Factorization patterns and fields of definition of $\ell$-torsion points on the Jacobians of genus 3 hyperelliptic curves

Abstract

Motivated by Schoof-Pila-type point-counting algorithms, we study the fields of definition and factorization patterns of Galois orbits of $\ell$-torsion points on the Jacobians of genus-3 hyperelliptic curves over finite fields. We show that the degree of the field of definition of the $\ell$-torsion points can be bounded by $O(\ell^4)$, improving the previously expected $O(\ell^6)$ bound (which reflects the size of the $\ell$-torsion subgroup, of order $\ell^6$ for a genus 3 Jacobian). Moreover, we establish a precise correspondence between the rank of the $\ell$-torsion subgroup and the Galois orbits of $\ell$-torsion divisors. Our approach relies on a detailed analysis of the Jordan decomposition of the Frobenius action on $J$ and its nilpotent part.