{"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/an-effective-open-image-theorem-for-products","title":"An effective open image theorem for products of principally polarized abelian varieties","arxiv_id":"2212.11472","date":"2022-12-22","proceeding":null,"authors":["Jacob Mayle","Tian Wang"],"abstract":"Let $A = \\prod_{1\\leq i\\leq n} A_i$ be the product of principally polarized abelian varieties $A_1, \\ldots, A_n$ of dimensions $g_1, \\ldots, g_n$, respectively, each defined over a number field $K$, and pairwise nonisogenous over $\\overline{K}$. We make effective an open image theorem for $A$ due to Hindry and Ratazzi. More specifically, we give an explicit bound of the constant $c(A)$ under GRH, in terms of standard invariants of $K$ and each $A_i$, where $c(A)$ is defined to be the smallest positive integer such that for any prime $\\ell>c(A)$, the image of the $\\ell$-adic Galois representation of $A$ is \"as large as possible\" in a suitable sense.","url_abs":"https://arxiv.org/abs/2212.11472v5","url_pdf":"https://arxiv.org/pdf/2212.11472v5.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":"an-effective-open-image-theorem-for-products","repo_url":"https://github.com/maylejacobj/productppavs","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"an-effective-open-image-theorem-for-products","repo_url":"https://github.com/maylejacobj/productsecs","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}