English

Extremal quasimodular forms of lower depth with integral Fourier coefficients

Number Theory 2021-03-31 v2

Abstract

We show that, based on Grabner's recent results on modular differential equations satisfied by quasimodular forms, there exist only finitely many normalized extremal quasimodular forms of depth rr that have all Fourier coefficients integral for each of r=1,2,3,4r=1,2,3,4, and partly classifies them, where the classification is complete for r=2,3,4r=2,3,4; in fact, we show that there exists no normalized extremal quasimodular forms of depth 44 with all Fourier coefficients integral. Our result disproves a conjecture by Pellarin.

Keywords

Cite

@article{arxiv.2101.09897,
  title  = {Extremal quasimodular forms of lower depth with integral Fourier coefficients},
  author = {Tsudoi Kaminaka and Fumiharu Kato},
  journal= {arXiv preprint arXiv:2101.09897},
  year   = {2021}
}

Comments

14 pages; to appear in Kyushu Journal of Mathematics Vol.75 no.2 (2021)