Title: Arithmetic representations of fundamental groups I,
Abstract: Let $X$ be a normal algebraic variety over a finitely generated field $k$ of
characteristic zero, and let $\ell$ be a prime. Say that a continuous
$\ell$-adic representation $\rho$ of $\pi_1^{\text{ét}}(X_{\bar k})$ is
arithmetic if there exists a representation $\tilde \rho$ of a finite index
subgroup of $\pi_1^{\text{ét}}(X)$, with $\rho$ a subquotient of
$\tilde\rho|_{\pi_1(X_{\bar k})}$. We show that there exists an integer $N=N(X,
\ell)$ such that every nontrivial, semisimple arithmetic representation of
$\pi_1^{\text{ét}}(X_{\bar k})$ is nontrivial mod $\ell^N$. As a corollary,
we prove that any nontrivial semisimple representation of
$\pi_1^{\text{ét}}(X_{\bar k})$, which arises from geometry, is nontrivial
mod $\ell^N$.