We characterize the abelian varieties arising as absolutely simple factors of -type varieties over a number field . In order to obtain this result, we study a wider class of abelian varieties: the -varieties satisfying that is a maximal subfield of . We call them Ribet–Pyle varieties over . We see that every Ribet–Pyle variety over is isogenous over to a power of an abelian -variety and, conversely, that every abelian -variety occurs as the absolutely simple factor of some Ribet–Pyle variety over . We deduce from this correspondence a precise description of the absolutely simple factors of the varieties over of -type.
Cite this article
Xavier Guitart, Abelian varieties with many endomorphisms and their absolutely simple factors. Rev. Mat. Iberoam. 28 (2012), no. 2, pp. 591–601DOI 10.4171/RMI/686