\(L_p(1,\chi)\bmod p\)
type: Item
instance of: scholarly article
title: \(L_p(1,\chi)\bmod p\)
zbMATH Open document ID: 1046.11080
author: Jangheon Oh
published in: Proceedings of the Japan Academy. Series A
publication date: 2003-01-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1116442546
review text: Let \(k\) be a number field, \(\chi\) a Dirichlet character of conductor \(\Delta\), and \(p\) an odd prime coprime to \(\Delta\). The main result of this article is the congruence \[ \chi(p)^{-1} L_p(1,\chi) \equiv \sum_{t=1}^{p-1} (1 + \tfrac12 + \ldots + \tfrac1t) (\chi(rt) + \chi(rt+1) + \ldots + \chi(rt+r-1)) \bmod p, \] where \(r\) and \(s\) are integers with \(rp + s\Delta = 1\). If \(K = \mathbb Q(\sqrt{D}\,)\) is a real quadratic number field with associated Dirichlet character \(\chi\) in which \(p\) splits, and if the right hand side of the congruence above is nonzero, then the Iwasawa \(\mu\)- and \(\lambda\)-invariant of \(K\) vanish.
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11S40
zbMATH DE Number: 2024711
zbMATH Keywords: \(p\)-adic \(L\)-function
zbMATH Keywords: Iwasawa invariants
zbMATH Keywords: Greenberg's conjecture
MaRDI profile type: Publication
OpenAlex ID: W2031725217
cites work: Residue at \(s=1\) of \(p\)-adic zeta functions
cites work: On the Iwasawa Invariants of Totally Real Number Fields
cites work: l-adic L-functions and rational function measures
cites work: On \(p\)-adic zeta functions and \(\mathbb{Z}_p\)-extensions of certain totally real number fields
cites work: Q3943047
DOI: 10.3792/PJAA.79.11
description: scientific article; zbMATH DE number 2024711