Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-23T02:33:18.107Z Has data issue: false hasContentIssue false
  • English
  • Français

COMPLETENESS OF THE LIST OF SPINOR REGULAR TERNARY QUADRATIC FORMS

Part of: Forms and linear algebraic groups

Published online by Cambridge University Press:  05 December 2018

A. G. Earnest  and
Anna Haensch
A. G. Earnest
Affiliation:
Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, U.S.A. email [email protected]
Anna Haensch
Affiliation:
Department of Mathematics and Computer Science, Duquesne University, Pittsburgh, PA 15282, U.S.A. email [email protected]
Get access

Abstract

Extending the notion of regularity introduced by Dickson in 1939, a positive definite ternary integral quadratic form is said to be spinor regular if it represents all the positive integers represented by its spinor genus (that is, all positive integers represented by any form in its spinor genus). Jagy conducted an extensive computer search for primitive ternary quadratic forms that are spinor regular, but not regular, resulting in a list of 29 such forms. In this paper, we will prove that there are no additional forms with this property.

MSC classification

Primary: 11E20: General ternary and quaternary quadratic forms; forms of more than two variables 11E12: Quadratic forms over global rings and fields
Type
Research Article
Information
Mathematika , Volume 65 , Issue 2 , 2019 , pp. 213 - 235
Copyright
Copyright © University College London 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The research of the second author was supported by an Association for Women in Mathematics Mentoring Travel Grant.

References

Benham, J. W., Earnest, A. G., Hsia, J. S. and Hung, D. C., Spinor regular positive ternary quadratic forms. J. Lond. Math. Soc. (2) 42 1990, 1–10.Google Scholar
Chan, W. K. and Earnest, A. G., Discriminant bounds for spinor regular ternary quadratic lattices. J. Lond. Math. Soc. (2) 69 2004, 545–561.Google Scholar
Hsia, J. S., Spinor norms of local integral rotations I. Pacific J. Math. 57(1) 1975, 199–206.Google Scholar
Jagy, W. C., Kaplansky, I. and Schiemann, A., There are 913 regular ternary forms. Mathematika 44 1997, 332–341.Google Scholar
Jagy, W. C., Integer coefficient positive ternary quadratic forms that are spinor regular but are not regular. Catalogue of Lattices,http://www.math.rwth-aachen.de/Gabriele.Nebe/LATTICES/Jagy.txt.Google Scholar
Kaplansky, I., Personal correspondence, February 5, 1997.Google Scholar
Lemke Oliver, R. J., Representation by ternary quadratic forms. Bull. Lond. Math. Soc. 46 2014, 1237–1247.Google Scholar
Oh, B.-K., Regular positive ternary quadratic forms. Acta Arith. 147 2011, 233–243.Google Scholar
O’Meara, O. T., Introduction to Quadratic Forms, Springer (New York, 1963).Google Scholar
The Sage Developers, SageMath, the Sage Mathematics Software System (Version 8.0), http://www.sagemath.org, 2017.Google Scholar
Watson, G. L., Some problems in the theory of numbers. PhD Thesis, University College London, 1953.Google Scholar