Group-theoretical independence of $\ell $-adic Galois representations
Volume 176 / 2016
Abstract
Let $K/\mathbb Q$ be a finitely generated field of characteristic zero and $X/K$ a smooth projective variety. Fix $q\in\mathbb N$. For every prime number $\ell$ let $\rho_\ell$ be the representation of $\operatorname{Gal}(K)$ on the étale cohomology group $H^q(X_{\overline{K}}, \mathbb Q_\ell)$. For a field $k$ we denote by $k_{\rm ab}$ its maximal abelian Galois extension. We prove that there exist finite Galois extensions $k/\mathbb Q$ and $F/K$ such that the restricted family of representations $(\rho_\ell|\operatorname{Gal}(k_{\rm ab} F))_\ell$ is group-theoretically independent in the sense that $\rho_{\ell_1}(\operatorname{Gal}(k_{\rm ab} F))$ and $\rho_{\ell_2}(\operatorname{Gal}(k_{\rm ab} F))$ do not have a common finite simple quotient group for all prime numbers $\ell_1\neq \ell_2$.
