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

TL;DR

This paper establishes a deep connection between the Fourier coefficients of weakly holomorphic Poincaré series for modular forms and the single-valued periods of associated motives, revealing their shared arithmetic nature.

Contribution

It proves that the field generated by these Fourier coefficients matches that of the single-valued periods, generalizing previous explicit formulas and using harmonic lifts of Poincaré series.

Findings

Fourier coefficients generate the same field as single-valued periods.

Generalization of Brown and Acres--Broadhurst formulas.

Utilization of harmonic lifts of Poincaré series in proof.

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.