The trace formula of GL(3)

Abstract

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.