Hirzebruch sums and Hecke operators
type: Item
instance of: scholarly article
title: Hirzebruch sums and Hecke operators
zbMATH Open document ID: 0734.11030
DOI: 10.1016/0022-314X(91)90082-M
author: Hongwen Lu
published in: Journal of Number Theory
publication date: 1991-01-01T00:00:00Z
review text: Let p be a rational prime congruent to 3 modulo 4. \textit{F. Hirzebruch} [Hilbert Modular Surfaces, Enseign. Math., II. Ser. 19, 183-281 (1973; Zbl 0285.14007)] showed that, under certain conditions, the class number of \({\mathbb{Q}}(\sqrt{-p})\) is related to a sum over the partial quotients of the continued fraction of \(\sqrt{p}\). This sum, defined in general for a real quadratic irrational and dubbed the Hirzebruch sum, is the subject of this paper. The author shows (Theorem 7) that this sum satisfies a simple, but notationally cumbersome, identity which on one side resembles a weighted form of the action of the Hecke operators from modular form theory. In those cases involving irrational numbers from fields of discriminant p or 4p, p prime, this identity reduces to the result that this Hirzebruch sum is an eigenfunction for the Hecke operators themselves. The proof of the identity itself depends on an earlier result of the author on the transformation formula for the Dedekind \(\eta\)- function [Acta Math. Sci. 8, 249-262 (1988; Zbl 0665.10016)] and an identity of \textit{M. Knopp} involving Dedekind sums [J. Number Theory 12, 2-9 (1980; Zbl 0423.10015)].
Mathematics Subject Classification ID: 11F20
Mathematics Subject Classification ID: 11A55
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11F25
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 4214119
zbMATH Keywords: class number
zbMATH Keywords: continued fraction
zbMATH Keywords: irrational numbers
zbMATH Keywords: identity
zbMATH Keywords: Hirzebruch sum
zbMATH Keywords: eigenfunction
zbMATH Keywords: Hecke operators
zbMATH Keywords: Dedekind \(\eta\)-function
zbMATH Keywords: Dedekind sums
reviewed by: James Lee Hafner
MaRDI profile type: Publication
cites work: The class-number of real quadratic number fields
cites work: Hecke operators and an identity for the Dedekind sums
cites work: Q3816142
cites work: On a Pellian equation conjecture
cites work: Q5663963
full work available at URL: 0022-314x(91)90082-m
OpenAlex ID: W2017416212
description: scientific article; zbMATH DE number 4214119