Counting Rational Points on K3 Surfaces

For any algebraic variety X defined over a number field K, and height functionHD onX corresponding to an ample divisorD, one can define the counting functionNX,D(B) = #{P ∈ X(K) | HD(P ) ≤ B}. In this paper, we calculate the counting function for hyperelliptic K3 surfaces X which admit a generically two-to-one cover of P1 × P1 branched over a singular curve. In particular, we effectively construct a finite union Y = ∪Ci of curves Ci ⊂ X such that NX−Y,D(B) NY,D(B); that is, Y is an accumulating subset of X. In the terminology of Batyrev and Manin [4], this amounts to proving that Y is the first layer of the arithmetic stratification of X. We prove a more precise result in the special case where X is a Kummer surface whose associated abelian surface is a product of elliptic curves.