Counting Rational Points on K3 Surfaces

Abstract

For any algebraic variety $V$ defined over a number field $k$, and ample height function $H$ on $V$, one can define the counting function $N_V(B) = #{P\in V(k) \mid H(P)\leq B}$. In this paper, we calculate the counting function for Kummer surfaces $V$ whose associated abelian surface is the product of elliptic curves. In particular, we effectively construct a finite union $C = \cup C_i$ of curves $C_i$ on $V$ such that $N_{V-C}(B)\ll N_C(B)$; that is, $C$ is an accumulating subset of $V$. In the terminology of Batyrev and Manin, this amounts to proving that $C$ is the first layer of the arithmetic stratification of $V$.