Séminaire de Géométrie Arithmétique Paris-Pékin-Tokyo
# Goren-Oort stratification and Tate cycles on Hilbert modular varieties

##
by

→
Europe/Paris

Centre de conférences Marilyn et James Simons (IHES)
### Centre de conférences Marilyn et James Simons

#### IHES

Le Bois Marie
35, route de Chartres
91440 Bures-sur-Yvette

Description

Let B be a quaternionic algebra over a totally real field F, and *p* be a prime at least 3 unramified in F. We consider a Shimura variety X associated to B^{*} of level prime to *p*. A generalization of Deligne-Carayol's "modèle étrange" allows us to define an integral model for X. We will then define a Goren-Oort stratification on the characteristic *p* fiber of X, and show that each closed Goren-Oort stratum is an iterated P^{1}-fibration over another quaternionic Shimura variety in characteristic *p*. Now suppose that [F:Q] is even and that *p* is inert in F. An iteration of this construction gives rise to many algebraic cycles of middle codimension on the characteristic *p* fibre of Hilbert modular varieties of prime-to-*p* level. We show that the cohomological classes of these cycles generate a large subspace of the Tate cycles, which, in some special cases, coincides with the prediction of the Tate conjecture for the Hilbert modular variety over finite fields. This is a joint work with Liang Xiao.

Contact