The circle method and non lacunarity of Modular Functions
Number Theory
2012-10-23 v1
Abstract
Serre proved that any holomorphic cusp form of weight one for is lacunary while a holomorphic modular form for 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 is lacunary, it is necessarily holomorphic on the upper-half plane, finite at the cusps and has non-negative weight.
Cite
@article{arxiv.1210.5608,
title = {The circle method and non lacunarity of Modular Functions},
author = {Sanoli Gun and Joseph Oesterlé},
journal= {arXiv preprint arXiv:1210.5608},
year = {2012}
}