On a conjecture of Gross on special values of L-functions
type: Item
instance of: scholarly article
title: On a conjecture of Gross on special values of L-functions
zbMATH Open document ID: 0689.12002
DOI: 10.1007/BF01214904
author: Masakazu Yamagishi
published in: Mathematische Zeitschrift
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174061
review text: The author studies the case \(k={\mathbb{Q}}\) of a conjecture of \textit{B. H. Gross} [J. Fac. Sci., Univ. Tokyo, Sect. I A 35, No.1, 177-197 (1988; Zbl 0681.12005)]: \(\theta_ G\equiv h_ T\cdot \det_ G \lambda\) mod \(I^{n+1}\), where \(\theta_ G\in {\mathbb{Z}}[G]\) is determined by \(\chi (\theta_ G)=L_ T(\chi,0)\) for all \(\chi\in \hat G\), \(L_ T(\chi,s)\) is the Hecke-Tate L-function relative to S and modified at T, S and T are sets of places of k, \(G=Gl(K/k)\), K an Abelian extension unramified outside S, \(\det_ G \lambda \in I^ n/I^{n+1}\) is the regulator defined over S-units, I the augmentation ideal of \({\mathbb{Z}}[G]\), \(h_ T\) is the element number of the group of invertible A-modules with trivialization at T, A the S-integers. He proves the conjecture for \(n=1\) in an elementary way and shows \(\theta_ G\in I^ 2\) for \(n\geq 2\). He also gives an example in case \(n=2\) for which the conjecture holds.
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 4128935
zbMATH Keywords: class number
zbMATH Keywords: Hecke-Tate L-function
zbMATH Keywords: Abelian extension
zbMATH Keywords: regulator
zbMATH Keywords: units
Wikidata QID: Q123308830
MaRDI profile type: Publication
cites work: A Generalization of a Theorem of Gauss on Sums Involving [ x ]
cites work: Q4729878
cites work: Q3259107
description: scientific article; zbMATH DE number 4128935