Counting rational points on k3 surfaces

For any algebraic variety X defined over a number field K, and height funct ion H-D on X corresponding to an ample divisor D, one can define the counti ng function N-X,N-D(B) = # {P is an element of X(K)\ H-D(P) less than or eq ual to B}. In this paper, we calculate the counting function for hyperellip tic K3 surfaces; X which admit a generically two-to-one cover of P-1 x P-1 branched over a singular curve. In particular. we effectively construct a f inite union Y = boolean OR C-i of curves C-i subset of X such that N-X-Y,N- D(B) much less than N-Y,N-D(B); that is, Y is an accumulating subset of X. In the terminology of Batyrev and Mnin [4], this amounts to proving that Y is the first layer of the arithmetic stratification of X. We prove a more p recise result in the special case where X is a Kummer surface whose associa ted Abelian surface is a product of elliptic curves. (C) 2000 Academic Pres s.