On the Fourier Transform of Bessel Functions over Complex Numbers---II: the General Case

TL;DR

This paper derives exponential integral formulas for the Fourier transform of Bessel functions over complex numbers, facilitating advanced spectral analysis in number theory and extending key formulas to broader number fields.

Contribution

It introduces new exponential integral formulas for Bessel functions over complex numbers, enabling the development of complex spectral theory and extending the Waldspurger formula.

Findings

Proves an exponential integral formula for the Fourier transform of Bessel functions.

Develops a radial exponential integral formula for Bessel functions.

Extends the Waldspurger formula to arbitrary number fields.

Abstract

In this paper, we prove an exponential integral formula for the Fourier transform of Bessel functions over complex numbers, along with a radial exponential integral formula. The former will enable us to develop the complex spectral theory of the relative trace formula for the Shimura-Waldspurger correspondence and extend the Waldspurger formula from totally real fields to arbitrary number fields.