Counting Rational Points on K3 Surfaces Original Research Article

For any algebraic variety X defined over a number field K, and height function HD on X corresponding to an ample divisor D, one can define the counting function NX, D(B)=#{Pset membership, variantX(K) mid HD(P)less-than-or-equals, slantB}. 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=union or logical sum Ci of curves Cisubset ofX such that NX−Y, D(B)much less-thanNY, 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.