Basis for the space of weakly holomorphic modular forms in higher level cases
type: Item
instance of: scholarly article
title: Basis for the space of weakly holomorphic modular forms in higher level cases
zbMATH Open document ID: 1282.11027
author: Chang Heon Kim
author: SoYoung Choi
published in: Journal of Number Theory
publication date: 2013-02-21T00:00:00Z
review text: Let \(p\) be either \(1\) or a prime number. Let \(\Gamma_0(p)^+\) be the group generated by the group \(\Gamma_0(p)\) and the Fricke involution \(W_p\) and \(M_k^!(\Gamma_0(p)^+)\) be the space of weakly holomorphic modular forms (that is, meromorphic with poles only at the cusps) of even integral weight \(k\) with respect to \(\Gamma_0(p)^+\). In this article, the results of \textit{W. Duke} and \textit{P. Jenkins} [Pure Appl. Math. Q. 4, No. 4, 1327--1340 (2008; Zbl 1200.11027)] on a canonical basis of \(M_k^!(\Gamma_0(1)^+)\)~(\(\Gamma_0(1)^+= \mathrm{SL}_2(\mathbb Z))\) are extended to those of \(M_k^!(\Gamma_0(p)^+)\), where \(\Gamma_0(p)^+\) is of genus \(0\). These groups have only one cusp \(i\infty\) up to equivalence and have the Hauptmodul with a pole at the cusp. Let \(m_k\) be the maximal value of order of Fourier expansions at the cusp of non-zero forms belong to \(M_k^!(\Gamma_0(p))\). The authors give explicitly the form \(F_k\in M_k^!(\Gamma_0(p)^+)\) of order \(m_k\) with the leading coefficient \(1\). By using \(F_k\) and the Hauptmodul, a canonical basis \(\{f_{k,n}\} (n\in \mathbb Z,-n\leq m_k)\) of the space \(M_k^!(\Gamma_0(p))\) is constructed such that \(f_{k,n}=q^{-n}+\sum_{m>m_k}a_k(n,m)q^m,~a_k(n,m)\in\mathbb Z,~q=e^{2\pi i z}\) and \(\{f_{k,n}\}\) have a generating function and the duality \(a_k(n,m)=-a_{2-k}(m,n)\) of Fourier coefficients.
reviewed by: Noburo Ishii
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F37
zbMATH DE Number: 6138271
zbMATH Keywords: weakly holomorphic modular form
zbMATH Keywords: Fourier coefficient
MaRDI profile type: Publication
full work available at URL: j.jnt.2012.09.015
OpenAlex ID: W2084228835
cites work: Monstrous moonshine and monstrous Lie superalgebras
cites work: The Values of Modular Functions and Modular Forms
cites work: Monstrous Moonshine
cites work: p-Adic Properties of Coefficients of Weakly Holomorphic Modular Forms
cites work: On the zeros and coefficients of certain weakly holomorphic modular forms
cites work: LINEAR CONGRUENCES AND RELATIONS ON SPACES OF CUSP FORMS
cites work: FOURIER EXPANSIONS WITH MODULAR FORM COEFFICIENTS
cites work: Borcherds products associated with certain Thompson series
cites work: Q4693061
cites work: Q5625265
cites work: Q4284459
DOI: 10.1016/J.JNT.2012.09.015
Recommended article: The zeros of certain weakly holomorphic Drinfeld modular forms
Recommended article: Arithmetic of weakly holomorphic modular forms for Hecke groups
Recommended article: Arithmetic properties for the minus space of weakly holomorphic modular forms
Recommended article: Interlacing of zeros of certain weakly holomorphic modular forms for \(\Gamma_0^+(2)\)
Recommended article: Weakly holomorphic modular forms for some moonshine groups
Recommended article: \(p\)-adic properties of coefficients of basis for the space of weakly holomorphic modular forms of weight 2
Recommended article: Divisibility properties of coefficients of modular functions in genus zero levels
Recommended article: Zeros of weakly holomorphic modular forms of level 5
Recommended article: Zeros of modular forms of half integral weight
Recommended article: The transcendence of zeros of canonical basis elements of the space of weakly holomorphic modular forms for \({\Gamma}\)\(_{0}\)(2)
description: scientific article; zbMATH DE number 6138271