Trisection for genus 2 curves in odd characteristic

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 $\ell \in \{5,7\}$.

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 $\ge 1$.