Geometrization of Trace Formulas

Recently, a new approach to functoriality of automorphic representations [L1] has been proposed in [FLN] following earlier work [L2, L3, L4] by Robert Langlands. The idea may be roughly summarized as follows. Let G and H be two reductive algebraic groups over a global field F (which is either a number field or a function field; that is, the field of functions on a smooth projective curve X over a finite field), and assume that G is quasi-split. Let LG and LH be the Langlands dual groups as defined in [L1]. The functoriality principle states that for each homomorphism LH → LG there exists a transfer of automorphic representations from H(A) to G(A), where A is the ring of adéles of F . In other words, to each L-packet of automorphic representations of H(A) should correspond an L-packet of automorphic representations of G(A), and this correspondence should satisfy some natural properties. Functoriality has been established in some cases, but is unknown in general. In [L2, L3, L4, FLN] the following strategy for proving functoriality was proposed. In the space of automorphic functions on G(F )\G(A) one should construct a family of integral operators which project onto those representations which come by functoriality from other groups H. One should then use the trace formula to decompose the traces of these operators as sums over these H (there should be sufficiently many operators to enable us to separate the different H). The hope is that analyzing the orbital side of the trace formula and comparing the corresponding orbital integrals for G and H, one should ultimately be able prove functoriality. In the present paper we make the first steps in developing geometric methods for analyzing these orbital integrals in the case of the function field of a curve X over a finite field Fq. We also suggest a conjectural framework of “geometric trace formulas” in the case of curves defined over the complex field.