Refinements of the trace formula for GL(2)

TL;DR

This paper refines the trace formula for GL(2), expressing noncuspidal terms via orbital integrals, and explores implications for the Ramanujan conjecture and beyond endoscopy methods.

Contribution

It provides a geometric expansion of the trace formula's cuspidal part and links it to the tempered spectrum under the Ramanujan conjecture, offering an alternative approach to beyond endoscopy.

Findings

Geometric expansion of the trace formula for GL(2)

Conditional results assuming Ramanujan conjecture

Framework for an $r$-trace formula

Abstract

We express the discrete noncuspidal terms in the spectral side of the trace formula for GL(2) in terms of orbital integrals, obtaining a geometric expansion for the cuspidal part of the trace formula. Assuming the Ramanujan conjecture for GL(2), this is equal to the tempered part of the trace formula, providing an alternate approach to the method of Frenkel-Langlands-Ngo in the context of Beyond Endoscopy. Using this, we establish a formula which in principle leads to an $r$-trace formula, and conclude with some remarks regarding the primitivization of the trace formula.