CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues

  Published:2006-12-01
 Printed: Dec 2006
  • Ronald van Luijk
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   HTML   LaTeX   MathJax   PDF   PostScript  

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 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
 

© Canadian Mathematical Society, 2014 : http://www.cms.math.ca/