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

Fonseca, T

## Journal:

Research in the Mathematical Sciences

## Last Updated:

2021-06-12T20:47:23.207+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.

1077537

Submitted

Journal Article