10/30/2008

A.Vasiu
Binghampton University

On the Tate and Langlands--Rapoport conjectures for Shimura varieties of Hodge type

Let p be a prime. Let F be an algebraic closure of the finite field F_p with p elements. An integral canonical model N of a Shimura variety Sh(G,X) of Hodge type is a regular, closed subscheme of a suitable pull back of the Mumford moduli tower M over Z_{(p)}. We recall that M parametrizes isomorphism classes of principally polarized abelian schemes over Z_{(p)}-schemes which have a fixed relative dimension and which have level-m symplectic similitude structures for all m prime to p. Deep conjectures of Tate and Langlands--Rapoport pertain to points of N with values in an algebraic closure of the field with p elements. We report on the proof of the Langlands--Rapoport conjecture for those Sh(G,X) with the property that each simple factor of the adjoint Shimura pair (G^{ad},X^{ad}) has compact factors and it is not of D_n^{H} type. As a key ingredient we get an ad\'elic version of the Tate conjecture for many supersingular abelian varieties which are associated to F-valued points of certain N.