Extremal quasimodular forms of lower depth with integral Fourier coefficients

Tsudoi Kaminaka; Fumiharu Kato

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arXiv · 2101.09897

Extremal quasimodular forms of lower depth with integral Fourier coefficients

Tsudoi Kaminaka·Fumiharu Kato

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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.

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Tsudoi Kaminaka, Fumiharu Kato. 2021-01-25. Extremal quasimodular forms of lower depth with integral Fourier coefficients. https://arxiv.org/abs/2101.09897

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