Trisection for genus 2 curves in odd characteristic

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 \(\ell \in \{5,7\}\).