Beyond endoscopy for the relative trace formula I: local theory

TL;DR

This paper establishes a nonstandard matching between two relative trace formulas for PGL(2), using explicit integral transforms, paving the way for new proofs of classical results like Waldspurger's theorem.

Contribution

It proves nonstandard matching and the fundamental lemma between Jacquet's relative trace formula and the Kuznetsov trace formula for PGL(2), introducing explicit integral transforms.

Findings

Established nonstandard matching between trace formulas

Proved fundamental lemma for PGL(2)

Set groundwork for reproof of Waldspurger's theorem

Abstract

For the group G = PGL(2) we prove nonstandard matching and the fundamental lemma between two relative trace formulas: on one hand, the relative trace formula of Jacquet for the quotient T\G/T, where T is a nontrivial torus; on the other, the Kuznetsov trace formula with nonstandard test functions. The matching is nonstandard in the sense that orbital integrals are related to each other not one-by-one, but via an explicit integral transform. These results will be used in the sequel to compare the corresponding global trace formulas and reprove the celebrated result of Waldspurger on toric periods.