{"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/brumer-stark-units-and-hilbert-s-12th-problem","title":"Brumer-Stark Units and Explicit Class Field Theory","arxiv_id":"2103.02516","date":"2021-03-03","proceeding":null,"authors":["Samit Dasgupta","Mahesh Kakde"],"abstract":"Let $F$ be a totally real field of degree $n$ and $p$ an odd prime. We prove the $p$-part of the integral Gross--Stark conjecture for the Brumer--Stark $p$-units living in CM abelian extensions of $F$. In previous work, the first author showed that such a result implies an exact $p$-adic analytic formula for these Brumer--Stark units up to a bounded root of unity error, including a ``real multiplication'' analogue of Shimura's celebrated reciprocity law from the theory of Complex Multiplication. In this paper we show that the Brumer--Stark units, along with $n-1$ other easily described elements (these are simply square roots of certain elements of $F$) generate the maximal abelian extension of $F$. We therefore obtain an unconditional construction of the maximal abelian extension of any totally real field, albeit one that involves $p$-adic integration for infinitely many primes $p$. Our method of proof of the integral Gross--Stark conjecture is a generalization of our previous work on the Brumer--Stark conjecture. We apply Ribet's method in the context of group ring valued Hilbert modular forms. A key new construction here is the definition of a Galois module $\\nabla_{\\!\\sL}$ that incorporates an integral version of the Greenberg--Stevens $\\sL$-invariant into the theory of Ritter--Weiss modules. This allows for the reinterpretation of Gross's conjecture as the vanishing of the Fitting ideal of $\\nabla_{\\!\\sL}$. This vanishing is obtained by constructing a quotient of $\\nabla_{\\!\\sL}$ whose Fitting ideal vanishes using the Galois representations associated to cuspidal Hilbert modular forms..","url_abs":"https://arxiv.org/abs/2103.02516v2","url_pdf":"https://arxiv.org/pdf/2103.02516v2.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":"brumer-stark-units-and-hilbert-s-12th-problem","repo_url":"https://github.com/liuyj8526/Computation-of-Elliptic-Units","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}