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A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues |
Published:2006-12-01
Printed: Dec 2006
Printed: Dec 2006
- Ronald van Luijk
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 Néron-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 |