Fields of definition of elliptic k-curves and the realizability of all\n genus 2 Sato--Tate groups over a number field

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)

paper · pdf · doi:10.48550/arxiv.1511.02322

openalex publication_date 2015/11/07 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

Let A/\ℚ be an abelian variety of dimension g\≥ 1 that is\nisogenous over \\ℚ to E^g, where E is an elliptic\ncurve. If E does not have complex multiplication (CM), by results of Ribet\nand Elkies concerning fields of definition of elliptic \ℚ-curves E\nis isogenous to a curve defined over a polyquadratic extension of \ℚ.\nWe show that one can adapt Ribet's methods to study the field of definition of\nE up to isogeny also in the CM case. We find two applications of this\nanalysis to the theory of Sato--Tate groups: First, we show that 18 of the\n34 possible Sato--Tate groups of abelian surfaces over \ℚ occur\namong at most 51 \\ℚ-isogeny classes of abelian surfaces\nover \ℚ; Second, we give a positive answer to a question of Serre\nconcerning the existence of a number field over which abelian surfaces can be\nfound realizing each of the 52 possible Sato--Tate groups of abelian\nsurfaces.\n

Citations

• Twisting commutative algebraic groups

Related

• Sato-Tate groups of genus 2 curves
• An algebraic Sato-Tate group and Sato-Tate conjecture
• Abelian varieties over Q and modular forms
• Abelian varieties isogenous to a power of an elliptic curve over a\n Galois extension
• K-varieties and Galois representations