Ekedahl–Oort strata of hyperelliptic curves in characteristic 2

Elkin, Arsen; Pries, Rachel

doi:10.2140/ant.2013.7.507

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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

DOI: 10.2140/ant.2013.7.507

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 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

Milestones

Received: 7 July 2010

Revised: 11 April 2012

Accepted: 16 April 2012

Published: 23 August 2013

Authors

Arsen Elkin
Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom

Rachel Pries
Department of Mathematics Colorado State University Fort Collins, CO 80523 United States

Vol. 7, No. 3, 2013