On the Hasse invariant of Hilbert modular varieties mod $p$

TL;DR

This paper investigates the properties of the Hasse invariant on Hilbert modular varieties in characteristic p, relating its order of vanishing to the conjugate and Hodge filtrations, and connecting it to Ekedahl-Oort strata.

Contribution

It establishes a precise relation between the order of vanishing of the Hasse invariant and the conjugate filtration, extending Ogus' results to Hilbert modular varieties.

Findings

Order of vanishing equals the largest m with the conjugate filtration in the m-th Hodge filtration.

Order of vanishing at a point equals the codimension of the Ekedahl-Oort stratum.

The result generalizes Ogus' theorem to the setting of Hilbert modular varieties.

Abstract

Let $F$ be a totally real field and let $S$ denote the geometric special fiber of a Hilbert modular variety associated to $F$, at a prime unramified in $F$. We show that the order of vanishing of the Hasse invariant on $S$ is equal to the largest integer $m$ such that the smallest piece of the conjugate filtration lies in the $m^{\text{th}}$ piece of the Hodge filtration. This result is a direct analogue of Ogus' on families of Calabi-Yau varieties in positive characteristic. We also show that the order of vanishing at a point is the same as the codimension of the Ekedahl-Oort stratum containing it.