{"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/on-the-galois-invariant-part-of-the-weyl","title":"On the Galois-invariant part of the Weyl group of the Picard lattice of a K3 surface","arxiv_id":"2304.14686","date":"2023-04-28","proceeding":null,"authors":["Wim Nijgh","Ronald van Luijk"],"abstract":"Let $X$ denote a K3 surface over an arbitrary field $k$. Let $k^\\text{s}$ denote a separable closure of $k$ and let $X^\\text{s}$ denote the base change of $X$ to $k^\\text{s}$. The action of the absolute Galois group Gal($k^\\text{s}/k$) of $k$ on Pic $X^\\text{s}$ respects the intersection pairing, which gives Pic $X^\\text{s}$ the structure of a lattice. Let O(Pic $X$) and O(Pic $X^\\text{s}$) denote the group of isometries of Pic $X$ and Pic $X^\\text{s}$, respectively. Let $R_X$ denote the Galois invariant part of the Weyl group of O(Pic $X^\\text{s}$). One can show that each element in $R_X$ can be restricted to an element of O(Pic $X$). The following question arises: Is the image of the restriction map $R_X \\to $O(Pic $X$) a normal subgroup of O(Pic $X$) for every K3 surface $X$? We show that the answer is negative by giving counterexamples over $k=\\mathbb{Q}$.","url_abs":"https://arxiv.org/abs/2304.14686v1","url_pdf":"https://arxiv.org/pdf/2304.14686v1.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":"on-the-galois-invariant-part-of-the-weyl","repo_url":"https://github.com/wimnijgh/k3-and-weyl-group","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}