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A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues

Published:2006-12-01
Printed: Dec 2006
• Ronald van Luijk
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Abstract

In this article we will show that there are infinitely many symmetric, integral $3 \times 3$ 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 N\'eron--Severi group of this surface and find all low degree curves on it.
 Keywords: symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, Néron--Severi group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theory
 MSC Classifications: 14G05 - Rational points 14J28 - $K3$ surfaces and Enriques surfaces 11D41 - Higher degree equations; Fermat's equation