Generalisation of the Asai-Kaneko-Ninomiya identity to higher level

Abstract

Let j be the classical modular invariant, which is a modular function of weight zero for SL2(Z). Then for every z,τ in the upper half-plane, the Asai-Kaneko-Ninomiya (AKN) identity relates j(z)−j(τ) to the generating function of the Hecke system jm(τ) of modular functions for SL2(Z). In [7] the modular functions jm(z) were generalised to modular functions JN,m for higher level using harmonic weak Maass forms. In this paper, we show that under certain conditions on JN,m, the AKN identity can be generalised to higher level Γ0(N) using Poincaré series of weight 2 and level N. We also prove an infinite product formula for JN,1 which generalises the denominator formula for the Monster Lie algebra.