Polynesian Journal of Mathematics

Volume 4, Issue 10
Special volume AJAX in honor of René Schoof

Factorization pattern and fields of definition of ℓ-torsion points on the Jacobians of genus-3 hyperelliptic curves

Amalia Pizarro-Madariaga and Edgardo Riquelme Faúndez

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 $O\left({\ell }^4\right)$, improving the previously expected $O\left({\ell }^6\right)$ 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.

Keywords: Hyperelliptic curve cryptosystems, hyperelliptic curves, Jacobians, $\ell$-adic point counting algorithms, torsion points.

Citation:

Amalia Pizarro-Madariaga and Edgardo Riquelme Faúndez. Factorization pattern and fields of definition of ℓ-torsion points on the Jacobians of genus-3 hyperelliptic curves. Polynesian Journal of Mathematics, Volume 4, Issue 10, Pages 1–24. (Sept. 2026) DOI: 10.69763/polyjmath.4.10

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Milestones:

Received 26 Feb 2026
Revised 29 Jun 2026
Accepted 20 Aug 2026
Published 28 Sep 2026
Communicated by Elisa Lorenzo García

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