2020
Fonseca, Tiago J.
On coefficients of Poincaré series and single-valued periods of modular forms Journal Article
Em: Research in the Mathematical Sciences volume, vol. 7, 2020.
Resumo | Links | BibTeX | Tags: Fourier coefficients, Poincaré series
@article{nokey,
title = {On coefficients of Poincar\'{e} series and single-valued periods of modular forms},
author = {Tiago J. Fonseca},
url = {https://link.springer.com/article/10.1007/s40687-020-00232-5},
doi = {https://doi.org/10.1007/s40687-020-00232-5},
year = {2020},
date = {2020-11-05},
journal = {Research in the Mathematical Sciences volume},
volume = {7},
abstract = {We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincar\'{e} series of a given level Γ0(N) and integral weight k≥2 coincides with the field generated by the single-valued periods of a certain motive attached to Γ0(N). This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of Brown and Acres\textendashBroadhurst giving explicit series expansions for the single-valued periods of some modular forms. Our proof is based on Bringmann\textendashOno’s construction of harmonic lifts of Poincar\'{e} series.},
keywords = {Fourier coefficients, Poincar\'{e} series},
pubstate = {published},
tppubtype = {article}
}
We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level Γ0(N) and integral weight k≥2 coincides with the field generated by the single-valued periods of a certain motive attached to Γ0(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 Poincaré series.