## Algebra & Number Theory

### Ekedahl–Oort strata of hyperelliptic curves in characteristic 2

#### 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 $JX[2]$ 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

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

Dates
Revised: 11 April 2012
Accepted: 16 April 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513729964

Digital Object Identifier
doi:10.2140/ant.2013.7.507

Mathematical Reviews number (MathSciNet)
MR3095219

Zentralblatt MATH identifier
1282.11065

#### Citation

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. https://projecteuclid.org/euclid.ant/1513729964

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