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

# 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 adelic version of the Tate conjecture for many supersingular abelian varieties which are associated to $F$-valued points of certain $N$.