Modular abelian varieties over number fields
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 cha...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2014 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/208182 |
| Acceso en línea: | https://hdl.handle.net/2445/208182 |
| Access Level: | acceso abierto |
| Palabra clave: | Funcions holomorfes Varietats abelianes Holomorphic functions Abelian varieties |
| Sumario: | 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. |
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