On the number of point of given order on odd degree hyperelliptic curves

TL;DR

This paper investigates bounds on the number of points of a given order on hyperelliptic curves of odd degree, revisiting and strengthening classical division polynomial results to better understand their structure.

Contribution

It introduces new bounds for points of specific orders on hyperelliptic curves and refines Cantor's division polynomial divisibility results.

Findings

Established tighter bounds on points of order dividing N

Revisited and strengthened Cantor's division polynomial divisibility

Discussed multiple examples illustrating theoretical results

Abstract

For integers $N\geq 3$ and $g\geq 1$, we study bounds on the cardinality of the set of points of order dividing $N$ lying on a hyperelliptic curve of genus $g$ embedded in its jacobian using a Weierstrass point as base point. This leads us to revisit division polynomials introduced by Cantor in 1995 and strengthen a divisibility result proved by him. Several examples are discussed.