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 ${\pi }_1^{\text{eˊt}}(X_{\bar{k}})$ is arithmetic if there exists a representation $\tilde{\rho }$ of a finite index subgroup of ${\pi }_1^{\text{eˊt}}(X)$, 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 ${\pi }_1^{\text{eˊt}}(X_{\bar{k}})$ is nontrivial mod ${\ell }^N$. As a corollary, we prove that any nontrivial semisimple representation of ${\pi }_1^{\text{eˊt}}(X_{\bar{k}})$, which arises from geometry, is nontrivial mod ${\ell }^N$.

arXiv commentThis paper gives a streamlined proof of the main theorem of arXiv:1607.05740, with some minor improvements, and is intended for publication. That paper proves much more, some of which will appear in the sequel to this paper. Comments welcome; 27 pages, 1 figure

References in corpus (1)

• Revêtements étales et groupe fondamental (SGA 1)