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

## TL;DR

This paper proves that for a smooth projective variety over a finitely generated field, the associated family of $ ext{ell}$-adic Galois representations become group theoretically independent after restricting to certain finite Galois extensions, ensuring no common finite simple quotients.

## Contribution

It establishes the existence of finite Galois extensions making the family of Galois representations group theoretically independent, a novel structural result.

## Key findings

- Existence of finite Galois extensions with independence property.
- Family of Galois representations has no common finite simple quotients after restriction.
- Provides a new structural understanding of $ ext{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 \'etale 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 $\ell_1\neq \ell_2$.