Points on Shimura varieties over finite fields: the conjecture of Langlands and Rapoport

The purpose of this article is to state a strong form of the conjecture of Langlands and Rapoport (1987, §5), and to outline a proof of it. For Shimura varieties of PEL type, the outline is complete, and for Shimura varieties of Hodge type it becomes complete if one accepts certain statements, stated as conjectures in the original version of this article, whose proof has been announced by Vasiu and Kisin. The methods of M. Pfau then show that the conjecture is true for all Shimura varieties except possibly those defined by groups of type E6, E7, and certain types D. As a consequence, for these Shimura varieties, we obtain the integral formula for the number of points conjectured by Langlands and Kottwitz. This article is a revised and updated version of Section 4 of Milne 1994b and of Milne 1995. It remains a work in progress.