{"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/zeta-values-of-one-dimensional-arithmetic","title":"Zeta-values of one-dimensional arithmetic schemes at strictly negative integers","arxiv_id":"2111.13398","date":"2021-11-26","proceeding":null,"authors":["Alexey Beshenov"],"abstract":"Let $X$ be an arithmetic scheme (i.e., separated, of finite type over $\\operatorname{Spec} \\mathbb{Z}$) of Krull dimension $1$. For the associated zeta function $\\zeta (X,s)$, we write down a formula for the special value at $s = n < 0$ in terms of the \\'{e}tale motivic cohomology of $X$ and a regulator. We prove it in the case when for each generic point $\\eta \\in X$ with $\\operatorname{char} \\kappa (\\eta) = 0$, the extension $\\kappa (\\eta)/\\mathbb{Q}$ is abelian. We conjecture that the formula holds for any one-dimensional arithmetic scheme. This is a consequence of the Weil-\\'{e}tale formalism developed by the author in [arXiv:2012.11034] and [arXiv:2102.12114], following the work of Flach and Morin (Doc. Math. 23 (2018), 1425--1560). We also calculate the Weil-\\'{e}tale cohomology of one-dimensional arithmetic schemes and show that our special value formula is a particular case of the main conjecture from [arXiv:2102.12114].","url_abs":"https://arxiv.org/abs/2111.13398v1","url_pdf":"https://arxiv.org/pdf/2111.13398v1.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":"zeta-values-of-one-dimensional-arithmetic","repo_url":"https://github.com/alexey-beshenov/weil-etale","is_official":0,"mentioned_in_paper":0,"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}