Construction of normal bases by special values of Hilbert modular functions
type: Item
instance of: scholarly article
title: Construction of normal bases by special values of Hilbert modular functions
zbMATH Open document ID: 0877.11058
author: Keiichi Komatsu
published in: Proceedings of the Japan Academy. Series A
publication date: 1997-11-20T00:00:00Z
review text: Let \(K\) be a cyclic imaginary extension of \(\mathbb{Q}\). In this paper, the author shows how to construct elements \(\theta\) generating a normal basis of certain class fields over \(K\), by virtue of Hilbert modular functions. The details are very intricate, but (as the author points out) the general line becomes quite clear when one looks at earlier work of \textit{T. Okada} [J. Lond. Math. Soc., II. Ser. 22, 221-225 (1980; Zbl 0448.12009)]\ for the case that \(K\) is imaginary quadratic. Quite generally, a given element \(\theta\) generates a normal basis of the abelian \(G\)-Galois extension \(L/K\) iff none of the resolvents \((\theta,\chi)=\sum_{\sigma\in G}\chi(\sigma) \theta^\sigma\) vanishes, where \(\chi\) runs over all complex characters of \(G\). In Okada's work, as well as in the present article, the resolvents which come up can be identified with special values of Hecke L-series, not for \(\chi\) but for a modified character \(\chi\phi\) (note \(\chi\) is an ordinary character, and \(\chi\phi\) will be a true Hecke character). The general theory of Hecke L-series then allows to conclude that these values do not vanish. In Okada's paper, the modular forms which were used are, up to a factor, division values of derivatives of the Weierstrass \(\wp\)-function. In the setting of the paper under review, explicit Hilbert modular forms on the \(m\)-fold power of the upper half plane are used instead (with \(2m=[K:{\mathbb{Q}}]\)). In the main theorem, there are several technical conditions on the class field \(L\) for which a normal basis is being constructed, and these conditions are nontrivial to check. The author concludes with a nice explicit example where \(K={\mathbb{Q}}(\zeta_5)\); \(L\) turns out to be just the ray class field modulo 7, and with some effort the reviewer succeeded in verifying that all required conditions are met in the example. This is a well-written but also a demanding paper.
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 11F80
zbMATH DE Number: 1057073
zbMATH Keywords: normal bases
zbMATH Keywords: Hecke L-series
zbMATH Keywords: Hilbert modular forms
MaRDI profile type: Publication
full work available at URL: pjaa.73.42
OpenAlex ID: W2001553102
cites work: L-functions of elliptic curves with complex multiplication, II
cites work: Cyclic Galois extensions of commutative rings
cites work: Normal bases and \(\mathbb{Z}_ p\)-extensions
cites work: Modular construction of normal basis
cites work: Normal basis and Greenberg's conjecture
cites work: Models of certain automorphic function fields
cites work: Normal Bases of Class Fields over Gauss' Number Field
cites work: Galoismodulstruktur und elliptische Funktionen. (Galois module structure and elliptic functions)
cites work: Construction of class fields and zeta functions of algebraic curves
cites work: On some arithmetic properties of modular forms of one and several variables
cites work: Q5631179
cites work: Relative Galois module structure of rings of integers and elliptic functions. II
cites work: Relative Galois Module Structure of Rings of Integers and Elliptic Functions III
DOI: 10.3792/PJAA.73.42
description: scientific article; zbMATH DE number 1057073