On Hilbert modular forms. III
type: Item
instance of: scholarly article
title: On Hilbert modular forms. III
zbMATH Open document ID: 0555.10015
author: Shoyu Nagaoka
published in: Proceedings of the Japan Academy. Series A
publication date: 1983-01-01T00:00:00Z
review text: Let \(A_{{\mathbb{Z}}}(\Gamma_ K)_ k\) be the \({\mathbb{Z}}\)-module of symmetric Hilbert modular forms of (integral) weight k with Fourier coefficients in \({\mathbb{Z}}\), \(A_{{\mathbb{Z}}}(\Gamma_ K):=\oplus_{r\geq 0}A_{{\mathbb{Z}}}(\Gamma_ K)_{2r}\) and \(A^ a_{{\mathbb{Z}}}(\Gamma_ K):=\oplus_{k\geq 0}A_{{\mathbb{Z}}}(\Gamma_ K)_ k\). In part II [ibid. 58, 44-46 (1982; Zbl 0507.10021)], the author has described the struture of the graded ring \(A_{{\mathbb{Z}}}(\Gamma_ K)\) for \(K={\mathbb{Q}}(\sqrt{2})\), \({\mathbb{Q}}(\sqrt{5})\), in terms of modular forms explicitly given by Eisenstein series. This paper, concerned first with \(A^ a_{{\mathbb{Z}}}(\Gamma_ K)\) for \(K={\mathbb{Q}}(\sqrt{2})\), adapts Resnikoff's method in the case \(K={\mathbb{Q}}(\sqrt{5})\) (based on Igusa-Hammond's modular imbedding and) giving the existence of a symmetric Hilbert modular form of weight 15. The author presents a minimal set of 4 generators for \(A^ a_{{\mathbb{Z}}}(\Gamma_ K)\) for \(K={\mathbb{Q}}(\sqrt{2})\); he also describes a minimal set of 5 generators (including the W above) for the case \(K={\mathbb{Q}}(\sqrt{5})\). The results are consistent with earlier work of Hirzebruch on the ring of Hilbert modular forms and also of H. Cohn, for \(K={\mathbb{Q}}(\sqrt{2})\).
Mathematics Subject Classification ID: 11F41
zbMATH DE Number: 3884248
zbMATH Keywords: symmetric Hilbert modular forms
zbMATH Keywords: graded ring
zbMATH Keywords: Resnikoff's method
zbMATH Keywords: existence
zbMATH Keywords: weight 15
zbMATH Keywords: generators
MaRDI profile type: Publication
full work available at URL: pjaa.59.346
OpenAlex ID: W4238424456
cites work: An Explicit Modular Equation in Two Variables and Hilbert's Twelfth Problem
cites work: Q4148051
cites work: On Hilbert modular forms
cites work: On the ring of Hilbert modular forms over \(\mathbb Z\)
cites work: On the graded ring of Hilbert modular forms associated with \(Q (\sqrt 5)\)
DOI: 10.3792/PJAA.59.346
description: scientific article; zbMATH DE number 3884248