Modular Abelian Varieties Over Number Fields

descriptionPublicationkeyboard_double_arrow_right Article , Preprint , Other literature type 01 Feb 2014Embargo end date: 01 Jan 2009 Spain English Publisher:Canadian Mathematical SocietyJournal:Canadian Journal of Mathematics, volume 66, pages 170-196 (issn: 0008-414X, eissn: 1496-4279,

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Authors: Guitart Morales, Xavier; Quer Bosor, Jordi;

doi: 10.4153/cjm-2012-040-2 , 10.48550/arxiv.0905.2550

arXiv: 0905.2550

handle: 2117/21433 , 2445/208182 , 21.11116/0000-0004-151D-5 , 21.11116/0000-0004-151F-3 , 21.11116/0000-0004-1520-0

Modular Abelian Varieties Over Number Fields

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Abstract

AbstractThe main result of this paper is a characterization of the abelian varieties B/K defined over Galois number fields with the property that the L-function L(B/K; s) is a product of L-functions of non-CM newforms over ℚ for congruence subgroups of the form Γ1(N). The characterization involves the structure of End(B), isogenies between the Galois conjugates of B, and a Galois cohomology class attached to B/K.We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illustration of the ways in which the general results of the paper can be applied, we prove the strong modularity of some particular abelian surfaces belonging to that family, and we show how to find nontrivial examples of strongly modular varieties by twisting.

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Spain

Related Organizations

Max Planck Institute for Mathematics
Germany
University of Barcelona
Spain
Universitat Politècnica de Catalunya
Spain
Universitat Polite`cnica de Catalunya
Spain
Max Planck Society
Germany

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Keywords

11F11, 11G18, :Matemàtiques i estadística::Geometria::Geometria algebraica [Àrees temàtiques de la UPC], Classificació AMS::14 Algebraic geometry::14K Abelian varieties and schemes, Classificació AMS::11 Number theory::11G Arithmetic algebraic geometry (Diophantine geometry), Abelian varieties, Holomorphic functions, FOS: Mathematics, Geometria algebraica aritmètica, Number Theory (math.NT), Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria algebraica, :Matemàtiques i estadística::Àlgebra::Teoria de nombres [Àrees temàtiques de la UPC], Funcions holomorfes, Modular abelian varieties, Mathematics - Number Theory, Modular forms, Varietats abelianes, :14 Algebraic geometry::14K Abelian varieties and schemes [Classificació AMS], :11 Number theory::11G Arithmetic algebraic geometry (Diophantine geometry) [Classificació AMS], GL2-type varieties, Esquemes (Geometria algebraica), Arithmetical algebraic geometry, Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de nombres

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