Extremal quasimodular forms of lower depth with integral Fourier coefficients

TL;DR

This paper proves that only finitely many normalized extremal quasimodular forms of depths 1 to 4 have all integral Fourier coefficients, fully classifies depths 2 to 4, and disproves a previous conjecture.

Contribution

It establishes finiteness and classification results for extremal quasimodular forms with integral Fourier coefficients across multiple depths, and refutes a prior conjecture.

Findings

Finitely many extremal quasimodular forms of depths 1-4 have integral Fourier coefficients.

Complete classification of such forms for depths 2, 3, and 4.

No extremal quasimodular forms of depth 4 with integral Fourier coefficients exist.

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 $r$ that have all Fourier coefficients integral for each of $r=1,2,3,4$, and partly classifies them, where the classification is complete for $r=2,3,4$; in fact, we show that there exists no normalized extremal quasimodular forms of depth $4$ with all Fourier coefficients integral. Our result disproves a conjecture by Pellarin.