Points on some Shimura varieties over finite fields

The Hasse-Weil zeta functions of varieties over number fields are conjecturally products (and quotients) of automorphic L-functions.For a Shimura variety S associated to a connected reductive group Gover Q one can hope to be more specific about which automorphic L-functions appear in the zeta function.In fact Eichler, Shimura, Kuga, Sato, and Ihara, who studied GL 2 and its inner forms, found in those cases that it was enough to use automorphic L-functions for the group G itself.In the general case Langlands [L2-L4] has given a conjectural description of the zeta function in terms of automorphic L-functions for G and its endoscopic groups [LS] (see also [KS] for the contribution of non-tempered representations), and a description of this type has been verified in certain cases, beginning with [L3].For GL 2 it was possible to use the Eichler-Shimura congruence relation in order to make the connection between the zeta function and automorphic L- functions.In general one needs more information than the Eichler-Shimura congruence relation gives, and it seems to be necessary to describe the points on S over finite fields in terms of group-theoretical data (for the group G), in a way that is adapted to an eventual comparison of the number of points modulo p with the Selberg trace formula for G (actually with the stable trace formulas for G and its endoscopic groups), as is explained in [L2, L3, KS].Ihara [11, 12] gave such a group-theoretical description of points modulo p in the case of GL 2 (Q) and its inner forms, and Deligne [D!] gave a related description of the category of ordinary abelian varieties over a finite field.Langlands [Ll] conjectured a group-theoretical description in the general case, based on a detailed though incomplete study of Shimura varieties of PEL type [S]; Milne [Mil] gave a simplified exposition of this work of Langlands in a special case, using the description of the category of abelian varieties up to isogeny over a finite field due to Honda [H] and Tate [T2, T3].Zink [Z2] gave complete proofs for part of Langlands's conjectures for Shimura varieties of PEL type, but by that time it was clear that a new idea was needed to give a complete proof for the full conjecture, even for the case of the group of symplectic similitudes.In fact the conjecture itself needed some refinement; this was one of the objects of some work by Langlands-Rapoport [LR], whose main goal, however, was to put the conjecture into a Tannakian framework, in which it became conceptually clearer.The final step was taken independently by