\centerline{\bigtitle The Langlands-Rapoport conjecture}
\centerline{Abstract} Let $(G,X)$ be a pair defining a Shimura variety of preabelian type. Let $p$ be an odd rational prime such that $G$ is unramified over $\dbQ_p$. Let $\dbF$ be the algebraic closure of the field with $p$ elements. Let $H$ be a hyperspecial subgroup of $G(\dbQ_p)$, and let $v$ be a prime of the reflex field $E(G,X)$ dividing $p$. It is known that ${\text Sh}_H(G,X)$ has an integral canonical model $\scrN$ over the localization of $E(G,X)$ with respect to $v$. In 1976 (i.e. long before the integral canonical models were defined rigurously and proved to exist) Langlands and Rapoport predicted the existence of a combinatorial description of the set $\scrN(\dbF)$ acted on by $G(\dbA_f^p)\times\Phi\dbZ$ (here $\Phi\dbZ$ is the infinite cyclic group generated by the Frobenius automorphism of $\dbF$ fixing the residue field $k(v)$ of $\dbF$).
In this talk we outline the proof of the following theorem:
\smallskip {\bf Theorem.} The Langlands-Rapoport conjecture is true if one of the following two conditions is satisfied: -- all the factors of $(G^{\text{ad}},X^{\text{ad}})$ are of $A_n$, $B_n$ or $D_n^{\dbR}$ type;
-- $k(v)$ is the field with $p$ elements.
\smallskip The proof of this Theorem is obtained by combining the previous work of Milne, Pfau and Zink, useful tricks involving injective maps between Shimura pairs, and techniques from the crystalline cohomology.
The talk is organized as follows. First we have a review on Shimura varieties of preabelian type. Second we review the content of the conjecture and the previous work done on it. Third we state the result and its consequences on the ordinary reduction conjecture of abelian varieties and $K3$ surfaces over number fields. Fourth, we outline the proof. \enddocument