The trace formula of GL(3)

The trace formula constitutes a fundamental tool in the Langlands program. In general, Arthur introduced a truncation operator to render both the geometric and spectral sides of the formula convergent. This paper focuses on the case of $\mathrm{GL}(3)$. We first prove that the divergent terms on the geometric and spectral sides are equal, leading to their cancellation. We derive an explicit formula for ramified orbital integrals, showing they are limits of unramified ones and that Arthur’s definition yields a universal object, agreeing with that of Hoffmann-Wakatsuki. Finally, on the spectral side, we apply normalized intertwining operators to present the expansion in a form parallel to that of the geometric side.

💡 Research Summary

The paper “The Trace Formula of GL(3)” gives a complete and explicit treatment of Arthur’s trace formula in the concrete case of the general linear group of rank three. After a brief historical introduction, the authors fix the global field ℚ, the adele ring 𝔸, and the standard minimal parabolic subgroup P₀ with Levi component M₀. They recall Arthur’s truncation operator Λ_T, which is a finite linear combination of characteristic functions depending on a parameter T, and explain how it renders both the geometric and spectral sides of the trace formula convergent.

The geometric side is first decomposed according to the conjugacy classes O in G(ℚ). For GL(3) the set O splits into five families: o_G, o_{21}, o_{0 111} (unramified) and o_{2 111}, o_{3 111} (ramified). The authors describe how each orbit determines a minimal Levi subgroup M and a semisimple representative γ. They distinguish unramified orbits, where the centralizer G(γ) coincides with M(γ), from ramified ones, where G(γ) is strictly larger.

A central technical achievement is Theorem 1.1 (Theorem 9.4), which proves that the divergent contributions on the geometric side, denoted J^{d geo}(f), exactly match the divergent contributions on the spectral side, J^{d spec}(f). This equality is established by a careful analysis of the truncated kernels K_{P,o} and K_{P,χ}, together with explicit polynomial dependence on the truncation parameter T. Consequently, after subtraction of the divergent part, the remaining terms are polynomially bounded and converge absolutely.

The paper then tackles ramified orbital integrals, a topic where Arthur’s original work gave only an implicit definition. The authors introduce a limiting process: for a ramified element γ they choose a suitable element a∈A_M such that G(aγ)=M(aγ) and define the ramified distribution as a limit of unramified distributions, \