Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

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


Authors: Francesc Fité and Xavier Guitart
Journal: Trans. Amer. Math. Soc. 370 (2018), 4623-4659
MSC (2010): Primary 11G10, 11G15, 14K22
DOI: https://doi.org/10.1090/tran/7074
Published electronically: January 18, 2018
MathSciNet review: 3812090
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ A/\mathbb{Q}$ be an abelian variety of dimension $ g\geq 1$ that is isogenous over  $ \overline {\mathbb{Q}}$ to $ E^g$, 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 $ \mathbb{Q}$-curves, $ E$ is isogenous to a curve defined over a polyquadratic extension of $ \mathbb{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  $ \mathbb{Q}$ occur among at most $ 51$ $ \overline {\mathbb{Q}}$-isogeny classes of abelian surfaces over  $ \mathbb{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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11G10, 11G15, 14K22

Retrieve articles in all journals with MSC (2010): 11G10, 11G15, 14K22


Additional Information

Francesc Fité
Affiliation: Departament de Matemàtiques, Universitat Politècnica de Catalunya, Edifici Omega, C/Jordi Girona 1–3, 08034 Barcelona, Catalonia
Email: francesc.fite@gmail.com

Xavier Guitart
Affiliation: Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran via de les Corts Catalanes, 585, 08007 Barcelona, Catalonia
Email: xevi.guitart@gmail.com

DOI: https://doi.org/10.1090/tran/7074
Received by editor(s): February 11, 2016
Received by editor(s) in revised form: September 19, 2016
Published electronically: January 18, 2018
Additional Notes: The first author was funded by the German Research Council via SFB/TR 45
The second author was partially funded by MTM2012-33830 and MTM2012-34611.
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 682152).
Article copyright: © Copyright 2018 by Francesc Fité and Xavier Guitart