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

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

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.
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
References:
[1]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language,, J. Symbolic Comput., 24 (1997), 235. doi: 10.1006/jsco.1996.0125. Google Scholar

[2]

B. Cais, J. Ellenberg and D. Zureick-Brown, Random Dieudonné modules, random $p$-divisible groups, and random curves over finite fields,, J. Inst. Math. Jussieu, 12 (2013), 651. doi: 10.1017/S1474748012000862. Google Scholar

[3]

W. Castryck, Moving out the edges of a lattice polygon,, Discrete Comp. Geometry, 47 (2012), 496. doi: 10.1007/s00454-011-9376-2. Google Scholar

[4]

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

[5]

W. Castryck, J. Denef and F. Vercauteren, Computing zeta functions of nondegenerate curves,, Int. Math. Res. Pap., 2006 (2006), 1. Google Scholar

[6]

W. Castryck, A. Folsom, H. Hubrechts and A. V. Sutherland, The probability that the number of points on the Jacobian of a genus $2$ curve is prime,, Proc. London Math. Soc., 104 (2012), 1235. doi: 10.1112/plms/pdr063. Google Scholar

[7]

W. Castryck and J. Voight, On nondegeneracy of curves,, Algebra Number Theory, 3 (2009), 255. doi: 10.2140/ant.2009.3.255. Google Scholar

[8]

D. Cox, J. Little and H. Schenck, Toric Varieties,, Springer, (2011). Google Scholar

[9]

J. Denef and F. Vercauteren, Computing zeta functions of hyperelliptic curves over finite fields of characteristic $2$,, in Proc. Adv. Cryptology - CRYPTO 2002, (2002), 308. doi: 10.1007/3-540-45455-1_25. Google Scholar

[10]

J. Denef and F. Vercauteren, Computing zeta functions of $C_{a,b}$ curves using Monsky-Washnitzer cohomology,, Finite Fields App., 12 (2006), 78. doi: 10.1016/j.ffa.2005.01.003. Google Scholar

[11]

A. Elkin and R. Pries, Ekedahl-Oort strata of hyperelliptic curves in characteristic $2$,, Algebra Number Theory, 7 (2013), 507. doi: 10.2140/ant.2013.7.507. Google Scholar

[12]

S. Farnell and R. Pries, Families of Artin-Schreier curves with Cartier-Manin matrix of constant rank,, Linear Algebra Appl., 439 (2013), 2158. doi: 10.1016/j.laa.2013.06.012. Google Scholar

[13]

C. Haase and J. Schicho, Lattice polygons and the number $2i+7$,, Amer. Math. Monthly, 116 (2009), 151. doi: 10.4169/193009709X469913. Google Scholar

[14]

R. Hartshorne, Generalized divisors on Gorenstein curves and a theorem of Noether,, J. Math. Kyoto Univ., 26 (1986), 375. Google Scholar

[15]

N. Katz and P. Sarnak, Random Matrices, Frobenius Eigenvalues and Monodromy,, AMS, (1998). Google Scholar

[16]

N. Koblitz, Algebraic aspects of cryptography,, in Algorithms and Computation in Mathematics, (1999). doi: 10.1007/978-3-662-03642-6. Google Scholar

[17]

R. Koelman, The Number of Moduli of Families of Curves on Toric Surfaces,, Ph.D thesis, (1991). Google Scholar

[18]

Y. Manin, The Hasse-Witt matrix of an algebraic curve,, Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 153. Google Scholar

[19]

D. Mumford, Theta characteristics of an algebraic curve,, Ann. Sci. de l'É.N.S., 4 (1971), 181. Google Scholar

[20]

B. Poonen, Varieties without extra automorphisms. II. Hyperelliptic curves,, Math. Res. Lett., 7 (2000), 77. doi: 10.4310/MRL.2000.v7.n1.a7. Google Scholar

[21]

B. Poonen, Bertini theorems over finite fields,, Ann. Math., 160 (2004), 1099. doi: 10.4007/annals.2004.160.1099. Google Scholar

[22]

R. Pries and H. Zhu, The $p$-rank stratification of Artin-Schreier curves,, Ann. l'Institut Fourier, 62 (2012), 707. doi: 10.5802/aif.2692. Google Scholar

[23]

J. Scholten and H. Zhu, Hyperelliptic curves in characteristic $2$,, Int. Math. Res. Not., 2002 (2002), 905. doi: 10.1155/S1073792802111160. Google Scholar

[24]

J.-P. Serre, Sur la topologie des variétés algébriques en caractéristique $p$,, in Oeuvres (collected papers), (1986), 544. Google Scholar

[25]

K.-O. Stöhr and J. F. Voloch, A formula for the Cartier operator on plane algebraic curves,, J. reine angew. Math., 377 (1987), 49. doi: 10.1515/crll.1987.377.49. Google Scholar

[26]

H. Zhu, Hyperelliptic curves over $\mathbf F_2$ of every $2$-rank without extra automorphisms,, Proc. Amer. Math. Soc., 134 (2006), 323. doi: 10.1090/S0002-9939-05-08294-8. Google Scholar

show all references

References:
[1]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language,, J. Symbolic Comput., 24 (1997), 235. doi: 10.1006/jsco.1996.0125. Google Scholar

