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

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