On the Hasse invariant of Hilbert modular varieties mod p

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. & COPY; 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).

Automated summary

This article studies the geometric special fiber of a Hilbert modular variety associated to a totally real field, at a prime unramified in the field. The authors show that the order of vanishing of the Hasse invariant on the fiber 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, which is analogous to Ogus' result on families of Calabi-Yau varieties in positive characteristic. They also demonstrate that the order of vanishing at a point is the same as the codimension of the Ekedahl-Oort stratum containing it.