Beyond Endoscopy for $\mathrm{GL}(3, \mathbb{Q})$: Isolation of the residual spectrum via Poisson Summation

Abstract

We extend to $\mathrm{GL}(3,\mathbb Q)$ the Poisson-summation method developed by Altuğ for $\mathrm{GL}(2,\mathbb Q)$ in Beyond Endoscopy. For a fixed prime and a family of factorizable test functions, we isolate the contribution of the trivial representation from the regular elliptic part of the trace formula and obtain an explicit expansion of \[ \mathrm{I}_{\mathrm{ell}}(f^{p,k})-\mathrm{Tr}(\mathbf{1}(f^{p,k})). \] We rewrite the finite orbital integrals in terms of an overorder zeta function attached to the monogenic cubic orders defined by the characteristic polynomials. Its identification with Yun's zeta function supplies the functional equation needed for an approximate functional equation. A periodicity theorem then makes Poisson summation in the two polynomial coefficients possible. The zero frequency is governed by Kloosterman-type sums and a Dirichlet series that admits a uniform evaluation. The residues at the origin recover the trace of the trivial representation and one third of the trace of the normalized representation induced from the trivial character of the Borel subgroup.