# American Institute of Mathematical Sciences

November  2014, 8(4): 375-387. doi: 10.3934/amc.2014.8.375

## Trisection for supersingular genus $2$ curves in characteristic $2$

 1 Departament de Matemàtica, Universitat de Lleida, Jaume II 69, Lleida 25001, Spain 2 Departamento de Matemática, Universidad del Bío-Bío, Avenida Collao 1202, Concepción, Chile

Received  January 2014 Revised  June 2014 Published  November 2014

By reversing reduction in divisor class arithmetic we provide efficient trisection algorithms for supersingular Jacobians of genus $2$ curves over finite fields of characteristic $2$. With our technique we obtain new results for these Jacobians: we show how to find their $3$-torsion subgroup, we prove there is none with $3$-torsion subgroup of rank $3$ and we prove that the maximal $3$-power order subgroup is isomorphic to either $\mathbb{Z}/3^{v}\mathbb{Z}$ or $(\mathbb{Z}/3^{\frac v2}\mathbb{Z})^2$ or $(\mathbb{Z}/3^{\frac v4}\mathbb{Z})^4$, where $v$ is the $3$-adic valuation $v_{3}$(#Jac(C)$(\mathbb{F}_{2^m})$). Ours are the first trisection formulae available in literature.
Citation: Josep M. Miret, Jordi Pujolàs, Nicolas Thériault. Trisection for supersingular genus $2$ curves in characteristic $2$. Advances in Mathematics of Communications, 2014, 8 (4) : 375-387. doi: 10.3934/amc.2014.8.375
##### References:
 [1] D. Cantor, Computing in the Jacobian of a Hyperelliptic curve, Math. Comp., 48 (1987), 95-101. doi: 10.1090/S0025-5718-1987-0866101-0. [2] I. Kitamura, M. Katagi and T. Takagi, A complete divisor class halving algorithm for hyperelliptic curve cryptosystems of genus two, in Information Security and Privacy, Springer, 2005, 146-157. doi: 10.1007/11506157_13. [3] J. Miret, J. Pujolàs and A. Rio, Explicit 2-power torsion of genus $2$ curves over finite fields, Adv. Math. Commun., 4 (2010), 155-165. doi: 10.3934/amc.2010.4.155. [4] F. Oort, Subvarieties of moduli spaces, Invent. Math., 24 (1974), 95-119. doi: 10.1007/BF01404301. [5] R. Schoof, Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A, 46 (1987), 183-211. doi: 10.1016/0097-3165(87)90003-3. [6] C. Xing, On supersingular abelian varieties of dimension two over finite fields, Finite Fields Appl., 2 (1996), 407-421. doi: 10.1006/ffta.1996.0024.

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##### References:
 [1] D. Cantor, Computing in the Jacobian of a Hyperelliptic curve, Math. Comp., 48 (1987), 95-101. doi: 10.1090/S0025-5718-1987-0866101-0. [2] I. Kitamura, M. Katagi and T. Takagi, A complete divisor class halving algorithm for hyperelliptic curve cryptosystems of genus two, in Information Security and Privacy, Springer, 2005, 146-157. doi: 10.1007/11506157_13. [3] J. Miret, J. Pujolàs and A. Rio, Explicit 2-power torsion of genus $2$ curves over finite fields, Adv. Math. Commun., 4 (2010), 155-165. doi: 10.3934/amc.2010.4.155. [4] F. Oort, Subvarieties of moduli spaces, Invent. Math., 24 (1974), 95-119. doi: 10.1007/BF01404301. [5] R. Schoof, Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A, 46 (1987), 183-211. doi: 10.1016/0097-3165(87)90003-3. [6] C. Xing, On supersingular abelian varieties of dimension two over finite fields, Finite Fields Appl., 2 (1996), 407-421. doi: 10.1006/ffta.1996.0024.
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