Group theoretical independence of $\ell$-adic Galois representations

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 $\mathrm{Gal}(K)$ on the étale cohomology group $H^q(X_{\overline{K}},{\mathbb{Q}}_{\ell })$. For a field $k$ we denote by $k_{\mathrm{ab}}$ 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 $({\rho }_{\ell }|\mathrm{Gal}(k_{\mathrm{ab}}F){)}_{\ell }$ is group theoretically independent in the sense that ${\rho }_{{\ell }_1}(\mathrm{Gal}(k_{\mathrm{ab}}F))$ and ${\rho }_{{\ell }_2}(\mathrm{Gal}(k_{\mathrm{ab}}F))$ do not have a common finite simple quotient group for all prime numbers ${\ell }_1\ne {\ell }_2$.

References in corpus (1)

• Un critère d'indépendance pour une famille de représentations l-adiques