Stratifying Lie Strata of Hilbert Modular Varieties

Abstract

In this survey we explain a stratification of a Hilbert modular variety $\mathscr{M}_{E}$ in characteristic $p \gt 0$ attached to a totally real number field $E$. This stratification refines the stratification of $\mathscr{M}_{E}$ by Lie type, and has the property that many strata are central leaves in $\mathscr{M}_{E}$, called distinguished central leaves.

In the case when the totally real field $E$ is unramified above $p$, this stratification reduces to the stratification of $\mathscr{M}_{E}$ by $\alpha$-type first introduced by Goren and Oort and studied by Yu, and coincides with the EO stratification on $\mathscr{M}_{E}$. Moreover it is known that every non-supersingular $\alpha$-stratum of $\mathscr{M}_{E}$ 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 $\mathcal{O}_E$ in characteristic $p$. For every Lie stratum $\mathcal{N}_{\underline{e}}$ on $\mathscr{M}_{E}$, this new invariant defines a finite number of locally closed subsets $\mathcal{Q}_{\underline{c}}(\mathcal{N}_{\underline{e}})$, and $\mathcal{N}_{\underline{e}}$ is the disjoint union of these Lie-congruity strata $\mathcal{Q}_{\underline{c}}(\mathcal{N}_{\underline{e}})$ in $\mathcal{N}_{\underline{e}}$.

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 $\mathscr{M}_{E}$, 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 $\mathscr{M}_{E}$ is irreducible. The last irreducibility result is a key ingredient of the proof the Hecke orbit conjecture for Siegel modular varieties.