Images of ℓ-adic representations and automorphisms of abelian varieties (original) (raw)

On the image of l-adic Galois representations for abelian varieties of type I and II

2004

In this paper we investigate the image of the lll-adic representation attached to the Tate module of an abelian variety over a number field with endomorphism algebra of type I or II in the Albert classification. We compute the image explicitly and verify the classical conjectures of Mumford-Tate, Hodge, Lang and Tate, for a large family of abelian varieties of type I and II. In addition, for this family, we prove an analogue of the open image theorem of Serre.

On the Image of l-Adic Galois Representations for Abelian Varieties of Type I and II Dedicated to John Coates on the occasion of his 60-th birthday

Documenta Mathematica Extra Volume: : John H. Coates’ Sixtieth Birthday (2006), pp.35-75., 2006

In this paper we investigate the image of the l-adic representation attached to the Tate module of an abelian variety over a number field with endomorphism algebra of type I or II in the Albert classification. We compute the image explicitly and verify the classical conjectures of Mumford-Tate, Hodge, Lang and Tate for a large family of abelian varieties of type I and II. In addition, for this family, we prove an analogue of the open image theorem of Serre. 2000 Mathematics Subject Classification: 11F80, 11G10

Independence of ℓ-adic Galois representations over function fields

2012

Let K be a finitely generated extension of Q. We consider the family of ℓ-adic representations (ℓ varies through the set of all prime numbers) of the absolute Galois group of K, attached to ℓ-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.

Independence of `-adic representations of geometric Galois groups

2013

Let k be an algebraically closed field of arbitrary characteristic, let K/k be a finitely generated field extension and let X be a separated scheme of finite type over K. For each prime `, the absolute Galois group of K acts on the `-adic etale cohomology modules of X. We prove that this family of representations varying over ` is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure of K of the kernels of the representations for all ` become linearly disjoint over a finite extension of K. In doing this, we also prove a number of interesting facts on the images and on the ramification of this family of representations. 1

A support problem for the intermediate Jacobians of l-adic representations

Eprint Arxiv Math 0212420, 2002

This is a revised version of ANT-0332: "A support problem for the intermediate Jacobians of l-adic representations", by G. Banaszak, W. Gajda & P. Krason, which was placed on these archives on the 29th of January 2002. Following a suggestion of the referee we have subdivided the paper into two separate parts: "Support problem for the intermediate Jacobians of l-adic representations", and "On Galois representations for abelian varieties with complex and real multiplications". Our results on the image of Galois and the Mumford-Tate conjecture for some RM abelian varieties are contained in the second paper. Both papers were accepted for publication.

Connectedness results for lll-adic representations associated to abelian varieties

Compositio Mathematica, 1995

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

arXiv (Cornell University), 2011

Let K be a finitely generated extension of Q. We consider the family of ℓ-adic representations (ℓ varies through the set of all prime numbers) of the absolute Galois group of K, attached to ℓ-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.