Arithmetic representations of fundamental groups I

Let $X$ be a normal algebraic variety over a finitely generated field $k$ of characteristic zero, and let $\mathrm{\ell }$ be a prime. Say that a continuous $\mathrm{\ell }$-adic representation $\rho$ of ${\pi }_1^{\acute{\text{e}}\text{t}}(X_{\bar{k}})$ is arithmetic if there exists a representation $\tilde{\rho }$ of a finite index subgroup of ${\pi }_1^{\acute{\text{e}}\text{t}}(X)$, with $\rho$ a subquotient of $\tilde{\rho }{\mathrm{\mid }}_{{\pi }_1(X_{\bar{k}})}$. We show that there exists an integer $N=N(X, \ell)$suchthateverynontrivial,semisimplearithmeticrepresentationof$\pi_1^{$\acute{\text{e}}\text{t}$}(X_{\bar k})$isnontrivialmod$\ell^N$. As a corollary, we prove that any nontrivial semisimple representation of ${\pi }_1^{\acute{\text{e}}\text{t}}(X_{\bar{k}})$, which arises from geometry, is nontrivial mod ${\mathrm{\ell }}^N$.