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Goren-Oort stratification and Tate cycles on Hilbert modular varieties
(Morningside Center for Mathematics)
Centre de conférences Marilyn et James Simons (IHES)
Centre de conférences Marilyn et James Simons
Le Bois Marie
35, route de Chartres
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 P1-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.