On the Hasse invariant of Hilbert modular varieties mod p

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 mth piece of the Hodge filtration. This result is a direct analogue of Ogus' on families of Calabi–Yau varieties in positive characteristic (see [15]). We also show that the order of vanishing at a point is the same as the codimension of the Ekedahl-Oort stratum containing it.