Ekedahl-Oort strata of hyperelliptic curves in characteristic 2

TL;DR

This paper classifies the possible 2-torsion group schemes of Jacobians of hyperelliptic curves over algebraically closed fields of characteristic 2, linking their structure to ramification invariants and Ekedahl-Oort types.

Contribution

It provides a complete classification of the 2-torsion group schemes for hyperelliptic Jacobians in characteristic 2, based on ramification data and Ekedahl-Oort stratification.

Findings

Decomposition of de Rham cohomology indexed by branch points

Explicit computation of the 2-torsion group scheme class

Classification of all possible 2-torsion group schemes for hyperelliptic curves

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 $J_X[2]$ of the Jacobian of $X$ in terms of the Ekedahl-Oort type. The interesting feature is that $J_X[2]$ 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 which occur as the $2$-torsion group schemes of Jacobians of hyperelliptic $k$-curves of arbitrary genus.