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Modular Abelian Varieties Over Number Fields

Published online by Cambridge University Press:  20 November 2018

Xavier Guitart  and
Jordi Quer
Xavier Guitart
Affiliation:
Max Planck Institute forMathematics, Vivatsgasse 7, 53111 Bonn, Germany and Departament de Matemàtica Aplicada II, Universitat Politécnica de Catalunya, C. Jordi Girona 1-3, 08034 Barcelona, Spain e-mail: xevi.guitart@gmail.com
Jordi Quer
Affiliation:
Departament de Matemàtica Aplicada II, Universitat Politécnica de Catalunya, C. Jordi Girona 1-3, 08034 Barcelona, Spain e-mail: jordi.quer@upc.edu
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Abstract

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The 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 $L\left( B/K;\,s \right)$ is a product of $L$-functions of non-$\text{CM}$ newforms over $Q$ for congruence subgroups of the form ${{\Gamma }_{1}}\,\left( N \right)$. The characterization involves the structure of End$\left( B \right)$, 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.

Keywords

11G1011G1811F11modular abelian varietiesGLsub2/sub-type varietiesmodular forms
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 1 , February 2014 , pp. 170 - 196
Copyright
Copyright © Canadian Mathematical Society 2014

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