On the Fourier transform of regularized Bessel functions on complex numbers and Beyond Endoscopy over number fields

TL;DR

This paper derives integral formulas for Bessel functions over complex numbers and applies these results to extend Venkatesh's Beyond Endoscopy method from totally real to arbitrary number fields.

Contribution

It introduces new integral formulas for Bessel functions over complex numbers and extends Beyond Endoscopy techniques to general number fields.

Findings

Derived Weber-Schafheitlin type integral formulas for Bessel functions over complex numbers.

Extended Beyond Endoscopy for Sym^2 on GL_2 from totally real to arbitrary number fields.

Abstract

In this article, we prove certain Weber-Schafheitlin type integral formulae for Bessel functions over complex numbers. A special case is a formula for the Fourier transform of regularized Bessel functions on complex numbers. This is applied to extend the work of A. Venkatesh on Beyond Endoscopy for $\mathrm{Sym}^2$ on $\mathrm{GL}_2$ from totally real to arbitrary number fields.