On coefficients of Poincaré series and single-valued periods of modular forms

Author: 

Fonseca, T

Journal: 

Research in the Mathematical Sciences

Last Updated: 

2020-10-09T14:45:21.997+01:00

DOI: 

10.1007/s40687-020-00232-5

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.

Symplectic id: 

1077537

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Submitted to ORA: 

Submitted

Publication Type: 

Journal Article