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PreprintCC BY 4.0arXiv

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

  • Amalia Pizarro-Madariaga
  • Edgardo Riquelme
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Abstract

Motivated by Schoof-Pila-type point-counting algorithms, we study the fields of definition and factorization patterns of Galois orbits of ℓ-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 ℓ-torsion points can be bounded by O(ℓ4), improving the previously expected O(ℓ6) bound (which reflects the size of the ℓ-torsion subgroup, of order ℓ6 for a genus 3 Jacobian). Moreover, we establish a precise correspondence between the rank of the ℓ-torsion subgroup and the Galois orbits of ℓ-torsion divisors. Our approach relies on a detailed analysis of the Jordan decomposition of the Frobenius action on J and its nilpotent part.

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Cite

BibTeXRIS
BibTeX
@misc{pizarromadariaga2026factorization,
  title = {{Factorization patterns and fields of definition of $\ell$-torsion points on the Jacobians of genus 3 hyperelliptic curves}},
  author = {Amalia Pizarro-Madariaga and Edgardo Riquelme},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03171},
  eprint = {2609.03171},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03171},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Factorization patterns and fields of definition of ℓ-torsion points on the Jacobians of genus 3 hyperelliptic curves
AU  - Amalia Pizarro-Madariaga
AU  - Edgardo Riquelme
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.03171
UR  - https://arxiv.org/abs/2609.03171
AB  - Motivated by Schoof-Pila-type point-counting algorithms, we study the fields of definition and factorization patterns of Galois orbits of ℓ-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 ℓ-torsion points can be bounded by O(ℓ^4), improving the previously expected O(ℓ^6) bound (which reflects the size of the ℓ-torsion subgroup, of order ℓ^6 for a genus 3 Jacobian). Moreover, we establish a precise correspondence between the rank of the ℓ-torsion subgroup and the Galois orbits of ℓ-torsion divisors. Our approach relies on a detailed analysis of the Jordan decomposition of the Frobenius action on J and its nilpotent part.
ER  -