On the universal power series for Jacobi sums and the Vandiver conjecture
type: Item
instance of: scholarly article
title: On the universal power series for Jacobi sums and the Vandiver conjecture
zbMATH Open document ID: 0686.12003
DOI: 10.1016/0022-314X(89)90076-0
author: Humio Ichimura
author: Masanobu Kaneko
published in: Journal of Number Theory
publication date: 1989-01-01T00:00:00Z
review text: This paper studies Ihara's series \(F_{\rho}(u,v)\), a unit in \({\mathbb{Z}}_{\ell}u,v\) related to an element \(\rho\) of the Galois group of \({\mathbb{Q}}\). It relates the image of \(\rho \to F_{\rho}\) to the Vandiver conjecture at \(\ell\). This image is contained in a multiplicative subgroup \({\mathcal F}\) defined by five conditions. The surjectivity of Gal(\({\bar {\mathbb{Q}}}/{\mathbb{Q}})\to {\mathcal F}\) is equivalent to it (theorem 1). The relation of the cokernel to a classical Iwasawa module is given in theorem 2. In conclusion, theorem 3 studies \(h_{\rho}(u)\), the first term in the v-expansion of \(F_{\rho}(u,v)\) à la Coleman.
reviewed by: Roland Gillard
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 4123839
zbMATH Keywords: class number of real cyclotomic field
zbMATH Keywords: Galois representations
zbMATH Keywords: power series representation
zbMATH Keywords: Galois action on Fermat curves of l-power degrees
zbMATH Keywords: Ihara's series
zbMATH Keywords: Iwasawa module
Wikidata QID: Q122856135
MaRDI profile type: Publication
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full work available at URL: 0022-314x(89)90076-0
OpenAlex ID: W2008837442
description: scientific article; zbMATH DE number 4123839