Stickelberger elements over rational function fields
type: Item
instance of: scholarly article
title: Stickelberger elements over rational function fields
zbMATH Open document ID: 1264.11095
author: Po-Yi Huang
published in: Journal of Number Theory
publication date: 2010-01-08T00:00:00Z
review text: For an abelian extension \(L/K\) of global fields, Gross has conjectured a refined analytic class number formula relating the Stickelberger element \(\theta\) of the extension to a certain refined regulator \(\mathcal{R}\). The refined regulator, defined as an element of \(I^n/I^{n+1}\) where \(I\) is the augmentation ideal \(I\) of \(G = \mathrm{Gal}(L/K)\) and \(n\) is a certain integer, is given in terms of local reciprocity maps. To explain the integer \(n\), we remark that the setup requires one to choose a finite set \(S\) of places of \(K\) containing the infinite and ramified ones and a finite set \(T\) of places of \(K\) such that \(S \cap T = \emptyset\) and the group of \((S,T)\)-units of \(L\) is \(\mathbb{Z}\)-torsion-free. The Stickelberger element and the refined regulator both depend on \(S\) and \(T\), and \(n = \# S - 1\). The conjecture predicts that \[ \theta \equiv \pm h \mathcal{R} \mod I^{n+1} \] where \(h\) is the \((S,T)\)-class-number of \(K\). The article under review proves Gross' Conjecture (in its formulation for infinite abelian extensions) when \(L\) is the maximal unramified-outside-\(S\) abelian extension of \(K = \mathbb{F}_q(x)\) in the case where \(q \not\equiv 1 \mod 4\), \(S\) consists of places of degree \(1\), and \(T\) consists of places for which the greatest common divisor of their degrees is coprime to \(q - 1\). The method uses the notions of ``tame family'' and ``universal polynomial'', which are too technical to discuss here. A key ingredient is that Stickelberger elements form a tame family and that this family is ``represented'' by a universal polynomial. In the function field case, Gross' Conjecture had already been proven in 2007; see Theorem A of \textit{D. Burns} [Invent. Math. 169, No. 3, 451--499 (2007; Zbl 1133.11063)]. (Note, however, that Burns' proof in that paper used the truth, established by Tan, of Gross' Conjecture for extensions of \(p\)-power degree where \(p\) is the residue characteristic of the gi…
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 5654991
zbMATH Keywords: Stickelberger element
zbMATH Keywords: special values of abelian \(L\)-functions
zbMATH Keywords: conjecture of Gross
zbMATH Keywords: class numbers
reviewed by: Paul Buckingham
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.06.007
OpenAlex ID: W1966638060
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DOI: 10.1016/J.JNT.2009.06.007
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description: scientific article; zbMATH DE number 5654991