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

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 JX[2] of the Jacobian of X in terms of the Ekedahl–Oort type. The interesting feature is that JX[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.

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
Received: 7 July 2010
Revised: 11 April 2012
Accepted: 16 April 2012
Published: 23 August 2013
Arsen Elkin
Mathematics Institute
University of Warwick
United Kingdom
Rachel Pries
Department of Mathematics
Colorado State University
Fort Collins, CO
United States