Modular forms arising from divisor functions
type: Item
instance of: scholarly article
title: Modular forms arising from divisor functions
zbMATH Open document ID: 1244.11038
author: Bumkyu Cho
author: Daeyeoul Kim
author: Ja Kyung Koo
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-06-18T00:00:00Z
review text: Klein forms are defined by means of Weierstrass \(\sigma\)-functions. They can be expressed by infinite products, and certain products of Klein forms are holomorphic modular forms on congruence subgroups of the modular group \(\text{SL}_2(\mathbb{Z})\) whose zeros are supported only at the cusps. The theory of basic hypergeometric series provides many identities between infinite \(q\)-products and infinite \(q\)-series whose coefficients are combinations of divisor sums of the type \[ E_r(n, m) = \sum_{{d\mid n}\atop{d\equiv r\pmod n}}^{} 1 - \sum_{{d\mid n}\atop {d\equiv - r \pmod n}}^{} 1. \] These identities can be restated in terms of Klein forms. In this way, the authors obtain holomorphic modular forms for many congruence groups whose Fourier coefficients are expressed by divisor sums. For several groups, these functions constitute bases of the spaces of holomorphic modular forms of weights 1 and 2. As an application, the representation numbers of integers by binary quadratic forms with discriminants \(-7\) and \(-11\) are expressed by means of divisor sums \(E_r(n, m)\). Reviewer's remark: In some cases the (combinations of) divisor sums \(E_r(n, m)\) are character sums, and the corresponding modular forms can be identified with Hecke theta series on certain imaginary quadratic fields.
reviewed by: Günter Köhler
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F30
zbMATH DE Number: 5566254
zbMATH Keywords: modular forms
zbMATH Keywords: basic hypergeometric series
zbMATH Keywords: Klein forms
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.03.003
OpenAlex ID: W1999594834
cites work: Q4040189
cites work: Q3953865
cites work: Q4693061
cites work: Q5625265
cites work: Q4317891
DOI: 10.1016/J.JMAA.2009.03.003
description: scientific article; zbMATH DE number 5566254