Theoretical Aspects of the Trace Formula for GL ( 2 )

The Selberg-Arthur trace formula is one of the tools available for approaching the conjecture of global functoriality in the Langlands program. Global functoriality is described within this volume in [Kn2]. We start with reductive groups G and H, say over the rationals Q for simplicity. We assume that G is quasisplit, and we suppose that we are given an L homomorphism ψ : H → G. From an automorphic representation of the adeles of H, we use ψ to construct, place-byplace from the Local Langlands Conjecture (or at almost every place without the conjecture), an irreducible representation of the adeles of G. The question of global functoriality is whether the latter representation is automorphic (or, in the case that it is defined only at almost every place, whether it can be completed to an automorphic representation). If it is automorphic, then we want to know also what conditions ensure that a cuspidal representation of the adeles of H yields a cuspidal representation of the adeles of G under this process. It is known that these questions capture various deep conjectures in classical algebraic number theory, arithmetic algebraic geometry, and representation theory and that they unify and generalize such conjectures significantly. The trace formula for the reductive group G gives information about the multiplicity of the occurrence of an irreducible representation of the adeles of G in the cuspidal spectrum. If Z denotes the center of G, the quotient Z(A)G(Q)\G(A) is almost compact in the sense that it has finite volume. If Z(A)G(Q)\G(A) is actually compact and if R denotes the right regular representation of G(A) on L2(Z(A)G(Q)\G(A)), then the trace formula will assert the equality of two expressions for Tr(R(φ)) on this L space, φ being a suitably regular function of compact support on G(A). In the notation of [Ar4], the formula in the compact case has the shape ∑