Beyond Endoscopy: Poisson Summation Formula and Kuznetsov Trace Formula on $\mathrm{GL}_2$

TL;DR

This paper provides a direct proof of a Poisson summation formula on the Whittaker space of GL2 and derives a Kuznetsov-type trace formula with non-standard test functions, linking the Poisson formula to L-function functional equations.

Contribution

It introduces a new direct proof of the Poisson summation formula on GL2's Whittaker space and develops a Kuznetsov trace formula with non-standard test functions.

Findings

Poisson summation formula underpins the local Hankel transform.

Derived Kuznetsov trace formula with non-standard test functions.

Poisson formula naturally yields the functional equation for GL2 L-functions.

Abstract

In the first part of this paper, we present a direct proof of a Poisson summation formula on the Whittaker space of $\mathrm{GL}_2$, which underlies the local Hankel transform computed by H. Jacquet. In the second part, we derive a Kuznetsov-type trace formula for incorporating non-standard test functions. This formulation reveals that the Poisson summation formula naturally yields the functional equation for the standard $L$-functions of $\mathrm{GL}_2$.