# Revista Matemática Iberoamericana

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**Volume 28, Issue 2, 2012, pp. 591–601**

**DOI: 10.4171/rmi/686**

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 GL_{2}-type, `k`-varieties, building blocks

Guitart Xavier: Abelian varieties with many endomorphisms and their absolutely simple factors. *Rev. Mat. Iberoamericana* 28 (2012), 591-601. doi: 10.4171/rmi/686