Level structure, arithmetic representations, and noncommutative Siegel linearization

Abstract

Let $\ell$ be a prime, $k$ a finitely generated field of characteristic different from $\ell$, and $X$ a smooth geometrically connected curve over $k$. Say a semisimple representation of $\pi_1^{\mathrm{et}}(X_{\bar k})$ is arithmetic if it extends to a finite index subgroup of $\pi_1^{\mathrm{et}}(X)$. We show that there exists an effective constant $N=N(X,\ell)$ such that any semisimple arithmetic representation of $\pi_1^{\mathrm{et}}(X_{\bar k})$ into $\mathrm{GL}_n(\bar{\mathbb{Z}_\ell})$, which is trivial mod $\ell^N$, is in fact trivial. This extends a previous result of the second author from characteristic zero to all characteristics. The proof relies on a new noncommutative version of Siegel's linearization theorem and the $\ell$-adic form of Baker's theorem on linear forms in logarithms.