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

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May 2010, 4(2): 155-168. doi: 10.3934/amc.2010.4.155

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

Josep M. Miret ^1, , Jordi Pujolàs ^1, and Anna Rio ^2,

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

Received May 2009 Revised March 2010 Published May 2010

Abstract
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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.

Keywords: finite fields., Jacobian, Genus 2 curves.

Mathematics Subject Classification: Primary: 11G20, 14H40; Secondary: 14H5.

Citation: Josep M. Miret, Jordi Pujolàs, Anna Rio. Explicit 2-power torsion of genus 2 curves over finite fields. Advances in Mathematics of Communications, 2010, 4 (2) : 155-168. doi: 10.3934/amc.2010.4.155

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