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](https://mathoverflow.net/questions/225468/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 $x$? 



Is an $\ell$-adic local system on $\mathbb A^2$ that restricts to a trivial local system on every vertical line a pullback via the first projection?
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!


