If we choose a geometric point $z$ of $Z_0$ we have a morphism
\[
\pi_1^{\et}(Z,z)\xrightarrow{f_{*}} \pi_1^{\et}(Y_0,f(z)).
\]
For every étale compatible system $\underline{\calV_0}$ over $Y_0$ we denote by $\{\rho_{\lambda,0}\}_{\lambda \in |E|_{\neq p}}$ the associated family of $\ell$-adic representations of $\pi_1^{\et}(Y_0,f(z))$. Let $\overline{\Image(f_*)}$ be the smallest normal closed subgroup of $\pi_1^{\et}(Y_0,f(z))$ containing the image of $f_*$. The following lemma is a direct consequence of the definition of a $\lambda$-uniform morphism.

\begin{lemm}\label{uniform-group-property:l}
A morphism $f_0$ is $\lambda$-uniform if and only if for every untwisted compatible system $\underline{\calV_0}$ over $Y_0$, if\/ $\overline{\Image(f_*)}\subseteq \Ker(\rho_{\lambda,0})$ for one $\lambda$ then the same is true for every other $\lambda\in |E|$. In particular, the property of a morphism of being $\lambda$-uniform depends only on the inclusion $\overline{\Image(f_*)}\subseteq \pi_1^{\et}(Y_0,y)$ as topological groups together with the assignment of the conjugacy classes of the Frobenii at closed points of $\pi_1^{\et}(Y_0,y)$ and their degrees.
\end{lemm}

As a consequence of the previous lemma, we prove an ``homotopic invariance'' of $\lambda$-uniformity. Let $T_0$ and $S_0$ be geometrically connected varieties over $\F_q$ and $h_0: T_0\to S_0$ a proper and flat morphism with connected and reduced geometric fibres. Let $s_0$ and $s'_0$ be closed points of $S_0$ and write $\iota_0:Z_0\hookrightarrow T_0$ and $\iota'_0:Z'_0\hookrightarrow T_0$ for the closed immersions of the fibres of $h_0$ above $s_0$ and $s'_0$ respectively.

\begin{theo}\label{homotopy:t}
For every morphism $\widetilde{f}_0: T_0\to Y_0$, the restriction $f_0:=\widetilde{f}_0|_{Z_0}$ is $\lambda$-uniform if and only if $f'_0:=\widetilde{f}_0|_{Z'_0}$ is $\lambda$-uniform.
\end{theo}

\begin{proof}
Let $z$ and $z'$ be geometric points of $Z_0$ and $Z'_0$ respectively. By~\cite[Tag 0C0J]{Stacks}, we have exact sequences
\begin{gather*}
\pie(Z,z)\xrightarrow{\iota_{*}} \pie(T,z)\xrightarrow{h_{*}} \pie(S,h(z))\to 1
\\
\pie(Z',z')\xrightarrow{\iota'_{*}} \pie(T,z')\xrightarrow{h_{*}} \pie(S,h(z'))\to 1.
\end{gather*}
The choice of an étale path $\gamma$ joining $z$ with $z'$ induces isomorphisms $\gamma:\pie(T,z)\iso \pie(T',z')$ and $h_*(\gamma):\pie(S,h(z))\iso \pie(S',h(z'))$. Thanks to the two exact sequences this implies that $\gamma$ restricts to an isomorphism $\overline{\Image(\iota_*)}\iso \overline{\Image(\iota'_*)}$. In turn, this implies that the induced isomorphism $f_*(\gamma):\pi_1^{\et}(Y_0,f(z))\iso \pi_1^{\et}(Y_0,f'(z'))$ restricts to an isomorphism $\overline{\Image(f_*)}\iso \overline{\Image(f'_*)}$. By construction, $f_*(\gamma)$ respects the conjugacy classes of Frobenii at closed points and their degrees. We conclude applying Lemma~\ref{uniform-group-property:l}.
\end{proof}



