On the refined class number formula of Gross
type: Item
instance of: scholarly article
title: On the refined class number formula of Gross
zbMATH Open document ID: 1067.11072
author: Joongul Lee
author: David Burns
published in: Journal of Number Theory
publication date: 2004-08-16T00:00:00Z
review text: For a finite abelian extension \(K/k\) of congruence function fields, let \(G:= \text{ Gal}(K/k)\), \(S\) a finite non-empty set of places of \(k\) containing all ramified places in \(K/k\) and \(L _ S (\chi,s)\) the associated \(S\)-truncated Dirichlet \(L\)-function. If \(T\) is any finite non-empty set of places of \(k\) which is disjoint from \(S\), \textit{B. Gross} [J. Fac. Sci., Univ. Tokyo, Sect. I A 35, No. 1, 177--197 (1988; Zbl 0681.12005)] proved that \(\theta _ {K/k, S, T} (0)\) belongs to \({\mathbb Z}[G]\) where \[ \theta _{K/k,S,T}(s) = \biggl( \prod _ {t\in T} (1-N t ^ {1-s} \sigma _ t)\biggr) \sum_ {\chi \in \widehat{G}} e _ \chi L_ S (\chi, s). \] Here \(\widehat {G}\) is the dual of \(G\), \(e_ \chi\) denotes the corresponding idempotent \({{1}\over{| G| }} \sum _ {g\in G} \chi(g^ {-1}) g\in {\mathbb C} [G]\) and \(\sigma _ t\) is the Frobenius automorphism. B. Gross conjectured that \(\theta _ {K/k,S,T}(0)\) is congruent with \(\pm\) the \((S, T)\) class number of \(k\) multiplied by an explicit factor modulo \(I _ G ^ {| S| }\) where \(I_ G\) is the augmentation ideal of \({\mathbb Z}[G]\). In the paper under review it is verified that the conjecture of Gross is valid for any finite group \(G\) of prime exponent.
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 2091711
zbMATH Keywords: class number
zbMATH Keywords: truncated Dirichlet \(L\)-function
zbMATH Keywords: augmentation ideal
reviewed by: Gabriel D. Villa Salvador
MaRDI profile type: Publication
full work available at URL: j.jnt.2004.02.002
OpenAlex ID: W2004473893
cites work: On the values of equivariant zeta functions of curves over finite fields
cites work: The analytic theory of multiplicative Galois structure
cites work: Q4729878
cites work: Q4361720
cites work: Q4435724
cites work: Q4698146
cites work: The Cohomology Groups of Tori in Finite Galois Extensions of Number Fields
DOI: 10.1016/J.JNT.2004.02.002
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description: scientific article; zbMATH DE number 2091711