The spectral side of Arthur's trace formula

Abstract

The trace formula is one of the most important tools in the theory of automorphic forms. It was invented in the 1950's by Selberg, who mostly studied the case of hyperbolic surfaces, and was later on developed extensively by Arthur in the generality of an adelic quotient of a reductive group over a number field. Here we provide an explicit expression for the spectral side, improving Arthur's fine spectral expansion. As a result, we obtain its absolute convergence for a wide class of test functions.