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arXiv · 1504.00844

Hyperbolic Fourier coefficients of Poincaré series

Cormac O'Sullivan·Karen Taylor
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Abstract

Poincaré in 1911 and Petersson in 1932 gave the now classical expression for the parabolic Fourier coefficients of holomorphic Poincaré series in terms of Bessel functions and Kloosterman sums. Later, in 1941, Petersson introduced hyperbolic and elliptic Fourier expansions of modular forms and the associated hyperbolic and elliptic Poincaré series. In this paper we express the hyperbolic Fourier coefficients of Poincaré series, of both parabolic and hyperbolic type, in terms of hypergeometric series and Good's generalized Kloosterman sums. In an explicit example for the modular group, we see that the hyperbolic Kloosterman sum corresponds to a sum over lattice points on a hyperbola contained in an ellipse. This allows for numerical computation of the hyperbolic Fourier coefficients.

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Cormac O'Sullivan, Karen Taylor. 2017-02-12. Hyperbolic Fourier coefficients of Poincaré series. https://doi.org/10.1007/s11139-016-9772-6

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