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Revista Matemática Iberoamericana

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Abelian varieties with many endomorphisms and their absolutely simple factors

Xavier Guitart (1)

(1) Departament Matemàtica Aplicada II, Universitat Politècnica de Catalunya, c/ Jordi Girona 1-3, 08034, BARCELONA, SPAIN

We characterize the abelian varieties arising as absolutely simple factors of $\operatorname{GL}_2$-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 $\operatorname{GL}_2$-type.

Keywords: Abelian varieties of GL₂-type, k-varieties, building blocks