Beyond endoscopy for the Rankin-Selberg L-function

TL;DR

This paper introduces a novel approach using trace formulas to analyze the poles of Rankin-Selberg L-functions, providing new insights and proofs in analytic number theory without relying on traditional endoscopic methods.

Contribution

It presents a new method for studying L-function poles via trace formulas, avoiding endoscopic techniques, and uncovers the convolution operation for Bessel transforms.

Findings

New proof of the analyticity of Rankin-Selberg L-function at s=1

Development of a convolution operation for Bessel transforms

Alternative approach to understanding L-function poles

Abstract

We try to understand the poles of L-functions via taking a limit in a trace formula. This technique avoids endoscopic and Kim-Shahidi methods. In particular, we investigate the poles of the Rankin-Selberg L-function. Using analytic number theory techniques to take this limit, we essentially get a new proof of the analyticity of the Rankin-Selberg L-function at $s=1.$ Along the way we discover the convolution operation for Bessel transforms.