Stratifying lie strata of hilbert modular varieties

In this survey we explain a stratification of a Hilbert modular variety ME in characteristic p > 0 attached to a totally real number field E. This stratification refines the stratification of ME by Lie type, and has the property that many strata are central leaves in ME, called distinguished central leaves. In the case when the totally real field E is unramified above p, this stratification reduces to the stratification of ME by α-type first introduced by Goren and Oort and studied by Yu, and coincides with the EO stratification on ME. Moreover it is known that every non-supersingular α-stratum of ME 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 OE in characteristic p. For every Lie stratum Ne on ME, this new invariant defines a finite number of locally closed subsets Qc(Ne), and Ne is the disjoint union of these Lie-congruity strata Qc(Ne) in Ne. 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 ME, 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 ME is irreducible. The last irreducibility result is a key ingredient of the proof the Hecke orbit conjecture for Siegel modular varieties.