Divisibility properties of values of partial zeta functions at non-positive integers
type: Item
instance of: scholarly article
title: Divisibility properties of values of partial zeta functions at non-positive integers
zbMATH Open document ID: 1292.11129
author: Barry R. Smith
published in: Mathematische Zeitschrift
publication date: 2012-05-08T00:00:00Z
review text: To an abelian extension \(K/k\) of number fields with group \(G\) one may attach an equivariant (imprimitive) \(L\)-value \(\theta_{K/k}\) at \(s=0\). This generalised Stickelberger element is an element of the group ring \(\mathbb Q[G]\). It may have denominators, but they are bounded by \(w\), the number of roots of unity in \(K\). If \(I\) is the annihilator of \(\mu_K\) in \(Z[G]\), then \(I\theta_{K/k}\) is conjectured to annihilate the class group (Brumer's conjecture); there is an analog for higher K-groups (Coates-Sinnott conjecture) where \(s=0\) is replaced by \(s=-n\), a negative integer. It is a natural question what the denominator really is; more precisely, what the denominators of the coefficients are, when \(\theta_{K/k}\) is written in the form \(\sum_{\sigma\in G}a_\sigma\,\sigma\) with rational \(a_\sigma\). \textit{D. R. Hayes} has observed [Burns, David (ed.) et al., Stark's conjectures: recent work and new directions, Contemp. Math. 358, 193--206 (2004; Zbl 1076.11057)] that all \(a_\sigma\) have the same denominator (assuming of course that we write them in lowest terms). He also formulated conjectures concerning these denominators. The author of the paper under review formulates a closely related one, called Conjecture 1: Take \(s=0\). If \(w=|\mu_K|\) is exactly divisible by \(p^r\), \(\mu_K\) is \(G\)-cohomologically trivial, and there exists an unramified cyclic degree \(p^r\) extension \(L/K\) such that the associated Kummer radical is fixed under \(G\), then \(\theta_{K/k}\) has no \(p\) in its denominators at all. There are two more conjectures, one of them bearing on the case \(s=-n\). The author indicates why these conjectures are close to those of Hayes (which we do not formulate here for lack of space), and he proceeds to show that the Coates-Sinnott conjecture implies Conjecture 2, and that the Brumer conjecture implies Conjecture 1 and 3. In a final section he discusses the ``dynamics'' of his conjectures, that is, their behaviour…
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 6031803
zbMATH Keywords: L-function
zbMATH Keywords: Brumer's conjecture
zbMATH Keywords: class group
zbMATH Keywords: higher K-group
zbMATH Keywords: Coates-Sinnott conjecture
MaRDI profile type: Publication
full work available at URL: s00209-010-0817-z
OpenAlex ID: W1997112733
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DOI: 10.1007/S00209-010-0817-Z
description: scientific article; zbMATH DE number 6031803