Bounds on the number of torsion points of given order on curves embedded in their jacobians

TL;DR

This paper establishes bounds on the number of torsion points of a given order on algebraic curves embedded in their Jacobians, providing an efficient computational method and extending previous hyperelliptic curve results to more general cases.

Contribution

It introduces new bounds and an efficient method for counting torsion points on embedded curves, generalizing prior hyperelliptic-specific results to broader classes of curves.

Findings

Bounds depend on the containment of multiples of the curve in the theta divisor

An efficient computational method for torsion points is proposed

Examples illustrate the bounds and methods

Abstract

Working over an algebraically closed field of arbitrary characteristic we study, for integers $N\geq 2$ and $g\geq 2$, the set of points of order dividing $N$ lying on an irreducible smooth proper curve of genus $g$ embedded in its jacobian using a fixed base point. We discuss bounds on its cardinality and describe an efficient method for computing the set. Our method uses wronskians similar to those used in the study of Weierstrass points and the strength of our bounds is related to whether or not a certain multiple of the curve is contained in the negative of the theta divisor. Several examples are discussed. This generalizes our previous work [https://doi.org/10.1216/rmj.2023.53.357] dealing with the case of hyperelliptic curves embedded in their jacobian using a Weierstrass point as base point.