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

Josep M. Miret; Jordi Pujolàs; Nicolas Thériault

doi:10.3934/amc.2014.8.375

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2014, Volume 8, Issue 4: 375-387. Doi: 10.3934/amc.2014.8.375

This issue Previous Article Editorial Next Article Comparison of scalar multiplication on real hyperelliptic curves

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

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

Received: December 31, 2013

Revised: May 31, 2014

Published: October 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 11G20, 14H40, 14H45.

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References

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