A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces

[摘要] Let $V$ be a $K3$ surface defined over a number field $k$. TheBatyrev-Manin conjecture for $V$ states that for every nonempty opensubset $U$ of $V$, there exists a finite set $Z_U$ of accumulatingrational curves such that the density of rational points on $U-Z_U$ isstrictly less than the density of rational points on $Z_U$. Thus,the set of rational points of $V$ conjecturally admits a stratificationcorresponding to the sets $Z_U$ for successively smaller sets $U$.In this paper, in the case that $V$ is a Kummer surface, we prove thatthe Batyrev-Manin conjecture for $V$ can be reduced to theBatyrev-Manin conjecture for $V$ modulo the endomorphisms of $V$induced by multiplication by $m$ on the associated abelian surface$A$. As an application, we use this to show that given some restrictionson $A$, the set of rational points of $V$ which lie on rational curveswhose preimages have geometric genus 2 admits a stratification of

[学科分类] 数学（综合）

[关键词] rational points;Batyrev-Manin conjecture;Kummer;surface;rational curve;abelian surface;height