A question of Stark
type: Item
instance of: scholarly article
title: A question of Stark
zbMATH Open document ID: 0905.11050
author: Brett A. Tangedal
published in: Pacific Journal of Mathematics
publication date: 1999-01-18T00:00:00Z
review text: The author studies a problem related to \textit{H. M. Stark}'s conjectures about values of Artin's \(L\)-functions and their derivatives. For these conjectures, see Stark's article [Adv. Math. 35, No. 3, 197-235 (1980; Zbl 0475.12018)]. Let \(K\) be a totally real abelian extension of a real quadratic field \(k\). For a finite set \(S\) of primes in \(k\), denote by \(L_S(s,\chi)\) the Artin \(L\)-function attached to a character \(\chi\) of \(\text{Gal} (K/k)\), with the Euler factors corresponding to the primes in \(S\) removed. The set \(S\) is assumed to contain at least the infinite primes plus the ramifying finite primes. Let \(S_K\) consist of all primes in \(K\) extending the primes in \(S\). The problem in question asks about the existence of two particular \(S_K\)-units (i.e., elements of \(K\) that are locally units outside \(S_K\)) \(\varepsilon_1\), \(\varepsilon_2\) with the following properties: (i) For every character \(\chi\), the second derivative of \(L_S(s,\chi)\) at \(s=0\) can be expressed as a certain linear combination of regulator-type \(2 \times 2\) determinants defined by \(\varepsilon_1\) and \(\varepsilon_2\); (ii) the fields \(K(\sqrt{\varepsilon_1})\) and \(K(\sqrt{\varepsilon_2})\) are abelian over \(k\) and coincide; (iii) the principal ideals generated by \(\varepsilon_1\) and by its conjugates (over \(k\)) are equal, and the same is true of \(\varepsilon_2\). The author shows that such \(S_K\)-units do exist in the case of a quadratic extension \(K/k\), provided \(S\) contains at least four primes. It seems that the hardest problem here is to have the condition \(K(\sqrt{\varepsilon_1})=K(\sqrt{\varepsilon_2})\) satisfied when \(| S| \) equals 4.
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R20
zbMATH DE Number: 1136642
zbMATH Keywords: values of Artin's \(L\)-functions
zbMATH Keywords: quadratic extensions
zbMATH Keywords: abelian extensions
zbMATH Keywords: \(S_K\)-units
MaRDI profile type: Publication
full work available at URL: pjm.1997.180.187
OpenAlex ID: W2104509608
DOI: 10.2140/PJM.1997.180.187
description: scientific article; zbMATH DE number 1136642