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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 thereexists a genus $2$ curve $C$ and a map $f: C\to A$ such that $x$ is in theimage 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 thealgebraic closure of a finite field, where $S=A(k)$, as shown by Bogomolov andTschinkel. We deduce that if $k=\bar{\mathbb{Q}}$, the Kummer surfaceassociated to $A$ has infinitely many $k$-points not contained in a rationalcurve arising from a genus $2$ curve in $A$, again in contrast to the situationover the algebraic closure of a finite field.

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