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]

Math. Comp., 48 (1987), 95-101. doi: 10.1090/S0025-5718-1987-0866101-0.  Google Scholar

[2]

in Information Security and Privacy, Springer, 2005, 146-157. doi: 10.1007/11506157_13.  Google Scholar

[3]

Adv. Math. Commun., 4 (2010), 155-165. doi: 10.3934/amc.2010.4.155.  Google Scholar

[4]

Invent. Math., 24 (1974), 95-119. doi: 10.1007/BF01404301.  Google Scholar

[5]

J. Combin. Theory Ser. A, 46 (1987), 183-211. doi: 10.1016/0097-3165(87)90003-3.  Google Scholar

[6]

Finite Fields Appl., 2 (1996), 407-421. doi: 10.1006/ffta.1996.0024.  Google Scholar

show all references

References:
[1]

Math. Comp., 48 (1987), 95-101. doi: 10.1090/S0025-5718-1987-0866101-0.  Google Scholar

[2]

in Information Security and Privacy, Springer, 2005, 146-157. doi: 10.1007/11506157_13.  Google Scholar

[3]

Adv. Math. Commun., 4 (2010), 155-165. doi: 10.3934/amc.2010.4.155.  Google Scholar

[4]

Invent. Math., 24 (1974), 95-119. doi: 10.1007/BF01404301.  Google Scholar

[5]

J. Combin. Theory Ser. A, 46 (1987), 183-211. doi: 10.1016/0097-3165(87)90003-3.  Google Scholar

[6]

Finite Fields Appl., 2 (1996), 407-421. doi: 10.1006/ffta.1996.0024.  Google Scholar

[1]

Huaning Liu, Xi Liu. On the correlation measures of orders $ 3 $ and $ 4 $ of binary sequence of period $ p^2 $ derived from Fermat quotients. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021008

[2]

Siting Liu, Levon Nurbekyan. Splitting methods for a class of non-potential mean field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021014

[3]

Takao Komatsu, Bijan Kumar Patel, Claudio Pita-Ruiz. Several formulas for Bernoulli numbers and polynomials. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021006

[4]

Xianjun Wang, Huaguang Gu, Bo Lu. Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron. Electronic Research Archive, , () : -. doi: 10.3934/era.2021023

[5]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089

[6]

Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021006

[7]

Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021011

[8]

Rafael López, Óscar Perdomo. Constant-speed ramps for a central force field. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3447-3464. doi: 10.3934/dcds.2021003

[9]

René Aïd, Roxana Dumitrescu, Peter Tankov. The entry and exit game in the electricity markets: A mean-field game approach. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021012

[10]

Jennifer D. Key, Bernardo G. Rodrigues. Binary codes from $ m $-ary $ n $-cubes $ Q^m_n $. Advances in Mathematics of Communications, 2021, 15 (3) : 507-524. doi: 10.3934/amc.2020079

[11]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[12]

Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017

[13]

Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513

[14]

Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017

[15]

Jia Li, Junxiang Xu. On the reducibility of a class of almost periodic Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3905-3919. doi: 10.3934/dcdsb.2020268

[16]

Sascha Kurz. The $[46, 9, 20]_2$ code is unique. Advances in Mathematics of Communications, 2021, 15 (3) : 415-422. doi: 10.3934/amc.2020074

[17]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044

[18]

Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265

[19]

Lei Zhang, Luming Jia. Near-field imaging for an obstacle above rough surfaces with limited aperture data. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021024

[20]

Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021026

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (3)

[Back to Top]