Vol. 7, No. 3, 2013

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Ekedahl–Oort strata of hyperelliptic curves in characteristic 2

Arsen Elkin and Rachel Pries

Vol. 7 (2013), No. 3, 507–532
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.

Keywords
curve, hyperelliptic, Artin–Schreier, Jacobian, $p$-torsion, $a$-number, group scheme, de Rham cohomology, Ekedahl–Oort strata
Mathematical Subject Classification 2010
Primary: 11G20
Secondary: 14K15, 14L15, 14H40, 14F40, 11G10