[2]

B. Cais, J. Ellenberg and D. Zureick-Brown, Random Dieudonné modules, random $p$-divisible groups, and random curves over finite fields,, J. Inst. Math. Jussieu, 12 (2013), 651. doi: 10.1017/S1474748012000862. Google Scholar

[3]

W. Castryck, Moving out the edges of a lattice polygon,, Discrete Comp. Geometry, 47 (2012), 496. doi: 10.1007/s00454-011-9376-2. Google Scholar

[4]

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

[5]

W. Castryck, J. Denef and F. Vercauteren, Computing zeta functions of nondegenerate curves,, Int. Math. Res. Pap., 2006 (2006), 1. Google Scholar

[6]

W. Castryck, A. Folsom, H. Hubrechts and A. V. Sutherland, The probability that the number of points on the Jacobian of a genus $2$ curve is prime,, Proc. London Math. Soc., 104 (2012), 1235. doi: 10.1112/plms/pdr063. Google Scholar

[7]

W. Castryck and J. Voight, On nondegeneracy of curves,, Algebra Number Theory, 3 (2009), 255. doi: 10.2140/ant.2009.3.255. Google Scholar

[8]

D. Cox, J. Little and H. Schenck, Toric Varieties,, Springer, (2011). Google Scholar

[9]

J. Denef and F. Vercauteren, Computing zeta functions of hyperelliptic curves over finite fields of characteristic $2$,, in Proc. Adv. Cryptology - CRYPTO 2002, (2002), 308. doi: 10.1007/3-540-45455-1_25. Google Scholar

[10]

J. Denef and F. Vercauteren, Computing zeta functions of $C_{a,b}$ curves using Monsky-Washnitzer cohomology,, Finite Fields App., 12 (2006), 78. doi: 10.1016/j.ffa.2005.01.003. Google Scholar

[11]

A. Elkin and R. Pries, Ekedahl-Oort strata of hyperelliptic curves in characteristic $2$,, Algebra Number Theory, 7 (2013), 507. doi: 10.2140/ant.2013.7.507. Google Scholar

[12]

S. Farnell and R. Pries, Families of Artin-Schreier curves with Cartier-Manin matrix of constant rank,, Linear Algebra Appl., 439 (2013), 2158. doi: 10.1016/j.laa.2013.06.012. Google Scholar

[13]

C. Haase and J. Schicho, Lattice polygons and the number $2i+7$,, Amer. Math. Monthly, 116 (2009), 151. doi: 10.4169/193009709X469913. Google Scholar

[14]

R. Hartshorne, Generalized divisors on Gorenstein curves and a theorem of Noether,, J. Math. Kyoto Univ., 26 (1986), 375. Google Scholar

[15]

N. Katz and P. Sarnak, Random Matrices, Frobenius Eigenvalues and Monodromy,, AMS, (1998). Google Scholar

[16]

N. Koblitz, Algebraic aspects of cryptography,, in Algorithms and Computation in Mathematics, (1999). doi: 10.1007/978-3-662-03642-6. Google Scholar

[17]

R. Koelman, The Number of Moduli of Families of Curves on Toric Surfaces,, Ph.D thesis, (1991). Google Scholar

[18]

Y. Manin, The Hasse-Witt matrix of an algebraic curve,, Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 153. Google Scholar

[19]

D. Mumford, Theta characteristics of an algebraic curve,, Ann. Sci. de l'É.N.S., 4 (1971), 181. Google Scholar

[20]

B. Poonen, Varieties without extra automorphisms. II. Hyperelliptic curves,, Math. Res. Lett., 7 (2000), 77. doi: 10.4310/MRL.2000.v7.n1.a7. Google Scholar

[21]

B. Poonen, Bertini theorems over finite fields,, Ann. Math., 160 (2004), 1099. doi: 10.4007/annals.2004.160.1099. Google Scholar

[22]

R. Pries and H. Zhu, The $p$-rank stratification of Artin-Schreier curves,, Ann. l'Institut Fourier, 62 (2012), 707. doi: 10.5802/aif.2692. Google Scholar

[23]

J. Scholten and H. Zhu, Hyperelliptic curves in characteristic $2$,, Int. Math. Res. Not., 2002 (2002), 905. doi: 10.1155/S1073792802111160. Google Scholar

[24]

J.-P. Serre, Sur la topologie des variétés algébriques en caractéristique $p$,, in Oeuvres (collected papers), (1986), 544. Google Scholar

[25]

K.-O. Stöhr and J. F. Voloch, A formula for the Cartier operator on plane algebraic curves,, J. reine angew. Math., 377 (1987), 49. doi: 10.1515/crll.1987.377.49. Google Scholar

[26]

H. Zhu, Hyperelliptic curves over $\mathbf F_2$ of every $2$-rank without extra automorphisms,, Proc. Amer. Math. Soc., 134 (2006), 323. doi: 10.1090/S0002-9939-05-08294-8. Google Scholar

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