Stratifying Lie Strata of Hilbert Modular Varieties

In this survey we explain a stratification of a Hilbert modular variety M-E in characteristic p > 0 attached to a totally real number field E. This stratification refines the stratification of M-E by Lie type, and has the property that many strata are central leaves in M-E, called distinguished central leaves. In the case when the totally real field E is unramified above p, this stratification reduces to the stratification of M-E by alpha-type first introduced by Goren and Oort and studied by Yu, and coincides with the EO stratification on M-E.( )Moreover it is known that every non-supersingular alpha-stratum of M-E is irreducible. To treat the general case where E may be ramified above p, a key ingredient is the notion of congruity, a p-adic numerical invariant for abelian varieties with real multiplication by O-E in characteristic p. For every Lie stratum N (e) under bar on M-E, this new invariant defines a finite number of locally closed subsets Q((c) under bar) (N-(e) under bar), and N (e) under bar is the disjoint union of these Liecongruity strata Q (c) under bar (N-(e) under bar) in N-(e) under bar. The incidence relation between the Lie-congruity strata enables one to show that the prime-to-p Hecke correspondences operate transitively on the set of all irreducible components of any distinguished central leaf in M-E, see Theorems 7.1, 8.1 and 9.1. The Hecke transitivity implies, according to the method of prime-to-p monodromy of Hecke invariant subvarieties, that every non-supersingular distinguished central leaf in a Hilbert modular variety M-E is irreducible. The last irreducibility result is a key ingredient of the proof the Hecke orbit conjecture for Siegel modular varieties.