\section{Some examples} In this section, we verify Conjecture~\ref{unif:c} in some cases. Note that by virtue of Proposition~\ref{normal-uniform:p} we already know the conjecture when $Z_0$ is normal.
\subsection{Simply connected dual complex}\label{semi-stab-curv:ss}
Let $Z_0$ be a geometrically connected variety over $\F_q$. Write $Z^{(i)}$ where ${1\leq i \leq n}$ for the irreducible components of $Z$. Suppose that for every $i$, the irreducible component $Z^{(i)}$ is normal. Let $z$ be a geometric point of $Z_0$ and for every $1\leq i \leq n$, let $z^{(i)}$ be a generic geometric point of $Z^{(i)}$. We denote by $\Gamma$ the dual complex of $Z$ and by $P$ the point of $\Gamma$ associated to the irreducible component where $z$ lies. Write $\Pi$ for the free product $\pie(Z^{(1)},z^{(1)})*\dots*\pie(Z^{(n)},z^{(n)})$ and ${\pi}_1(\Gamma,P)^\wedge$ for the profinite completion of the topological fundamental group of $\Gamma$ at $P$.
\begin{theo}[Stix]\label{Stix:t}
The choice of étale paths $\{\gamma^{(i)}\}_{1\leq i \leq n}$ joining $z$ to $z^{(i)}$ for every $i$ determines an exact sequence
\[
\Pi\xrightarrow{\alpha} \pie(Z,z)\xrightarrow{\beta} \pi_1(\Gamma,P)^\wedge\to 1,
\]
in the sense that the smallest normal closed subgroup containing the image of $\alpha$ is equal to the kernel of $\beta$.
\end{theo}

\begin{proof}
The result follows from~\cite[Cor.~3.3]{Stix}.
\end{proof}

\begin{theo}\label{tree:t}
If\/ $\Gamma$ is simply connected, every untwisted compatible system over $Z_0$ is $\lambda$-uniform.
\end{theo}

\begin{proof}
Let $\ucalVz$ be an untwisted compatible system over $Z_0$. If $\calV_{\lambda,0}$ is geometrically trivial for one $\lambda$, then it remains geometrically trivial when restricted to every irreducible component of $Z_0$. By Proposition~\ref{normal-uniform:p}, for every other $\lambda'\in |E|_{\neq p}$, the restriction of $\calV_{\lambda',0}$ to every irreducible component of $Z_0$ is geometrically trivial as well. By Theorem~\ref{Stix:t}, since $\Gamma$ is simply connected, the geometric étale fundamental group of $Z_0$ is the smallest normal subgroup containing the images of the étale fundamental groups of the irreducible components $Z^{(i)}$. This shows that for every $\lambda'$ the Weil lisse sheaf $\calV_{\lambda',0}$ is geometrically trivial over $Z_0$ and this yields the desired result.
\end{proof}

\begin{exem}\label{noda-curv:ex}
If $Z_0$ is an irreducible split nodal cubic curve over $\Fq$ with nodal point $z_0$, then $\pie(Z,z)$ is isomorphic to $\hZ$. In addition, the action of $\Gal(\F/\Fq)$ on $\pie(Z,z)$ is trivial. Therefore, $\pie(Z_0,z)$ is isomorphic to $\hZ\times \Gal(\F/\Fq)$, where the embedding $\Gal(\F/\Fq)\subseteq \pie(Z_0,z)$ is induced by the closed immersion of $z_0\hookrightarrow Z_0$. The Frobenius elements of $\pie(Z_0,z)$ correspond via this isomorphism to elements of $\hZ\times \Gal(\F/\Fq)$ of the form $(0,F^d)$, where $F$ is the geometric Frobenius of $\Fq$ and $d$ is some positive integer. This implies that every pair of étale lisse sheaves over $Z_0$ which are trivial at $z_0$ are $\Q$-compatible with all the eigenvalues at closed points equal to $1$.

On the other hand, for every prime number $\ell\neq p$ and every continuous group automorphism $\alpha$ of $\overline{\Z}_\ell^{\oplus r}$, where $\overline{\Z}_\ell$ is the ring of integers of $\Qlbar$, there exists an étale lisse $\Qlbar$-sheaf of this type such that the induced $\ell$-adic representation sends $(1,\mathrm{id})$ to $\alpha$. In particular, we may take $r=1$ and as $\alpha$ the multiplication by a root of unit. This construction produces lots of examples of untwisted compatible systems over $Z_0$ which are not $\lambda$-uniform.
\end{exem}

\subsection{Finite monodromy}

As in Proposition~\ref{normal-uniform:p}, we may use the theory of weights in order to prove that certain morphisms are $\lambda$-uniform. This strategy needs strong finiteness conditions.

