A note on Fourier coefficients of Poincar\'e series

TL;DR

This paper provides a simplified proof demonstrating the asymptotic orthogonality of Fourier coefficients in Poincaré series for classical and Siegel modular forms, highlighting their qualitative behavior.

Contribution

It offers a concise and accessible proof of the asymptotic orthogonality of Fourier coefficients for Poincaré series in modular forms.

Findings

Fourier coefficients of Poincaré series are asymptotically orthogonal.

The proof is short and 'soft', emphasizing qualitative understanding.

Applicable to both classical and Siegel cusp forms.

Abstract

We give a short and "soft" proof of the asymptotic orthogonality of Fourier coefficients of Poincar\'e series for classical modular forms as well as for Siegel cusp forms, in a qualitative form.