Rational points on K3 surfaces: A new canonical height

Abstract

A fundamental tenet of Diophantine Geometry is that the geometric properties of an algebraic variety should determine its basic arithmetic properties. This is certainly true for curves, where the sign of the Euler characteristic of C determines whether the set of rational points on C is finite (X (C) 0). For higher dimensional varieties there are some precise conjectures due to Bombieri, Lang, and Vojta [15] which predict when the rational points on a variety should be finite or degenerate (i.e. not Zariski dense), and some conjectures of Manin et al. [-2, 5] on the distribution of rational points in those cases when they are Zariski dense. But except for abelian varieties, their subvarieties, and some Fano varieties (varieties for which the anticanonical bundle is ample), there are very few general theorems. In this paper we will study the rational points on a certain class of K 3 surfaces defined over a number field K. The moduli space of marked algebraic K3 surfaces is a countable union of 19 dimensional quasi-projective varieties. We are going to look at the 18 dimensional family studied by Wehler 1-17]. Wehler's family consists of K3 surfaces S whose automorphism group Aut(S) contains a subgroup d isomorphic to the free product 7/,2"~ 2 of two cyclic groups of order 2. We will use the geometric information provided by this infinite automorphism group to study the K-rat ional points on S. For any point PeS, we can look at the orbit of P under the action of d,