Abelian varieties with many endomorphisms and their absolutely simple factors
We characterize the abelian varieties arising as absolutely simple factors of $\mathrm{GL}_2$-type varieties over a number field $k$. In order to obtain this result, we study a wider class of abelian varieties: the $k$ varieties $A / k$ satisfying that $\operatorname{End}_k^0(A)$ is a maximal subfie...
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| Formato: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2012 |
| País: | España |
| Recursos: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/193366 |
| Acesso em linha: | https://hdl.handle.net/2445/193366 |
| Access Level: | acceso abierto |
| Palavra-chave: | Geometria algebraica Teoria de nombres Varietats abelianes K-teoria Algebraic geometry Number theory Abelian varieties K-theory |
| Resumo: | We characterize the abelian varieties arising as absolutely simple factors of $\mathrm{GL}_2$-type varieties over a number field $k$. In order to obtain this result, we study a wider class of abelian varieties: the $k$ varieties $A / k$ satisfying that $\operatorname{End}_k^0(A)$ is a maximal subfield of $\operatorname{End}_{\bar{k}}^0(A)$. We call them Ribet-Pyle varieties over $k$. We see that every Ribet-Pyle variety over $k$ is isogenous over $\bar{k}$ to a power of an abelian $k$-variety and, conversely, that every abelian $k$-variety occurs as the absolutely simple factor of some Ribet-Pyle variety over $k$. We deduce from this correspondence a precise description of the absolutely simple factors of the varieties over $k$ of $\mathrm{GL}_2$-type. |
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