A note on Poisson summation for GL(2)

Using analytic number theory techniques, Altuğ showed that the contribution of the trivial representation to the Arthur-Selberg trace formula for GL(2) over $\Q$ could be cancelled by applying a modified Poisson summation formula to the regular elliptic contribution. Drawing on recent works, we re-examine these methods from an adelic perspective.

💡 Research Summary

The paper revisits the cancellation of the trivial representation in the Arthur‑Selberg trace formula for GL(2) over ℚ, a key step in Langlands’ “Beyond Endoscopy” program. Earlier works (FLN10, Alt15, EELKW24, Che25) showed that a modified Poisson summation applied to the regular elliptic part could eliminate the contribution of the trivial representation, but the arguments relied heavily on classical analytic number theory and faced three main obstacles: (i) the global volume term is defined only over the field and not adelically, (ii) the sum over the Steinberg‑Hitchin base A(F) is incomplete because it runs only over regular elliptic elements, and (iii) orbital integrals have singularities that must be controlled before a Poisson formula can be applied.

The author proposes an adelic reformulation that resolves these issues. Two independent adelic approaches are presented. The first follows Langlands’ analysis of stable orbital integrals on the Steinberg‑Hitchin base and Gordon’s work on measures. By distinguishing the geometric measure (derived from the differential form ω_{c^{-1}} on the fibers of the map c:G→A) from the canonical (quotient) measure, the paper obtains a clean expression for the normalized orbital integral θ_f(a) in terms of the Weyl discriminant D(γ) and an integral over T\G. Lemma 2.2 shows that θ_f is essentially compactly supported on A(F) and smooth except at two points where it is merely continuous. This regularity is sufficient to define a Fourier transform (\widehat{θ}_f) in the G_a‑direction.

The second obstacle—completion of the sum—is tackled by using the geometric measure rather than the canonical one. The canonical measure carries an Artin L‑value factor that diverges for split tori when evaluated at s=1. By expressing this L‑value through an approximate functional equation (a standard tool in analytic number theory), the author replaces the divergent factor with a well‑behaved Dirichlet series attached to a quadratic symbol. This allows the sum over A(F) to be extended to a full lattice without introducing divergence. Lemma 2.3 formalizes this completion, yielding the identity \