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This question is a reformulation of a special case of the question An $\ell$-adic local system which is trivial on every fiber of a morphism (this special case did not receive an answer on MathOverflow).

Let $\mathbb A^2$ be the affine plane over the algebraic closure of the finite field $\bar {\mathbb F}_q$. Let $L$ be an $\ell$-adic local system on this $\mathbb A^2$ which restricts to a trivial local system on every vertical line $\mathbb A^1 \subset \mathbb A^2$ (i.e. every line in the above $\mathbb A^2$ given by an equation of the form $x=c$ for $c \in \bar {\mathbb F_q}$). In other words, for any vertical line $\mathbb A^1$ as above, the restriction of the representation $\rho$ of the etale fundamental group $\pi_1^{et}(\mathbb A^2)$ (which corresponds to the local system $L$) via the group homomorphism $\pi_1^{et}(\mathbb A^1) \to \pi_1^{et}(\mathbb A^2)$ is a trivial representation of $\pi_1^{et}(\mathbb A^1)$.

Is it necessarily true that $L$ is the pullback of an $\ell$-adic local system on $\mathbb A^1$ via the projection map $\mathbb A^2 \to \mathbb A^1$ sending $(x,y)$ to $x$?

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First translate to the case of a continuous map $\rho : \pi_1(\mathbf{A}^2) \to G$ with $G$ finite. Using a limit argument reduce to the case where $\rho$ is defined on $\pi_1(\mathbf{A}^2_{\mathbf{F}_q})$ for some $q$. Let $\sigma : \mathbf{A}^1_{\mathbf{F}_q} \to \mathbf{A}^2_{\mathbf{F}_q}$, $x \mapsto (x, 0)$ be the zero section to the projection morphism of the question. Then let $C \to \mathbf{A}^1_{\mathbf{F}_q}$ be the finite \'etale covering corresponding to $\pi_1(\mathbf{A}^1_{\mathbf{F}_q}) \to G$ gotten by composing $\rho$ and $\pi_1(\sigma)$. Pulling back to $C \times \mathbf{A}_{\mathbf{F}_q}$ we see that now we have $$ \rho : \pi_1(C \times \mathbf{A}^1_{\mathbf{F}_q}) \longrightarrow G $$ which is trivial after composing with $\pi_1(\sigma) : \pi_1(C) \to \pi_1(C \times \mathbf{A}_{\mathbf{F}_q})$ and trivial when restricted to $\pi_1(\mathbf{A}^1_{\overline{\mathbf{F}}_q})$ via the inclusion of a fibre of the projection. We want to show that $\rho$ is trivial.

Let $X \to C \times \mathbf{A}^1_{\mathbf{F}_q}$ be the $G$-covering corresponding to $\rho$. Then looking at a fibre $X_c$ of the projection $X \to C$ we see that $X_c$ is geometrically a disjoint union of $|G|$ copies of the affine line. On the other hand, the fibre of $X_c \to \{c\} \times \mathbf{A}^1_{\mathbf{F}_q}$ over $(c, 0)$ is split, i.e., consists of $|G|$ points defined over the residue field $\kappa(c)$. Thus we see that $X_c$ is a disjoint union of $|G|$ copies of $\{c\} \times \mathbf{A}^1_{\mathbf{F}_q}$. Hence by Lang-Weil we see that $X$ must have $|G|$ irreducible components: just count rational points over $\mathbf{F}_{q^n}$ for some fixed large $n$ by using the above for all $c \in C(\mathbf{F}_{q^n})$.

Please write out all the details and check carefully. If this is correct I am sure you can find it in the literature. In particular, I suggest reading Katz work on these kinds of things. Cheers!

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