Explicit 2-power torsion of genus 2 curves over finite fields

Josep M. Miret; Jordi Pujolàs; Anna Rio

doi:10.3934/amc.2010.4.155

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2010, Volume 4, Issue 2: 155-168. Doi: 10.3934/amc.2010.4.155

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Explicit 2-power torsion of genus 2 curves over finite fields

1.
Departament de Matemàtica, Universitat de Lleida, Jaume II 69, Lleida 25001
2.
Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Jordi Girona 1–3, Barcelona 08134

Received: April 30, 2009

Revised: February 28, 2010

Published: April 2010

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Abstract

We give an efficient explicit algorithm to find the structure and generators of the maximal 2-subgroup of the Jacobian of a genus 2 curve over a finite field of odd characteristic. We use the 2-torsion points as seeds to successively perform a chain of halvings to find divisors of increasing 2-power order. The halving loop requires a solution to certain degree 16 polynomials over the base field, and the termination of the algorithm is based on the description of the graph structure of the maximal 2-subgroup. The structure of our algorithm is the natural extension of the even characteristic case.

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Mathematics Subject Classification: Primary: 11G20, 14H40; Secondary: 14H55.

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