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

Abstract

We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré 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 Poincaré series.

36 pages. Final version

References in corpus (3)

• Multiple Modular Values and the relative completion of the fundamental group of $M_{1,1}$
• Single-valued integration and double copy
• Algebraic de Rham theory for weakly holomorphic modular forms of level one