Independence of $\ell$-adic Galois representations over function fields

TL;DR

This paper proves that the fields fixed by the kernels of varying $ ext{ell}$-adic Galois representations over a finitely generated field are linearly disjoint, confirming a question posed by Serre in 1991.

Contribution

It establishes the independence of $ ext{ell}$-adic Galois representations over function fields, showing the associated fields are linearly disjoint over a finite extension.

Findings

Fields fixed by kernels are linearly disjoint over a finite extension.

Provides a positive answer to Serre's 1991 question.

Advances understanding of $ ext{ell}$-adic Galois representations over function fields.

Abstract

Let $K$ be a finitely generated extension of $\mathbb{Q}$. We consider the family of $\ell$-adic representations ($\ell$ varies through the set of all prime numbers) of the absolute Galois group of $K$, attached to $\ell$-adic cohomology of a smooth separated scheme of finite type over $K$. We prove that the fields cut out from the algebraic closure of $K$ by the kernels of the representations of the family are linearly disjoint over a finite extension of K. This gives a positive answer to a question asked by Serre in 1991.