Abstract
We provide trisection (division by 3) algorithms for Jacobians of genus 2 curves over finite fields \(\mathbb {F}_q\) of odd characteristic which rely on the factorization of a polynomial whose roots correspond (bijectively) to the set of trisections of the given divisor. We also construct a polynomial whose roots allow us to calculate the 3-torsion divisors. We show the relation between the rank of the 3-torsion subgroup and the factorization of this 3-torsion polynomial, and describe the factorization of the trisection polynomials in terms of the Galois structure of the 3-torsion subgroup. We also generalize these ideas for \(\ell \in \{5,7\}\).
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https://dl.dropboxusercontent.com/u/50859627/TrisecAAECC/TorsionAAECC.txt
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Acknowledgments
I wish to thank Jordi Pujolàs and Nicolas Thériault for several suggestions and comments.
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The author is supported by CONICYT/Chile doctoral scholarship.
Appendix
Appendix
In the following table we give the factorization types (over \(\mathbb {F}_q\)) of the 3-torsion and trisection polynomials \(T(a_0), P(a_1)\) according to the factorization of the characteristic polynomial over \(\mathbb {F}_3\) when \({\mathrm{Jac}}(\mathrm {C})(\mathbb {F}_q)[3]\) has rank \(\ge 1\).
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Riquelme, E. Trisection for genus 2 curves in odd characteristic. AAECC 27, 373–397 (2016). https://doi.org/10.1007/s00200-015-0282-3
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DOI: https://doi.org/10.1007/s00200-015-0282-3