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A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues

Published online by Cambridge University Press:  20 November 2018

Ronald van Luijk
Ronald van Luijk*
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720-3840 e-mail: rmluijk@math.berkeley.edu
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Abstract

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In this article we will show that there are infinitely many symmetric, integral 3 × 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 $\text{K3}$ surface are dense. We will also compute the entire Néron–Severi group of this surface and find all low degree curves on it.

Keywords

14G0514J2811D41symmetric matriceseigenvalueselliptic surfacesK3 surfacesNéron–Severi grouprational curvesDiophantine equationsarithmetic geometryalgebraic geometrynumber theory
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 4 , 01 December 2006 , pp. 560 - 577
Copyright
Copyright © Canadian Mathematical Society 2006

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