Algebra & Number Theory

Ekedahl–Oort strata of hyperelliptic curves in characteristic 2

Arsen Elkin and Rachel Pries

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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.

Article information

Algebra Number Theory, Volume 7, Number 3 (2013), 507-532.

Received: 7 July 2010
Revised: 11 April 2012
Accepted: 16 April 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G20: Curves over finite and local fields [See also 14H25]
Secondary: 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx] 14L15: Group schemes 14H40: Jacobians, Prym varieties [See also 32G20] 14F40: de Rham cohomology [See also 14C30, 32C35, 32L10] 11G10: Abelian varieties of dimension > 1 [See also 14Kxx]

curve hyperelliptic Artin–Schreier Jacobian $p$-torsion $a$-number group scheme de Rham cohomology Ekedahl–Oort strata


Elkin, Arsen; Pries, Rachel. Ekedahl–Oort strata of hyperelliptic curves in characteristic 2. Algebra Number Theory 7 (2013), no. 3, 507--532. doi:10.2140/ant.2013.7.507.

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