We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level \(\varGamma _0(N)\) and integral weight \(k\ge 2\) coincides with the field generated by the single-valued periods of a certain motive attached to \(\varGamma _0(N)\). 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.

中文翻译：

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庞加莱级数的系数和模形式的单值周期

我们证明了由给定水平\（\ varGamma _0（N）\）和积分权重\（k \ ge 2 \）的弱全纯庞加莱级数的傅立叶系数生成的场与单值生成的场重合附加到\（\ varGamma _0（N）\）的某个动机的周期。这阐明了这种傅立叶系数的算术性质，并推广了Brown和Acres–Broadhurst的先前公式，从而给出了某些模块化形式的单值周期的显式级数展开。我们的证明基于Bringmann–Ono构造的庞加莱系列谐波升降机。