On the Conjecture of Langlands and Rapoport

FORENOTE (2007): The remarkable conjecture of Langlands and Rapoport (1987) gives a purely group-theoretic description of the points on a Shimura variety modulo a prime of good reduction. In an article in the proceedings of the 1991 Motives conference (Milne 1994, §4), I gave a heuristic derivation of the conjecture assuming a sufficiently good theory of motives in mixed characteristic. I wrote the present article in order to examine what was needed to turn the heuristic argument into a proof, and I distributed it to a few mathematicians (including Vasiu). Briefly, for Shimura varieties of Hodge type (i.e., those embeddable into Siegel modular varieties) I showed that the conjecture is a consquence of three statements: (a) a good theory of rational Tate classes (see statements (a,b,c,d) in §3 below); (b) existence of an isomorphism between integral étale and de Rham cohomology for an abelian scheme over the Witt vectors (see 0.1, 5.4 below); (c) every point in Shp(F) lifts to a special point in Sh(Qal). At the time I wrote the article, I erroneously believed that my work on Lefschetz classes etc. (Milne 1999a,b, 2002, 2005) implied (a). This work does show that (a) (and much more) follows from the Hodge conjecture for complex abelian varieties of CM-type, and in Milne 2007, I discuss some (apparently) much more accessible statements that imply (a). Also, it is possible to prove a variant of (a), which should suffice (in the presence of (b) and (c)) to prove a variant of the conjecture of Langlands and Rapoport, which becomes the true conjecture in the presence of (a). The situation concerning (b) and (c) is better since Vasiu (2003b) and Kisin (2007) have announced proofs of (b), and Vasiu (2003a) has announced a proof of (c). Finally, I mention that Pfau (1993, 1996b,a) has shown that the conjecture of Langlands and Rapoport for Shimura varieties of Hodge type implies that it holds for all Shimura varieties of abelian type (i.e., except for those defined by groups of type E6, E7, and mixed type D). Thus, the case of Shimura varieties of Hodge type is the crucial one. It should be clear from what I have already written, that the present manuscript is only a rough working draft, and not a polished work — everything in it should be taken with a grain of salt.