{"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-iwasawa-l-invariants-for-abelian-number","title":"On Iwasawa $λ$-invariants for abelian number fields and random matrix heuristics","arxiv_id":"2207.06287","date":"2022-07-13","proceeding":null,"authors":["Daniel Delbourgo","Heiko Knospe"],"abstract":"Following both Ernvall-Mets\\\"{a}nkyl\\\"{a} and Ellenberg-Jain-Venkatesh, we study the density of the number of zeroes (i.e. the cyclotomic $\\lambda$-invariant) for the $p$-adic zeta-function twisted by a Dirichlet character $\\chi$ of any order. We are interested in two cases: (i) the character $\\chi$ is fixed and the prime $p$ varies, and (ii) $\\text{ord}(\\chi)$ and the prime $p$ are both fixed but $\\chi$ is allowed to vary. We predict distributions for these $\\lambda$-invariants using $p$-adic random matrix theory and provide numerical evidence for these predictions. We also study the proportion of $\\chi$-regular primes, which depends on how $p$ splits inside $\\mathbb{Q}(\\chi)$. Finally in an extensive Appendix, we tabulate the values of the $\\lambda$-invariant for every character $\\chi$ of conductor $\\leq 1000$ and for odd primes $p$ of small size.","url_abs":"https://arxiv.org/abs/2207.06287v2","url_pdf":"https://arxiv.org/pdf/2207.06287v2.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-iwasawa-l-invariants-for-abelian-number","repo_url":"https://github.com/knospe/iwasawa","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}