Curves in characteristic $2$ with non-trivial $2$-torsion

Wouter Castryck; Marco Streng; Damiano Testa

doi:10.3934/amc.2014.8.479

Article Contents

2014, Volume 8, Issue 4: 479-495. Doi: 10.3934/amc.2014.8.479

This issue Previous Article Smoothness testing of polynomials over finite fields Next Article On the irreducibility of the hyperplane sections of Fermat varieties in $\mathbb{P}^3$ in characteristic $2$

Curves in characteristic $2$ with non-trivial $2$-torsion

1.
Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, 3001 Leuven
2.
Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden
3.
Mathematics Institute, University of Warwick, Coventry CV4 7AL

Received: January 2014

Revised: September 2014

Published: November 2014

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Abstract

Cais, Ellenberg and Zureick-Brown recently observed that over finite fields of characteristic two, all sufficiently general smooth plane projective curves of a given odd degree admit a non-trivial rational $2$-torsion point on their Jacobian. We extend their observation to curves given by Laurent polynomials with a fixed Newton polygon, provided that the polygon satisfies a certain combinatorial property. We also show that in each of these cases, if the curve is ordinary, then there is no need for the words ``sufficiently general''. Our treatment includes many classical families, such as hyperelliptic curves of odd genus and $C_{a,b}$ curves. In the hyperelliptic case, we provide alternative proofs using an explicit description of the $2$-torsion subgroup.

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Mathematics Subject Classification: 14H25,14H45,14M25.

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