\begin{theo}\label{finite:t}
Let $f_0: Z_0 \to Y_0$ be a morphism of geometrically connected varieties over $\Fq$ with $Y_0$ normal. Let $z$ be a geometric point of $Z_0$. The morphism $f_0$ is $\lambda$-uniform in the following cases.
\begin{enumerate}\romanenumi
\item \label{16i} If the smallest closed normal subgroup of $\pi_1^{\et}(Y_0,f(z))$ containing the image of $\pi_1^{\et}(Z,z)$ is an open subgroup of $\pi_1^{\et}(Y,f(z))$.
\item \label{16ii} If $\pi_1^{\et}(Y,f(z))$ contains an open solvable profinite subgroup.
\end{enumerate}
\end{theo}

\begin{proof}
Let $\underline{\calV_0}$ be an untwisted compatible system over $Y_0$. For every $\lambda\in |E|_{\neq p}$, write $G_{\lambda}$ for the geometric monodromy group of $\calV_{\lambda,0}$.

\begin{proof}[(\ref{16i})]
If $f_0^*(\calV_{\lambda,0})$ is geometrically trivial over $Z_0$ for one $\lambda$, then $\rho_{\lambda}$ is trivial when restricted to the image of $\pi_1^{\et}(Z,z)$ in $\pi_1^{\et}(Y,f(z))$. Thanks to the hypothesis, we deduce that $\rho_{\lambda}$ factors through a finite quotient of $\pi_1^{\et}(Y,f(z))$. This implies that $G_{\lambda}$ is a finite algebraic group. By~\cite[Prop.~2.2]{LP5} and~\cite[Thm.~4.1.1]{Dad}, the subgroups
\[
\Ker(\rho_{\lambda'})\cap \pi_1^{\et}(Y,f(z))\subseteq \pi_1^{\et}(Y,f(z))
\]
are all equal when $\lambda'$ varies in $|E|_{\neq p}$. Since the image of $\pi_1^{\et}(Z,z)$ in $\pi_1^{\et}(Y,f(z))$ is contained in $\Ker(\rho_{\lambda})$, it is also contained in $\Ker(\rho_{\lambda'})$ for every $\lambda'\in |E|_{\neq p}$. Therefore, the lisse sheaf $f_0^*(\calV_{\lambda',0})$ is geometrically trivial for every $\lambda'\nmid p$, as we wanted.
\let\qed\relax
\end{proof}

\begin{proof}[(\ref{16ii})]
Since $\pi_1^{\et}(Y,f(z))$ contains an open solvable profinite subgroup, we know that for every $\lambda$ the neutral component $G_\lambda^\circ \subseteq G_\lambda$ is solvable. Combining this with the fact that each $\calV_{\lambda,0}$ is geometrically semi-simple, we deduce that $G_\lambda^\circ$ is a torus. By~\cite[Thm.~1.3.8]{Weil2}, this implies that $G_\lambda^\circ$ is trivial so that $G_\lambda$ is finite and we can proceed as in the previous case.
\end{proof}
\let\qed\relax
\end{proof}

\begin{coro}
A dominant morphism $f_0:Z_0\to Y_0$ is $\lambda$-uniform. In particular, if $Y_0$ is a smooth curve, every morphism with target $Y_0$ is $\lambda$-uniform.
\end{coro}

\begin{proof}
Let $\eta\hookrightarrow Y$ be the generic point of $Y$ and write $Z_{\eta}$ the preimage of $\eta$ via $f$. Since $f$ is dominant, the scheme $Z_{\eta}$ is a non-empty variety over the function field of $Y$. Fix a closed point $\eta'$ of $Z_{\eta}$ and choose a geometric point $\overline{\eta}$ over $\eta'$. We have the following commutative diagram
\[
\begin{tikzcd}
\pie(\eta',\overline{\eta}) \arrow[hook,r] \arrow[d] & \pie(\eta,f(\overline{\eta})) \arrow[d, two heads]\\
\pi_1^{\et}(Z,\overline{\eta}) \arrow[r] & \pi_1^{\et}(Y,f(\overline{\eta})).
\end{tikzcd}
\]
Note that $\pie(\eta',\overline{\eta})$ maps to a finite index subgroup of $\pie(\eta,f(\overline{\eta}))$. Therefore, the image of $\pi_1^{\et}(Z,\overline{\eta})$ in $\pi_1^{\et}(Y,f(\overline{\eta}))$ has finite index as well. Thanks to this, we may apply Theorem~\ref{finite:t}$\MK$\eqref{16i} to conclude.
\end{proof}


