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Abelian varieties with many endomorphisms and their absolutely simple factors Guitart Morales, Xavier Geometria algebraica Teoria de nombres Varietats abelianes K-teoria Algebraic geometry Number theory Abelian varieties K-theory 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. 2023-02-09T17:01:28Z 2023-02-09T17:01:28Z 2012 2023-02-09T17:01:28Z info:eu-repo/semantics/article info:eu-repo/semantics/acceptedVersion Versió postprint del document publicat a: https://doi.org/10.4171/rmi/686 Revista Matematica Iberoamericana, 2012, vol. 28, num. 2, p. 591-601 https://doi.org/10.4171/rmi/686 (c) European Mathematical Society Publishing House, 2012 info:eu-repo/semantics/openAccess European Mathematical Society Publishing House Articles publicats en revistes (Matemàtiques i Informàtica)