On the Fourier transform of regularized Bessel functions on complex numbers and Beyond Endoscopy over 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 Sym 2 on GL 2 from totally real to arbitrary number fields.

A Voronoï--Oppenheim summation formula for number fields

“…Proof of Lemma 4.4. By [GR,3.761 9] and [Qi2,Lemma 4.4], the integrals in (4.9) and (4.10), respectively, are equal to 2p2πyq ´2ν Γp2νq cospπνq and 2p2πyq ´2ν Γpνq 2 sinpπνq, and we arrive at the right-hand sides of (4.9) and (4.10) by the duplication and the reflection formulae for the gamma function.…”