# Vorono\"{i} summation via switching cusps

## TL;DR

This paper develops a method to compute Fourier coefficients of cusp forms at various cusps using p-adic Whittaker functions and establishes a Voronoi summation formula with applications to bounding Fourier coefficient sums and Atkin-Lehner relations.

## Contribution

It introduces a new approach to compute Fourier coefficients at arbitrary cusps and derives a generalized Voronoi summation formula for cusp forms of general level.

## Key findings

- Derived a classical Voronoi summation formula with additive twists.
- Provided a method to compute Fourier coefficients via local p-adic theory.
- Applied results to bounds on Fourier coefficient sums and Atkin-Lehner relations.

## Abstract

We consider the Fourier expansion of a Hecke (resp.\ Hecke--Maa\ss) cusp form of general level $N$ at the various cusps of $\Gamma_{0}(N)\bs\Hb$. We explain how to compute these coefficients via the local theory of $p$-adic Whittaker functions and establish a classical Vorono\"i summation formula allowing an arbitrary additive twist. Our discussion has applications to bounding sums of Fourier coefficients and understanding the (generalised) Atkin--Lehner relations.