Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
EnglishFrançais
Canadian Mathematical Bulletin Canadian Mathematical Bulletin

Article contents

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
Corresponding
E-mail address:
rmluijk@math.berkeley.edu
Save pdf (0.23 mb)
Rights & Permissions[Opens in a new window]

Abstract

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

14G05 14J28 11D41 symmetric matrices eigenvalues elliptic surfaces K3 surfaces Néron–Severi group rational curves Diophantine equations arithmetic geometry algebraic geometry number theory
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 4 , 01 December 2006 , pp. 560 - 577
Copyright
Copyright © Canadian Mathematical Society 2006

References

[Ar] Artin, M., On isolated rational singularities of surfaces. Amer. J. Math. 88(1966), 129–136.CrossRefGoogle Scholar
[BPV] Barth, W., Peters, C., and Van de Ven, A., Compact Complex Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete 4, Springer-Verlag, 1984.CrossRefGoogle Scholar
[Be] Beukers, F., Problem 10. Nieuw Archief Wiskd. 1(2000), no. 4, 413–417.Google Scholar
[BLV] Beukers, F., van Luijk, R., and Vidunas, R., A linear algebra exercise. Nieuw Archief Wiskd. 3(2002), no. 2, 139–140.Google Scholar
[BT] Bogomolov, F., F. and Tschinkel, Yu., Density of rational points on elliptic K3 surfaces. Asian J. Math. 4(2000), no. 2, 351–368.CrossRefGoogle Scholar
[Br] Bremner, A., On squares of squares. II. Acta Arith. 99(2001), no. 3, 289–308.CrossRefGoogle Scholar
[Ch] Chinburg, T., Minimal models of curves over Dedekind rings. In: Arithmetic Geometry, Springer, New York, 1986, pp. 309–326.CrossRefGoogle Scholar
[Gr1] Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas, Seconde partie, Inst. Hautes études Sci. Publ. Math. 24, 1965.CrossRefGoogle Scholar
[Gr2] Grothendieck, A., et al. Théorie des Intersections et Théorème de Riemann-Roch. Lect. Notes in Math. 225, Springer-Verlag, Berlin, 1971.Google Scholar
[Ha] Hartshorne, R., Algebraic Geometry. Graduate Texts in Mathematics 52, Springer-Verlag, New York, 1977.CrossRefGoogle Scholar
[In] Inose, H., On certain Kummer surfaces which can be realized as non-singular quartic surfaces ℙ 3 . J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23(1976), no. 3, 545–560.Google Scholar
[Lu] van Luijk, R., An elliptic K3 surface associated to Heron triangles. To appear, J. Number Theory.Google Scholar
[Ni] Nikulin, V., Integral symmetric bilinear forms and some of their applications. Math. USSR Izvestija 14(1980), no. 1, 103–167.CrossRefGoogle Scholar
[PS] Pjateckii-Shapiro, I., and Shafarevich, I., Torelli's theorem for algebraic surfaces of type K3. Izv. Akad. Nauk SSSR Ser. Mat. 35(1971), 530–572.Google Scholar
[SD] Saint-Donat, B., Projective models of K-3 surfaces. Amer. J. Math. 96(1974), 602–639.CrossRefGoogle Scholar
[Sh] Shioda, T., On the Mordell-Weil lattices. Comment. Math. Univ. St Paul. 39(1990), no. 2, 211–240.Google Scholar
[SI] Shioda, T., and Inose, H., On singular K3 surfaces. In: Complex Analysis and Algebraic Geometry, Iwanami Shoten, Tokyo, 1977, pp. 119–136.CrossRefGoogle Scholar
[Si1] Silverman, J. H., The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics 106, Springer-Verlag, New York, 1986.CrossRefGoogle Scholar
[Si2] Silverman, J. H., Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics 151, Springer-Verlag, New York, 1994.CrossRefGoogle Scholar
[Ta] Tate, J., Algorithm for determining the type of a singular fiber in an elliptic pencil. In: Modular Functions of One Variable. IV. Lect. Notes in Math. 476, Springer-Verlag, Berlin, 1975, pp. 33–52.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 18 *
View data table for this chart

* Views captured on Cambridge Core between 20th November 2018 - 22nd January 2021. This data will be updated every 24 hours.

Access
1 Cited by
Hostname: page-component-76cb886bbf-fv2z2 Total loading time: 0.736 Render date: 2021-01-22T23:51:37.471Z Query parameters: { "hasAccess": "1", "openAccess": "0", "isLogged": "0", "lang": "en" } Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false }

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues
Available formats
×
×

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *