On extremal quasi-modular forms after Kaneko and Koike

TL;DR

This paper investigates extremal quasi-modular forms introduced by Kaneko and Koike, providing a sharp multiplicity estimate, discussing related conjectures, and proving integrality of certain Fourier coefficients.

Contribution

It offers a new multiplicity estimate for extremal quasi-modular forms and addresses conjectures on their arithmetic properties, including a proof of integrality for specific cases.

Findings

Established a sharp multiplicity estimate for extremal quasi-modular forms.

Provided partial answers to conjectures on their arithmetic properties.

Proved integrality of Fourier coefficients for a specific extremal quasimodular form.

Abstract

Kaneko and Koike introduced the notion of extremal quasi-modular form and proposed conjectures on their arithmetic properties. The aim of this note is to prove a rather sharp multiplicity estimate for these quasi-modular forms. The note ends with discussions and partial answers around these conjectures and an appendix by G. Nebe containing the proof of the integrality of the Fourier coefficients of the normalised extremal quasimodular form of weight 14 and depth 1.