{"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/computing-isogeny-classes-of-typical","title":"Computing isogeny classes of typical principally polarized abelian surfaces over the rationals","arxiv_id":"2301.10118","date":"2023-01-24","proceeding":null,"authors":["Raymond van Bommel","Shiva Chidambaram","Edgar Costa","Jean Kieffer"],"abstract":"We describe an efficient algorithm which, given a principally polarized (p.p.) abelian surface $A$ over $\\mathbb{Q}$ with geometric endomorphism ring equal to $\\mathbb{Z}$, computes all the other p.p. abelian surfaces over $\\mathbb{Q}$ that are isogenous to $A$. This algorithm relies on explicit open image techniques for Galois representations, and we employ a combination of analytic and algebraic methods to efficiently prove or disprove the existence of isogenies. We illustrate the practicality of our algorithm by applying it to 1 440 894 isogeny classes of Jacobians of genus 2 curves.","url_abs":"https://arxiv.org/abs/2301.10118v2","url_pdf":"https://arxiv.org/pdf/2301.10118v2.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":"computing-isogeny-classes-of-typical","repo_url":"https://github.com/edgarcosta/genus2isogenies","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}