The arithmetic of vector-valued modular forms on $Γ_{0}(2)$

Abstract

Let $ρ$ denote an irreducible two-dimensional representation of $Γ_{0}(2)$. The collection of vector-valued modular forms for $ρ$, which we denote by $M(ρ)$, form a graded and free module of rank two over the ring of modular forms on $Γ_{0}(2)$, which we denote by $M(Γ_{0}(2))$. For a certain class of $ρ$, we prove that if Z is any vector-valued modular form for $ρ$ whose component functions have algebraic Fourier coefficients then the sequence of the denominators of the Fourier coefficients of both component functions of Z is unbounded. Our methods involve computing an explicit basis for $M(ρ)$ as a $M(Γ_{0}(2))$-module. We give formulas for the component functions of a minimal weight vector-valued form for $ρ$ in terms of the Gaussian hypergeometric series $_{2}F_{1}$, a Hauptmodul of $Γ_{0}(2)$, and the Dedekind $η$-function.