Arithmetic representations of fundamental groups I

TL;DR

This paper proves that nontrivial, semisimple arithmetic representations of the étale fundamental group of a normal algebraic variety over a finitely generated field are nontrivial modulo a fixed power of a prime, linking arithmetic and geometric representations.

Contribution

It establishes a uniform bound on the mod $\ell^N$ triviality of semisimple arithmetic representations, connecting geometric origin to nontriviality modulo a fixed power of $\ell$.

Findings

Existence of a universal integer N for nontrivial representations.

Nontrivial geometric representations are nontrivial mod $\ell^N$.

Provides a link between arithmetic representations and geometric origin.

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 $\ell^N$.