Rational points on K3 surfaces

In this thesis I consider several problems of a Diophantine nature that relate to algebraic surfaces.; Frits Beukers has asked whether there is an integral matrix 0ab a0c bc0 with all its eigenvalues integral and not in {lcub}0, +/-a, +/-b, +/-c {rcub}. Using the theory of elliptic surfaces, I show that up to scaling infinitely many such matrices exist.; A Heron triangle is a triangle with integral sides and integral area. There are pairs of nonsimilar Heron triangles with the same area and the same perimeter. The problem of finding three such triangles, brought to my attention by Richard Guy, can again be solved with the use of elliptic surfaces. I show that for each positive integer N there is in fact an infinite parametrized family of N such triangles.; In both cases, the solution involves showing that the set of rational points on a certain K3 surface is Zariski dense. I also compute the geometric Picard number of these surfaces. This important geometric invariant equals the rank of the Neron-Severi group of the surface over an algebraic closure of its base field. This group, consisting of divisor classes modulo algebraic equivalence, has rank at most 22 for K3 surfaces.; In general, little is known about the arithmetic of K3 surfaces, especially for those with geometric Picard number 1. I prove that in the moduli space of polarized K3 surfaces of degree 4, the set of surfaces defined over Q with geometric Picard number 1 and infinitely many rational points is dense in both the Zariski topology and the real analytic topology. This answers a question posed by Sir Peter Swinnerton-Dyer and Bjorn Poonen. Its effective proof, citing explicit examples, also disposes of an old challenge attributed to David Mumford.; For the convenience of the reader, I provide proofs of several theorems involving constructions of elliptic surfaces and the behavior of the Neron-Severi group under reduction. Some of these results are well known to experts, but a substantial search in the literature failed to reveal complete proofs. I also give a scheme-theoretic summary of the theory of elliptic surfaces, including a new proof of the classification of singular fibers.