Fourier Coefficients of Theta Functions at Cusps other than Infinity

TL;DR

This paper derives explicit formulas for the Fourier coefficients of theta functions associated with Dirichlet characters at various cusps, expanding understanding beyond the traditional cusp at infinity using adelic methods.

Contribution

It introduces a novel approach using adelic Schwartz space and metaplectic group actions to compute Fourier coefficients at non-infinity cusps.

Findings

Explicit formulas for Fourier coefficients at all cusps.

Method based on adelic Schwartz space and metaplectic group.

Restrictions on Dirichlet characters for simplicity.

Abstract

In this paper we study the Fourier coefficients of theta functions attached to Dirichlet characters at cusps other than infinity. The method is based on expressing them in terms of explicit elements of the adelic Schwartz space and studying the action of the adelic metaplectic group on these elements. We derive explicit formulae for the Fourier coefficients at all cusps. For the sake of simplicity, some restrictions are placed on the Dirichlet characters considered.