Fourier coefficients of meromorphic modular forms and a question of Petersson
Fourier coefficients of meromorphic modular forms and a question of Petersson
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亚纯模形式的傅立叶系数和Petersson问题
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
B. Kane
中科院分区：
文献类型：
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作者：
K. Bringmann;B. Kane
本文研究了Petersson在1950年研究的二元亚纯Poincar‘e级数。当时，他考虑了三种类型的Poincar‘e级数，其中一种在两个变量中都是模的，其中一种在两个变量中都是亚纯的，第三种类型来自于似乎不符合一般框架的差异。通过认识到这些函数之间的恒等式是所谓的极调和Maass形式的分裂，我们认识到第三类Poincar‘e级数是Poincar’e级数的非亚纯部分，它们在两个变量上都是模的。这就得到了这些Poincar‘e级数在一般情况下可表示为其他已研究好的Poincar’e级数的非全纯Eichler积分的公式。
这些Poincar‘e级数在极调和Maass理论中的作用也给出了亚纯模形式的傅里叶展开。多年来，亚纯模形式的一些特殊情况被证明具有与Hardy和Ramanujan计算的权$6$Eisenstein级数的倒数展开非常相似的傅立叶展开式。本文证明了所有负权亚纯模形式(以及所有拟亚纯模形式)都有这种类型的傅立叶展开式，假设它们是有界的。
In this paper, we investigate two-variable meromorphic Poincar'e series studied by Petersson in 1950. At that time, he considered three types of Poincar'e series, one of which was modular in both variables, one of which was meromorphic in both variables, and a third piece coming from the difference which did not seem to fit into a general framework. By realizing identities between these functions as splittings of so-called polar harmonic Maass forms, we recognize the Poincar'e series of the third type as the non-meromorphic parts of the Poincar'e series which are modular in both variables. This leads to a formula for these Poincar'e series as non-holomorphic Eichler integrals of other well-studied Poincar'e series in the general case.
The role of these Poincar'e series in the theory of polar harmonic Maass also yield Fourier expansions of meromorphic modular forms. Over the years, a number of special cases of meromorphic modular forms were shown to have Fourier expansions closely resembling the expansion of the reciprocal of the weight $6$ Eisenstein series which was computed by Hardy and Ramanujan. We prove in this paper that all negative-weight meromorphic modular forms (and furthermore all quasi-meromorpic modular forms) have Fourier expansions of this type, granted that they are bounded towards $iinfty$.
半积分权调和弱 Maass 形式系数的代数公式
影响因子：
1.7
作者：
J. H. Bruinier;K. Ono
通讯作者：
K. Ono