A K3 surface associated to certain integral matrices with integral eigenvalues

Abstract

In this article we will show that there are infinitely many symmetric, integral 3 x 3 matrices, with zeros on the diagonal, whose eigenvalues are all integral. We will do this by proving that the rational points on a certain non-Kummer, singular K3 surface are dense. We will also compute the entire Neron-Severi group of this surface and find all low degree curves on it.