# Abelian varieties with many endomorphisms and their absolutely simple factors

### Xavier Guitart

Universitat Politècnica de Catalunya, Barcelona, Spain

## Abstract

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.

## Cite this article

Xavier Guitart, Abelian varieties with many endomorphisms and their absolutely simple factors. Rev. Mat. Iberoam. 28 (2012), no. 2, pp. 591–601

DOI 10.4171/RMI/686