2015/11/07 by Francesc Fité, Fité, Francesc, Xavier Guitart +1
Computer Science · Mathematics · #11G10 #11G15 #14K22 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11G10 #msc:11G15 #msc:14K22
paper · pdf · doi:10.48550/arxiv.1511.02322
Erratum added at the end of the Introduction summarizing some small changes made on the published version in Trans. Amer. Math. Soc. None of the changes affects the main results of the paper
openalex publication_date 2015/11/07 · arxiv created 2020/03/15 · arxiv updated 2020/03/17 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
Let A/ℚ be an abelian variety of dimension g≥ 1 that is isogenous over ℚ to Eg, where E is an elliptic curve. If E does not have complex multiplication (CM), by results of Ribet and Elkies concerning fields of definition of elliptic ℚ-curves E is isogenous to a curve defined over a polyquadratic extension of ℚ. We show that one can adapt Ribet's methods to study the field of definition of E up to isogeny also in the CM case. We find two applications of this analysis to the theory of Sato--Tate groups: First, we show that 18 of the 34 possible Sato--Tate groups of abelian surfaces over ℚ occur among at most 51 ℚ-isogeny classes of abelian surfaces over ℚ; Second, we give a positive answer to a question of Serre concerning the existence of a number field over which abelian surfaces can be found realizing each of the 52 possible Sato--Tate groups of abelian surfaces.