Abelian varieties with many endomorphisms and their absolutely simple factors

2011/02/04 by Xavier Guitart, Guitart, Xavier
Mathematics · Medicine · #11G10 (Primary) #14K15 (Secondary) #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Berberine and alkaloids research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11G10 #msc:14K15

paper · pdf · doi:10.48550/arxiv.1102.0863

To appear in Revista Matemática Iberoamericana

arxiv created 2011/02/04 · openalex publication_date 2011/02/04 · arxiv updated 2011/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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_k⁰(A) is a maximal subfield of \End_k⁰(A). We call them Ribet-Pyle varieties over k. We see that every Ribet-Pyle variety over k is isogenous over 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.

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