Ramanujan-like formulas for Fourier coefficients of all meromorphic cusp forms

2016/03/30 by Bringmann, Kathrin, Kane, Ben
#11F11 #11F12 #11F30 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1603.09250

Abstract

In this paper, we investigate 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. By investigating meromorphic modular forms within a larger space of so-called polar harmonic Maass forms, 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 i∞.

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• Ramanujan and coefficients of meromorphic modular forms
• Harmonic weak Maass forms and periods II
• Arithmetic properties of Fourier coefficients of meromorphic modular forms
• Elliptic curves and Fourier coefficients of meromorphic modular forms
• Meromorphic analogues of modular forms generating the kernel of Shintani's lift