On modular forms for some noncongruence subgroups of \(\text{SL}_2(\mathbb Z)\) (Q958704)

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.