On modular forms for some noncongruence subgroups of \(\text{SL}_2(\mathbb Z)\)
type: Item
instance of: scholarly article
title: On modular forms for some noncongruence subgroups of \(\text{SL}_2(\mathbb Z)\)
zbMATH Open document ID: 1215.11043
author: Chris A. Kurth
author: Ling Long
published in: Journal of Number Theory
publication date: 2008-12-08T00:00:00Z
review text: Let \(\Gamma\) be a noncongruence subgroup of \(\text{PSL}_2(\mathbb Z)\). Let \(f\) be a holomorphic integral weight \(k\geq 2\) modular form for \(\Gamma\) but not for \(\Gamma^c\) the congruence closure of \(\Gamma\). Assume the Fourier coefficients of \(f\) at infinity are algebraic, then it is expected that these coefficients have unbounded denominators (the UBD condition). The paper gives some evidence that all noncongruence subgroups satisfy the UBD condition. The main result is as follows. We say a subgroup \(\Gamma\) of \(\Gamma_0\) is a character group of type (IIA) if there is a homomorphism \(\varphi: \Gamma^0\mapsto G\) where \(G\) is abelian, such that \(\Gamma=\ker(\varphi)\) and \(\Gamma\) contains all elliptic and parabolic elements of \(\Gamma^0\). Let \(\Gamma^0\) be a genus \(1\) congruence subgroup. If there exists a prime \(p\) such that every index \(p\) type (IIA) character groups of \(\Gamma^0\) satisfy the UBD condition, then a positive proportion of the type (IIA) character groups of \(\Gamma^0\) satisfy the UBD condition.
reviewed by: Zhengyu Mao
Mathematics Subject Classification ID: 11F06
Mathematics Subject Classification ID: 11F30
zbMATH DE Number: 5379230
zbMATH Keywords: noncongruence subgroup
zbMATH Keywords: holomorphic modular forms
describes a project that uses: ecdata
MaRDI profile type: Publication
full work available at URL: j.jnt.2007.10.007
OpenAlex ID: W2554204837
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DOI: 10.1016/J.JNT.2007.10.007
description: scientific article; zbMATH DE number 5379230