Beyond Endoscopy for $\mathrm{GL}(3, \mathbb{Q})$: Poisson Summation

TL;DR

This paper extends the Poisson Summation method to $ ext{GL}(3, ext{Q})$ within the Beyond Endoscopy framework, providing explicit formulas for the trace formula's elliptic part and isolating the trivial representation's contribution.

Contribution

It generalizes the Poisson Summation approach to $ ext{GL}(3, ext{Q})$ and introduces new techniques involving cubic orders, zeta functions, and arithmetic periodicity to analyze the trace formula.

Findings

Explicit expansion of elliptic trace terms

Evaluation of Kloosterman-type sums and Dirichlet series

Identification of the trivial representation contribution

Abstract

We generalize to $\mathrm{GL}(3,\mathbb{Q})$ the Poisson Summation method developed by Altu\u{g} for $\mathrm{GL}(2, \mathbb{Q})$ for the strategy of Beyond Endoscopy. Concretely, assuming Conjecture A, we isolate the contribution of the trivial representation from the regular elliptic part of the trace formula and obtain a concrete expansion of \[ \mathrm{I}_{\mathrm{ell}}(f)-\mathrm{Tr}(\mathbf{1}(f)). \] Our starting point is a reformulation of the regular elliptic part in terms of cubic orders attached to characteristic polynomials. To these orders we associate a zeta function, defined through their overorders, prove a functional equation for its completion, and apply an approximate functional equation to rewrite the elliptic term in a form suitable for Poisson Summation. A key arithmetic input is a periodicity theorem showing that the relevant coefficients depend only on the parameters modulo a finite modulus. This makes it possible to perform Poisson Summation on the integral parameters indexed by the cubic data. The resulting main terms are governed by Kloosterman-type sums and an associated double Dirichlet series, which we evaluate explicitly. From this evaluation we are able to apply the residue theorem and find the trace of the trivial representation as a residue. We also recover the contribution of the \say{special} representation.