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# Modular Abelian Varieties Over Number Fields

Published:2012-11-13
Printed: Feb 2014
• Xavier Guitart,
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
• Jordi Quer,
Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, C. Jordi Girona 1-3, 08034 Barcelona, Spain
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## Abstract

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 $L(B/K;s)$ is a product of $L$-functions of non-CM newforms over $\mathbb Q$ for congruence subgroups of the form $\Gamma_1(N)$. The characterization involves the structure of $\operatorname{End}(B)$, 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.
 Keywords: Modular abelian varieties, $GL_2$-type varieties, modular forms
 MSC Classifications: 11G10 - Abelian varieties of dimension $> 1$ [See also 14Kxx] 11G18 - Arithmetic aspects of modular and Shimura varieties [See also 14G35] 11F11 - Holomorphic modular forms of integral weight