具有积分傅立叶系数的下深度极值拟模形式

Pub Date : 2021-01-25 DOI:10.2206/kyushujm.75.351

Tsudoi Kaminaka, Fumiharu Kato

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引用次数: 1

摘要

基于Grabner关于准模形式满足的模微分方程的最新结果，我们证明了深度r的归一化极值拟模形式只存在有限多个，它们对r = 1,2,3,4的傅里叶系数都是积分的，并对它们进行了部分分类，其中对r = 2,3,4的分类是完全的;事实上，我们证明了深度4的归一化极值拟模形式不存在，且所有的傅里叶系数都是积分的。我们的结果反驳了佩拉林的一个猜想。

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EXTREMAL QUASIMODULAR FORMS OF LOWER DEPTH WITH INTEGRAL FOURIER COEFFICIENTS

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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