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November  2014, 8(4): 479-495. doi: 10.3934/amc.2014.8.479

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

Wouter Castryck 1, , Marco Streng 2, and  Damiano Testa 3,

1. 

Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium
2. 

Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, Netherlands
3. 

Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Received  January 2014 Revised  September 2014 Published  November 2014

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.
Keywords: even characteristic., Algebraic curves, toric surfaces.
Mathematics Subject Classification: 14H25,14H45,14M2.
Citation: Wouter Castryck, Marco Streng, Damiano Testa. Curves in characteristic $2$ with non-trivial $2$-torsion. Advances in Mathematics of Communications, 2014, 8 (4) : 479-495. doi: 10.3934/amc.2014.8.479
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show all references

References:
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J. Symbolic Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.  Google Scholar

[2]

J. Inst. Math. Jussieu, 12 (2013), 651-676. doi: 10.1017/S1474748012000862.  Google Scholar

[3]

Discrete Comp. Geometry, 47 (2012), 496-518. doi: 10.1007/s00454-011-9376-2.  Google Scholar

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W. Castryck and F. Cools, Linear pencils encoded in the Newton polygon,, preprint., ().   Google Scholar

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Int. Math. Res. Pap., 2006, (2006), 1-57.  Google Scholar

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Algebra Number Theory, 3 (2009), 255-281. doi: 10.2140/ant.2009.3.255.  Google Scholar

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Springer, 2011.  Google Scholar

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in Proc. Adv. Cryptology - CRYPTO 2002, 2003, 308-323. doi: 10.1007/3-540-45455-1_25.  Google Scholar

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Finite Fields App., 12 (2006), 78-102. doi: 10.1016/j.ffa.2005.01.003.  Google Scholar

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Algebra Number Theory, 7 (2013), 507-532. doi: 10.2140/ant.2013.7.507.  Google Scholar

[12]

Linear Algebra Appl., 439 (2013), 2158-2166. doi: 10.1016/j.laa.2013.06.012.  Google Scholar

[13]

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[14]

J. Math. Kyoto Univ., 26 (1986), 375-386.  Google Scholar

[15]

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[16]

in Algorithms and Computation in Mathematics, Springer, 1999. doi: 10.1007/978-3-662-03642-6.  Google Scholar

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Ph.D thesis, Katholieke Universiteit Nijmegen, 1991. Google Scholar

[18]

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[19]

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[20]

Math. Res. Lett., 7 (2000), 77-82. doi: 10.4310/MRL.2000.v7.n1.a7.  Google Scholar

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[24]

in Oeuvres (collected papers), Springer, 1986, 544-568. Google Scholar

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[26]

Proc. Amer. Math. Soc., 134 (2006), 323-331. doi: 10.1090/S0002-9939-05-08294-8.  Google Scholar

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