On the image of Galois lll-adic representations for abelian varieties of type III (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

Représentations galoisiennes et groupe de Mumford-Tate associé à une variété abélienne

2015

Soient KKK un corps de nombres et AAA une variete abelienne sur KKK dont nous notons ggg la dimension. Pour tout premier ellellell, le module de Tate ellellell-adique de AAA nous fournit une representation ellellell-adique du groupe de Galois absolu de KKK, et c'est a l'image de ces representations galoisiennes que l'on s'interesse dans cette these.Pour de nombreuses classes de varietes abeliennes on possede une description de ces images a une erreur finie pres : le premier but de ce travail est de quantifier explicitement cette erreur dans plusieurs cas differents. On parvient a resoudre completement le probleme pour une courbe elliptique sans multiplication complexe, ou plus generalement pour un produit de telles courbes elliptiques, et pour toute variete abelienne geometriquement simple admettant multiplication complexe. Pour d'autres classes de varietes abeliennes A/KA/KA/K on obtient seulement une description de l'image de Galois pour tout premier ellellell plus grand qu&#3...

On Galois representations for abelian varieties with complex and real multiplications

Journal of Number Theory, 2003

In this paper we study the image of l-adic representations coming from Tate module of an abelian variety defined over a number field. We treat abelian varieties with complex and real multiplications. We verify the Mumford-Tate conjecture for a new class of abelian varieties with real multiplication. J. P. Wintenberger, Démonstration d'une conjecture de Lang dans des cas particuliers, preprint (October 3, 2000), 1-25.

A class of p-adic Galois representations arising from abelian varieties over

Mathematische Annalen, 2005

Let V be a p-adic representation of the absolute Galois group G of Qp that becomes crystalline over a finite tame extension, and assume p = 2. We provide necessary and sufficient conditions for V to be isomorphic to the p-adic Tate module Vp(A) of an abelian variety A defined over Qp. These conditions are stated on the filtered (ϕ, G)module attached to V .

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.

On the modularity level of modular abelian varieties over number fields

Journal of Number Theory, 2010

Let f be a weight two newform for Γ 1 (N) without complex multiplication. In this article we study the conductor of the absolutely simple factors B of the variety A f over certain number fields L. The strategy we follow is to compute the restriction of scalars Res L/Q (B), and then to apply Milne's formula for the conductor of the restriction of scalars. In this way we obtain an expression for the local exponents of the conductor N L (B). Under some hypothesis it is possible to give global formulas relating this conductor with N. For instance, if N is squarefree we find that N L (B) belongs to Z and N L (B) f dim B L

Abelian Varieties and the Mordell{Lang Conjecture

Model Theory Algebra and Geometry 2000 Isbn 0 521 78068 3 Pags 199 227, 2000

This is an introductory exposition to background material useful to appreciate various formulations of the Mordell-Lang conjecture (now established by recent spectacular work due to Vojta, Faltings, Hrushovski, Buium, Voloch, and others). It gives an exposition of some of the elementary and standard constructions of algebro-geometric models (rather than model-theoretic ones) with applications (for example, via the method of Chabauty) relevant to Mordell-Lang. The article turns technical at one point (the step in the proof of the Mordell-Lang Conjecture in characteristic zero which passes from number fields to general fields). Two different procedures are sketched for doing this, with more details given than are readily found in the literature. There is also some discussion of issues of effectivity.