Stark's conjecture and Abelian \(L\)-functions with higher order zeros at \(s=0\)
type: Item
instance of: scholarly article
title: Stark's conjecture and Abelian \(L\)-functions with higher order zeros at \(s=0\)
zbMATH Open document ID: 0631.12006
DOI: 10.1016/0001-8708(87)90030-2
author: Jonathan W. Sands
published in: Advances in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The author proposes improvements of Stark's conjectures on Artin \(L\)-functions. Let \(K/k\) be an abelian extension of number fields, with Galois group \(G\); let \(T\subsetneq S\) be finite sets of places of \(k\) such that \(S\) contains the ramified ones in \(K/k\) and \(T\) splits completely in \(K/k\) (and contains the unramified infinite places). For \[ L_S(s,\chi)=\prod_{q\not\in S} (1-\chi (\text{Frob}(q))Nq^{-s})^{-1},\quad \chi \in \hat G,\;\text{Re}(s)>1, \] the following conjecture is proposed (where \(r=| T|\), \(w_K=| \text{tor}(K^\times)|\), \(c\in\mathbb Z)\): \[ (2^{-c}w_K/r!) L_S(0,\chi)=\theta (\chi) R(\chi,E),\text{ for\;all } \chi \in \hat G, \] where \(\theta (\chi)=<\chi,\mu >\in\mathbb Z[\chi]\), for a suitable \(\mu\in \mathbb Q[G]\), \(R(\chi,E)\) is a suitable \(\chi\)-regulator built ``essentially'' on independent \(T\)-units of \(K\). For \(r=1\), \(c=0\), \(\mu =1\), these are the Stark or Brumer-Stark conjectures [see \textit{J. Tate}, ``Les conjectures de Stark sur les fonctions \(L\) d'Artin en \(s=0\)'', Prog. Math. 47. Boston etc.: Birkhäuser (1984; Zbl 0545.12009)]. The author gives many remarks and observations to justify the conjecture, proves some essential functional properties in varying \(S\), \(k\). Finally, he proves some interesting cases of his conjecture, especially when the classical Stark's one is yet proved. Some aspects of \(S\)-genera theory are needed.
reviewed by: Georges Gras
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 4025537
zbMATH Keywords: Stark's conjectures
zbMATH Keywords: Artin L-functions
zbMATH Keywords: abelian extension of number fields
zbMATH Keywords: Brumer-Stark conjectures
zbMATH Keywords: functional properties
zbMATH Keywords: S-genera theory
Wikidata QID: Q122955290
MaRDI profile type: Publication
full work available at URL: 0001-8708(87)90030-2
OpenAlex ID: W1966277934
cites work: Values of abelian \(L\)-functions at negative integers over totally real fields
cites work: On the Iwasawa Invariants of Totally Real Number Fields
cites work: Über Den Bizyklischen Biquadratischen Zahlkörper
cites work: Q4063245
cites work: On the Stark-Shintani conjecture and cyclotomic \({\mathbb{Z}}_ p\)-extensions of class fields over real quadratic fields
cites work: Q3344069
cites work: Galois groups of exponent two and the Brumer-Stark conjecture.
cites work: Two cases of Stark's conjecture
cites work: On certain ray class invariants of real quadratic fields
cites work: On the Stickelberger ideal and the circular units of an abelian field
cites work: \(L\)-functions at \(s=1\). III: Totally real fields and Hilbert's twelfth problem
cites work: \(L\)-functions at \(s=1\). IV: First derivatives at \(s=0\)
cites work: Q3659870
cites work: The Stark conjectures on Artin \(L\)-functions at \(s=0\). Lecture notes of a course in Orsay edited by Dominique Bernardi and Norbert Schappacher.
cites work: On the Class Number of a Relatively Cyclic Number Field
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description: scientific article; zbMATH DE number 4025537