The Manin-Mumford conjecture in genus 2 and rational curves on K3 surfaces

TL;DR

This paper proves that for a simple abelian surface over an algebraically closed field of characteristic zero, the set of torsion points related to genus 2 curves and rational maps is finite, contrasting with the positive characteristic case.

Contribution

It establishes the finiteness of certain torsion points associated with genus 2 curves on abelian surfaces over characteristic zero fields, extending Manin-Mumford type results.

Findings

Set of torsion points is finite over characteristic zero fields.

Contrasts with infinite torsion points over algebraic closures of finite fields.

Kummer surface has infinitely many points outside rational curves from genus 2 curves.

Abstract

Let $A$ be a simple abelian surface over an algebraically closed field $k$. Let $S\subset A(k)$ be the set of torsion points $x$ of $A$ such that there exists a genus $2$ curve $C$ and a map $f: C\to A$ such that $x$ is in the image of $f$, and $f$ sends a Weierstrass point of $C$ to the origin of $A$. The purpose of this note is to show that if $k$ has characteristic zero, then $S$ is finite -- this is in contrast to the situation where $k$ is the algebraic closure of a finite field, where $S=A(k)$, as shown by Bogomolov and Tschinkel. We deduce that if $k=\bar{\mathbb{Q}}$, the Kummer surface associated to $A$ has infinitely many $k$-points not contained in a rational curve arising from a genus $2$ curve in $A$, again in contrast to the situation over the algebraic closure of a finite field.