The arithmetic of vector-valued modular forms on ${\Gamma }_0(2)$

Abstract

Let $\rho$ denote an irreducible two-dimensional representation of ${\Gamma }_0(2)$. The collection of vector-valued modular forms for $\rho$, which we denote by $M(\rho )$, form a graded and free module of rank two over the ring of modular forms on ${\Gamma }_0(2)$, which we denote by $M({\Gamma }_0(2))$. For a certain class of $\rho$, we prove that if Z is any vector-valued modular form for $\rho$ 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 $M(\rho )$ as a $M({\Gamma }_0(2))$-module. We give formulas for the component functions of a minimal weight vector-valued form for $\rho$ in terms of the Gaussian hypergeometric series ${}_2{F}_1$, a Hauptmodul of ${\Gamma }_0(2)$, and the Dedekind $\eta$-function.

To appear in The International Journal of Number Theory

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