Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T02:34:59.552Z Has data issue: false hasContentIssue false
  • English
  • Français

SIMPLE PROOFS FOR UNIVERSAL BINARY HERMITIAN LATTICES

Part of: Forms and linear algebraic groups

Published online by Cambridge University Press:  13 January 2010

POO-SUNG PARK
POO-SUNG PARK*
Affiliation:
Department of Mathematics Education, Kyungnam University, Masan, 631-701, Korea (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If a positive definite Hermitian lattice represents all positive integers, we call it universal. Several mathematicians, including the author, have between them found 25 universal binary Hermitian lattices. But their ad hoc proofs are complicated. We give simple and unified proofs of universality.

Keywords

universal Hermitian latticebinary Hermitian lattice

MSC classification

Secondary: 11E39: Bilinear and Hermitian forms 11E41: Class numbers of quadratic and Hermitian forms
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 2 , April 2010 , pp. 274 - 280
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Bhargava, M., ‘On the Conway–Schneeberger fifteen theorem’, Contemp. Math. 272 (2000), 27–37.CrossRefGoogle Scholar
[2]Bhargava, M. and Hanke, J., ‘Universal quadratic forms and the 290-theorem’, http://www.math.duke.edu/∼jonhanke/290/Universal-290.html.Google Scholar
[3]Brandt, H., Intrau, O. and Schiemann, A., ‘The Brandt–Intrau–Schiemann tables’,http://www.research.att.com/∼njas/lattices/Brandt_1.html,http://www.research.att.com/∼njas/lattices/Brandt_2.html .Google Scholar
[4]Cassels, J. W. S., Rational Quadratic Forms, London Mathematical Society Monographs, 13 (Academic Press, New York, 1978).Google Scholar
[5]Conway, J. H., ‘Universal quadratic forms and the fifteen theorem’, Contemp. Math. 272 (2000), 23–26.CrossRefGoogle Scholar
[6]Earnest, A. G. and Khosravani, A., ‘Universal binary Hermitian forms’, Math. Comp. 66(219) (1997), 1161–1168.CrossRefGoogle Scholar
[7]Iwabuchi, H., ‘Universal binary positive definite Hermitian lattices’, Rocky Mountain J. Math. 30(3) (2000), 951–959.CrossRefGoogle Scholar
[8]Kim, B. M., Kim, J. Y. and Park, P.-S., ‘The fifteen theorem for universal Hermitian lattices over imaginary quadratic fields’, Math. Comp., to appear.Google Scholar
[9]Kim, J.-H. and Park, P.-S., ‘A few uncaught universal Hermitian forms’, Proc. Amer. Math. Soc. 135 (2007), 47–49.CrossRefGoogle Scholar
[10]Lagrange, J. L., ‘Démonstration d’un théorème d’arithmétique’, Œuvres 3 (1770), 189–201.Google Scholar
[11]Nipp, G. L., ‘Gordon Nipp’s tables of quaternary quadratic forms’,http://www.research.att.com/∼njas/lattices/nipp.html.Google Scholar
[12]O’Meara, O. T., Introduction to Quadratic Forms (Spinger, New York, 1973).CrossRefGoogle Scholar
[13]Ramanujan, S., ‘On the expression of a number in the form ax 2+by 2+cz 2+dw 2’, Proc. Cambridge Philos. Soc. 19 (1917), 11–21.Google Scholar