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

Part of: Arithmetic algebraic geometry (Co)homology theory

Published online by Cambridge University Press:  25 April 2013

Wojciech Gajda  and
Sebastian Petersen
Wojciech Gajda
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61614 Poznań, Poland (email: gajda@amu.edu.pl)
Sebastian Petersen
Affiliation:
FB 10 - Mathematik und Naturwissenschaften, Universität Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany (email: petersen@mathematik.uni-kassel.de)
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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 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 of Serre.

Keywords

Galois representationétale cohomologyabelian varietyfinitely generated field

MSC classification

Secondary: 11G10: Abelian varieties of dimension $> 1$ 14F20: Étale and other Grothendieck topologies and (co)homologies
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 7 , July 2013 , pp. 1091 - 1107
Copyright
Copyright © 2013 The Author(s) 

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