The circle method and non lacunarity of Modular Functions

TL;DR

This paper investigates the lacunarity of modular functions of arbitrary real weight, establishing that lacunary functions must be holomorphic, finite at cusps, and have non-negative weight, extending Serre's results.

Contribution

It generalizes Serre's lacunarity results to modular functions of arbitrary real weight for finite index subgroups of SL(2,Z).

Findings

Lacunary modular functions are necessarily holomorphic.

Such functions are finite at the cusps.

They have non-negative weight.

Abstract

Serre proved that any holomorphic cusp form of weight one for $\Gamma_1(N)$ is lacunary while a holomorphic modular form for $\Gamma_1(N)$ of higher integer weight is lacunary if and only if it is a linear combination of cusp forms of CM-type (see Serre, subsections 7.6 and 7.7). In this paper, we show that when a non-zero modular function of arbitrary real weight for any finite index subgroup of the modular group ${\SL}_2(\Z)$ is lacunary, it is necessarily holomorphic on the upper-half plane, finite at the cusps and has non-negative weight.