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arXiv · 1912.02277

On coefficients of Poincar\'e series and single-valued periods of modular forms

Tiago J. Fonseca
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Abstract

We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincar\'e series of a given level $\Gamma_0(N)$ and weight $k\ge 2$ coincides with the field generated by the single-valued periods of a certain motive attached to $\Gamma_0(N)$. This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of Brown and Acres--Broadhurst giving explicit series expansions for the single-valued periods of some modular forms. Our proof is based on Bringmann--Ono's construction of harmonic lifts of Poincar\'e series.

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Tiago J. Fonseca. 2019-12-04. On coefficients of Poincar\'e series and single-valued periods of modular forms. https://doi.org/10.1007/s40687-020-00232-5

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