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

Author:

Stefan Reppen

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Number Theory (math.NT)

journal:

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date:

2023-06-26 16:00:00

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.