Torsion points of small order on hyperelliptic curves

TL;DR

This paper investigates torsion points of order 2g+1 on hyperelliptic curves, establishing finiteness results for pairs with many such points, extending understanding of torsion structures in Jacobians.

Contribution

It proves that for a fixed genus, only finitely many hyperelliptic curves with a marked Weierstrass point have at least six points of order 2g+1 in their Jacobian.

Findings

Finiteness of pairs with ≥6 torsion points of order 2g+1

Existence of infinitely many pairs with ≥4 such points

Characterization of torsion points on hyperelliptic curves

Abstract

Let $C$ be a hyperelliptic curve of genus $g>1$ over an algebraically closed field $K$ of characteristic zero and $O$ one of the $(2g+2)$ Weierstrass points in $C(K)$. Let $J$ be the jacobian of $C$, which is a $g$-dimensional abelian variety over $K$. Let us consider the canonical embedding of $C$ into $J$ that sends $O$ to the zero of the group law on $J$. This embedding allows us to identify $C(K)$ with a certain subset of the commutative group $J(K)$. A special case of the famous theorem of Raynaud (Manin--Mumford conjecture) asserts that the set of torsion points in $C(K)$ is finite. It is well known that the points of order 2 in $C(K)$ are exactly the "remaining" $(2g+1)$ Weierstrass points. One of the authors proved that there are no torsion points of order $n$ in $C(K)$ if $3\le n\le 2g$. So, it is natural to study torsion points of order $2g+1$ (notice that the number of such points in $C(K)$ is always even). Recently, the authors proved that there are infinitely many (for a given $g$) mutually nonisomorphic pairs $C,O)$ such that $C(K)$ contains at least four points of order $2g+1$. In the present paper we prove that (for a given $g$) there are at most finitely many (up to a isomorphism) pairs $(C,O)$ such that $C(K)$ contains at least six points of order $2g+1$.