{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/fields-of-definition-of-elliptic-k-curves-and","title":"Fields of definition of elliptic $k$-curves and the realizability of all genus 2 Sato--Tate groups over a number field","arxiv_id":"1511.02322","date":"2015-11-07","proceeding":null,"authors":["Francesc Fité","Xavier Guitart"],"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.","url_abs":"http://arxiv.org/abs/1511.02322v3","url_pdf":"http://arxiv.org/pdf/1511.02322v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"fields-of-definition-of-elliptic-k-curves-and","repo_url":"https://github.com/xguitart/sato-tate","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}