The Dieudonn\'{e} modules and Ekedahl-Oort types of Jacobians of hyperelliptic curves in odd characteristic

Given a principally polarized abelian variety $A$ of dimension $g$ over an algebraically closed field $k$ of characteristic $p$, the $p$ torsion $A[p]$ is a finite flat $p$-torsion group scheme of rank $p^{2g}$. There are exactly $2^g$ possible group schemes that can occur as some such $A[p]$. In this paper, we study which group schemes can occur as $J[p]$, where $J$ is the Jacobian of a hyperelliptic curve defined over $\mathbb{F}_p$. We do this by computing explicit formulae for the action of Frobenius and its dual on the de Rham cohomology of a hyperelliptic curve with respect to a given basis. A theorem of Oda's in [Oda69] allows us to relate these actions to the $p$-torsion structure of the Jacobian. Using these formulae and the computer algebra system Magma, we affirmatively resolve questions of Glass and Pries in [Cor05] on whether certain group schemes of rank $p^8$ and $p^{10}$ can occur as $J[p]$ of a hyperelliptic curve of genus $4$ and $5$ respectively.