On refined Stark conjectures in the non-abelian case
type: Item
instance of: scholarly article
title: On refined Stark conjectures in the non-abelian case
zbMATH Open document ID: 1184.11049
published in: Mathematical Research Letters
publication date: 2009-02-10T00:00:00Z
review text: The conjectures named after Stark come in two types: those ``over \(\mathbb Q\)'' and those ``over \(\mathbb Z\)''. In the former type, very roughly speaking, a regulator and a leading coefficient of an \(L\)-function at \(s=0\) are related by conjecturing (and in many cases proving) that their quotient is algebraic. In the latter, one additionally imposes integrality conditions (bounds on the denominator) on this quotient. In concrete cases this leads to the prediction of the existence of certain units, given by the logarithms of their conjugates. In Stark's seminal work, the setting was nonabelian (so one deals with arbitrary Galois extensions \(K/k\) of number fields), and the conjectures were mostly of the former type. (But in fact Tate's result on rational valued characters specify the rational quotient; these results are used in an essential way in the paper under review.) In later years, conjectures of the latter type were formulated in the abelian case, first for the case where the order of vanishing \(r_\chi\) of the relevant \(L\)-function \(L(s,\chi)\) at \(s=0\) is 1, and then in general (Rubin, Popescu). There is also a contribution of Stark and Chinburg, affording a conjecture over \(\mathbb Z\) in the nonabelian setting, where again \(r_\chi=1\) is assumed. Such a situation arises, by work of Deligne and Serre, from certain modular forms. In the paper under review, the author presents a ``Stark conjecture over \(\mathbb Z\)'' in considerable generality, that is, for any Galois extension \(K/k\) of global fields; the only persistent restriction is that the irreducible character \(\chi\) is assumed to be nontrivial. It is formulated as Conjecture 2.1, which has the form of an inclusion, going from left to right. The term involving the leading coefficient is on the left, and the regulator term on the right. Everything is stated at integral level. Apart from the standard quantities, the left hand side has a factor \(|G|^{r_\chi}\) (with \(G=\text{Gal}(K/k…
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 5505923
zbMATH Keywords: Stark conjectures
zbMATH Keywords: regulators
zbMATH Keywords: Stark units
zbMATH Keywords: representations
zbMATH Keywords: characters
author: David Burns
MaRDI profile type: Publication
Wikidata QID: Q123238261
DOI: 10.4310/MRL.2008.V15.N5.A2
description: scientific article; zbMATH DE number 5505923