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 symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, Néron--Severi group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theory

MSC Classifications:

14G05, 14J28, 11D41 show english descriptions Rational points
$K3$ surfaces and Enriques surfaces
Higher degree equations; Fermat's equation 14G05 - Rational points
14J28 - $K3$ surfaces and Enriques surfaces
11D41 - Higher degree equations; Fermat's equation

top of page | contact us | social media | privacy | site map