A conjectural product formula for Brumer-Stark units over real quadratic fields
type: Item
instance of: scholarly article
title: A conjectural product formula for Brumer-Stark units over real quadratic fields
zbMATH Open document ID: 1275.11140
author: Samit Dasgupta
published in: Journal of Number Theory
publication date: 2013-01-18T00:00:00Z
review text: Brumer-Stark units are conjectural elements in global fields whose role is to give algebraic meaning to special values of partial \(\zeta\)-functions. More precisely, the Brumer-Stark conjecture asserts that when \(H/F\) is an abelian extension of global fields, there should exist elements of \(H\) whose valuations are connected in a precise way with the associated partial \(\zeta\)-functions evaluated at \(0\). Those conjectural elements are the Brumer-Stark units. A version of the conjecture given in terms of special values of \(L\)-functions is often treated instead. In the function field case, the conjecture is a theorem thanks to work of Tate and Deligne. Significant progress has been made in the number field case: Greither and Popescu have proven it away from the \(2\)-primary component assuming the vanishing of certain classical Iwasawa \(\mu\)-invariants. The article under review presents two conjectural formulas involving the Brumer-Stark units in the case where \(F\) is a real quadratic field and \(H\) is a narrow ray class field of \(F\). The first formula, the ``product formula'' of the title, appears to be new and expresses a \textit{ratio} of Brumer-Stark units in terms of an infinite product. The product is taken over certain elements of \(F\) that map into a particular Shintani domain. The author points out that when \(F\) has narrow class number \(1\), one of the Brumer-Stark units appearing in the ratio is \(1\) and so the product formula is a formula for the remaining Brumer-Stark unit alone. The second conjectural formula, which the author presented for arbitrary totally real fields \(F\) in [Duke Math. J. 143, No. 2, 225--279 (2008; Zbl 1235.11102)], expresses a Brumer-Stark unit (not a ratio) in terms of a \(p\)-adic integral over the unit group of the completion of \(F\) at a prime \(p\) that is inert in \(F/\mathbb{Q}\). (We suppress, for brevity, the relationship of \(p\) to the extension \(H/F\) and to other arithmetic data.) It is worth no…
reviewed by: Paul Buckingham
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 6126841
zbMATH Keywords: Stark's conjectures
zbMATH Keywords: Brumer-Stark units
zbMATH Keywords: Shintani domains
MaRDI profile type: Publication
full work available at URL: j.jnt.2012.02.013
OpenAlex ID: W2046331599
cites work: Values at negative integers of zeta functions and \(p\)-adic zeta functions
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cites work: The refined \(\mathfrak p\)-adic abelian Stark conjecture in function fields
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DOI: 10.1016/J.JNT.2012.02.013
description: scientific article; zbMATH DE number 6126841