\subsection{Final comments}\label{fina-comm:ss}
For a Weil lisse sheaf, the property of being geometrically trivial can be thought as the combination of two properties: being geometrically unipotent\footnote{A geometrically unipotent Weil lisse sheaf is a Weil lisse sheaf $\calV_0$ such that $\calV$ is a successive extension of trivial lisse sheaves.} and geometrically semi-simple. For this reason, it seems pretty natural to split Conjecture~\ref{unif:c} in two parts. Let $f_0:Z_0\to Y_0$ be a morphism of varieties over $\Fq$ with $Y_0$ normal.

\begin{conj}[Unipotency]\label{unipotent:c}
For every pair $(\calV_0,\calW_0)$ of compatible Weil lisse sheaf over $Y_0$, if $f_0^*\calV_0$ is geometrically unipotent the same is true for $f^*_0\calW_0$.
\end{conj}

\begin{conj}[Semi-simplicity]\label{semi-simplicity:c}
For every geometrically semi-simple Weil lisse sheaf $\calV_0$ over $Y_0$, the inverse image $f_0^*\calV_0$ is geometrically semi-simple.
\end{conj}

These two conjectures have different natures. The first one is a numerical property of the traces of the elements of the geometric étale fundamental group. The second one can not be read looking at the traces and it would be a generalisation of~\cite[Cor.~3.6.7]{Dad}. Let us focus on this second conjecture. Suppose that $Z_0$ is a semi-stable curve with all the irreducible components isomorphic to $\PP^1_{\Fq}$. In light of~\cite[Thm.~1.1]{LR96}, one might be tempted to hope that for every Weil lisse sheaf over $Y_0$, the inverse image to $Z_0$ is geometrically finite, which would imply Conjecture~\ref{semi-simplicity:c} for this choice of $Z_0$. Unluckily, this does not seem to be true as pointed out by de Jong.

\begin{exem}[de Jong]\label{dJ:ex}
For an integer $n\geq 3$ such that $(n,p)=1$, we write $Y^{(n)}$ for the moduli scheme of principally polarized abelian surfaces over $\F$ with a symplectic level-$n$-structure. Let $Z^{(n)}\subseteq Y^{(n)}$ be the supersingular locus of $Y^{(n)}$. Since we are working with abelian surfaces, this coincides with the Moret--Bailly locus of $Y^{(n)}$. Therefore, by~\cite[Prop.~7.3]{Oor01}, the variety $Z^{(n)}$ is connected for every choice of $n$. Let $N$ be a positive multiple of $n$ prime to $p$. The preimage of the natural finite étale Galois cover $Y^{(N)}\to Y^{(n)}$ is $Z^{(N)}$. As $Z^{(N)}$ is connected, the restriction $Z^{(N)}\to Z^{(n)}$ is a finite étale Galois cover with the same Galois group as $\Gal(Y^{(N)}/Y^{(n)})$. If $z$ is a geometric point of $Z^{(n)}$, we have the following commutative diagram
\[
\begin{tikzcd}
\pi_1^{\et}(Z^{(n)},z) \arrow{r} \arrow[d,two heads] & \pi_1^{\et}(Y^{(n)},z)\arrow[d,two heads]\\
\Gal(Z^{(N)}/Z^{(n)}) \arrow[r,"\sim"] & \Gal(Y^{(N)}/Y^{(n)}).\
\end{tikzcd}
\]
When $N$ goes to infinity, the cardinality of $\Gal(Y^{(N)}/Y^{(n)})$ goes to infinity as well. This implies that the image of $\pi_1^{\et}(Z^{(n)},z)\to \pi_1^{\et}(Y^{(n)},z)$ is infinite. The varieties $Z^{(n)}$ and $Y^{(n)}$ descend to geometrically connected varieties $Z_0^{(n)}$ and $Y_0^{(n)}$ over some finite field $\F_q$.
\end{exem}

Example~\ref{dJ:ex} might be a nice concrete example to analyse for further developments on Conjecture~\ref{unif:c}.


\subsection*{Acknowledgements}

I am grateful to my advisor Hélène Esnault for introducing me to this topic and for all the time we spent talking about this problem.~I thank Emiliano Ambrosi, Raju Krishnamoorthy, and Jacob Stix for discussions and suggestions and Piotr Achinger and Daniel Litt for sharing de Jong's example~(Example~\ref{dJ:ex}). Finally, I thank the anonymous referees for some helpful comments which improved the exposition.

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