Ekedahl–Oort strata of hyperelliptic curves in characteristic 2

Abstract

Suppose $X$ is a hyperelliptic curve of genus $g$ defined over an algebraically closed field $k$ of characteristic $p=2$. We prove that the de Rham cohomology of $X$ decomposes into pieces indexed by the branch points of the hyperelliptic cover. This allows us to compute the isomorphism class of the $2$-torsion group scheme $J_X\left[2\right]$ of the Jacobian of $X$ in terms of the Ekedahl–Oort type. The interesting feature is that $J_X\left[2\right]$ depends only on some discrete invariants of $X$, namely, on the ramification invariants associated with the branch points. We give a complete classification of the group schemes that occur as the $2$-torsion group schemes of Jacobians of hyperelliptic $k$-curves of arbitrary genus, showing that only relatively few of the possible group schemes actually do occur.