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 $\pi_1^{\text{\'et}}(X_{\bar k})$ is arithmetic if there exists a representation $\tilde \rho$ of a finite index subgroup of $\pi_1^{\text{\'et}}(X)$, 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 $\pi_1^{\text{\'et}}(X_{\bar k})$ is nontrivial mod $\ell^N$. As a corollary, we prove that any nontrivial semisimple representation of $\pi_1^{\text{\'et}}(X_{\bar k})$, which arises from geometry, is nontrivial mod $\ell^N$.