Rational Torsion and Ramification Constraints on Odd Superelliptic Jacobians
Abstract
Let C: y^d = f(x) be a smooth projective superelliptic curve over Q, where d >= 3 is odd, f(x) in Q[x] is separable of degree n >= 3, and gcd(d,n) = 1. Let J = Jac(C) be its Jacobian. We study the rational torsion subgroup J(Q)_tors through the interaction of ramification geometry, Galois actions, reduction at primes of good reduction, and the geometry of divisors. First, we describe the subgroup R_d contained in J[d] generated by the ramification-point differences and determine its Galois-invariant subgroup explicitly in terms of the sizes of the Galois orbits of the roots of f. This yields a precise description of the rational d-torsion classes arising from ramification, without assuming that they account for all of J(Q)[d]. Second, we establish criteria based on the orders of reduced Jacobians that can prove the vanishing of the full rational torsion subgroup. Finally, we derive geometric constraints on divisor classes represented by effective divisors of degree smaller than the genus, including a restriction arising when distinct effective divisors represent the same class. Explicit examples illustrate how the Galois-theoretic and reduction-theoretic methods can be combined to study rational torsion on odd superelliptic Jacobians.
Related
- Rational torsion on simple genus two Jacobians
- Bounding Selmer Groups of Superelliptic Jacobians via Class Groups
- Factorization patterns and fields of definition of ℓ-torsion points on the Jacobians of genus 3 hyperelliptic curves
- On the Ramification of Quadratic Fields where there is a torsion growth
- Fields of definition of p-torsion points of elliptic curves and their ramification
BibTeX
@misc{das2026rational,
title = {{Rational Torsion and Ramification Constraints on Odd Superelliptic Jacobians}},
author = {Ankita Das},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06193},
eprint = {2611.06193},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06193},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rational Torsion and Ramification Constraints on Odd Superelliptic Jacobians
AU - Ankita Das
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06193
UR - https://arxiv.org/abs/2611.06193
AB - Let C: y^d = f(x) be a smooth projective superelliptic curve over Q, where d >= 3 is odd, f(x) in Q[x] is separable of degree n >= 3, and gcd(d,n) = 1. Let J = Jac(C) be its Jacobian. We study the rational torsion subgroup J(Q)_tors through the interaction of ramification geometry, Galois actions, reduction at primes of good reduction, and the geometry of divisors. First, we describe the subgroup R_d contained in J[d] generated by the ramification-point differences and determine its Galois-invariant subgroup explicitly in terms of the sizes of the Galois orbits of the roots of f. This yields a precise description of the rational d-torsion classes arising from ramification, without assuming that they account for all of J(Q)[d]. Second, we establish criteria based on the orders of reduced Jacobians that can prove the vanishing of the full rational torsion subgroup. Finally, we derive geometric constraints on divisor classes represented by effective divisors of degree smaller than the genus, including a restriction arising when distinct effective divisors represent the same class. Explicit examples illustrate how the Galois-theoretic and reduction-theoretic methods can be combined to study rational torsion on odd superelliptic Jacobians.
ER -