JournalsrmiVol. 28, No. 2pp. 591–601

Abelian varieties with many endomorphisms and their absolutely simple factors

  • Xavier Guitart

    Universitat Politècnica de Catalunya, Barcelona, Spain
Abelian varieties with many endomorphisms  and their absolutely simple factors cover
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Abstract

We characterize the abelian varieties arising as absolutely simple factors of GL2\operatorname{GL}_2-type varieties over a number field kk. In order to obtain this result, we study a wider class of abelian varieties: the kk-varieties A/kA/k satisfying that Endk0(A)\operatorname{End}_k^0(A) is a maximal subfield of Endkˉ0(A)\operatorname{End}_{\bar{k}}^0(A). We call them Ribet–Pyle varieties over kk. We see that every Ribet–Pyle variety over kk is isogenous over kˉ\bar{k} to a power of an abelian kk-variety and, conversely, that every abelian kk-variety occurs as the absolutely simple factor of some Ribet–Pyle variety over kk. We deduce from this correspondence a precise description of the absolutely simple factors of the varieties over kk of GL2\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