In this talk I will report on joint work with E. de Shalit (Hebrew U) on foliations on PEL Shimura varieties. The foliations that we construct are of two different types. One construction of foliations, “tautological foliations”, relies on the structure on the endomorphism ring and is an arithmetic construction. We will exemplify it for the case of Hilbert modular varieties. The other construction of foliations, “V foliations”, relies on properties of the Gauss-Manin connection in positive characteristic. We will illustrate this construction in the case of unitary Shimura varieties. We expect our methods to extend to a large class of Shimura varieties.

As will be clear from the results, the introduction of these foliations sheds light on the structure of the reduction of Shimura varieties with parahoric level structure, theta operators and generalized Hasse invariants.