Abelian varieties with many endomorphisms and their absolutely simple factors

We characterize the abelian varieties arising as absolutely simple factors of GL2-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 $\End_k^0(A)$ is a maximal subfield of $\End_k^0(A)$. 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 GL2-type.