On the Fourier coefficients of 2-dimensional vector-valued modular forms (Q2884412)

Let \(\Gamma=\text{SL}_2(\mathbb Z)\) be the modular group, \(\rho\) a complex two dimensional representation of \(\Gamma\). A holomorphic vector-valued modular form of weight \(k\) associated to \(\rho\) is a pair \(F=(f_1,f_2)\), so that under the action of \(\gamma\in \Gamma\), \(\gamma\circ F=F\rho(\gamma)\).NEWLINENEWLINEWhen the kernel of \(\rho\) contains a congruence subgroup of \(\Gamma\), we say \(\rho\) is modular. In this case the forms \(f_1\) and \(f_2\) must be classical modular forms with respect to that congruence subgroup. It is well known the Fourier coefficients of the classical modular forms have bounded denominator property. The paper studies the converse under the assumption that \(\rho\) has a finite projective level--meaning \(\rho(T)\) has finite order \(M\) in \(\text{PGL}_2(\mathbb C)\). (Here \(T\) is one of the standard generators of \(\Gamma\)).NEWLINENEWLINEThe paper states the conjecture that (under the above assumption of \(\rho\)) \(\rho\) is modular if and only if some \(F\) has bounded denominator property. It proves the conjecture under the assumption that the order \(M\) does not divide \(60\). Indeed in this case none of the \(\rho\) is modular and all \(F\) with rational Fourier coefficients have unbounded denominators.