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

Sebastian Petersen¹

¹ Fachbereich für Mathematik und Naturwissenschaften Universität Kassel Wilhelmshöher Allee 73 34121 Kassel, Germany

Acta Arithmetica, Tome 176 (2016) no. 2, pp. 161-176

Résumé

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 $\ell_1\neq \ell_2$.

Détail

DOI : 10.4064/aa8438-7-2016

Keywords: mathbb finitely generated field characteristic zero smooth projective variety fix mathbb every prime number ell rho ell representation operatorname gal tale cohomology group overline mathbb ell field denote its maximal abelian galois extension prove there exist finite galois extensions mathbb restricted family representations rho ell operatorname gal ell group theoretically independent sense rho ell operatorname gal rho ell operatorname gal have common finite simple quotient group prime numbers ell neq ell