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 ${\mathrm{GL}}_n(\overline{{\mathbb{Z}}_{\ell }})$, 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.

Various corrections in response to referee's comments; accepted for publication in Crelle's Journal