Arithmetic of weakly holomorphic modular forms for hecke groups

Abstract

Duke and Jenkins [4] constructed a nice canonical basis for the space of weakly holomorphic modular forms of weight k for SL2(Z) that are holomorphic away from the cusp at infinity. They investigated the arithmetic of these basis elements. Let Mk# (Γ0(p)) be the space of weakly holomorphic modular forms of weight k for Γ0(p) that are holomorphic away from the cusp at infinity. We generalize the results of Duke and Jenkins to the space Mk# (Γ0(p)) when the genus of Γ0(p) is zero. As applications, first, we find a basis which consists of Poincar? series, for the space of cusp forms for Γ0(p). Second, we show that the algebraicity of coefficients of the holomorphic part of a harmonic weak Maass form in H#2?k (Γ0(p)) # is determined by its first few coefficients. ? 2015 Pushpa Publishing House, Allahabad, India.