Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T18:58:05.472Z Has data issue: false hasContentIssue false
  • English
  • Français

Extreme forms and cyclotomy

Published online by Cambridge University Press:  26 February 2010

Maurice Craig
Maurice Craig
Affiliation:
Care of Prof. L. Schoenfeld, Department of Mathematics, State University of New York at Buffalo, Buffalo, N.Y., U.S.A.
Get access

Extract

Positive definite quadratic forms are associated with pointlattices in the following way. If x ∊ Zm and H = MT for a real m × m matrix M, then xTHx is the square of the distance from the origin to the point Mx of MZm (equally, to W Mx ∊ W MZm, for orthogonal W).

Type
Research Article
Information
Mathematika , Volume 25 , Issue 1 , June 1978 , pp. 44 - 56
Copyright
Copyright © University College London 1978

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.)

References

1.Barnes, E. S.. “Note on extreme forms”, Canad. J. Math., 7 (1955), 150–154.CrossRefGoogle Scholar
2.Barnes, E. S.. “The complete enumeration of extreme senary forms”, Phil. Trans. Roy. Sac. Lond. (A), 249 (1957), 461–506.Google Scholar
3.Barnes, E. S.. “The perfect and extreme senary forms”, Canad. J. Math., 9 (1957), 235–242.CrossRefGoogle Scholar
4.Barnes, H. S.. “The construction of perfect and extreme forms I”, Acta Arith., 5 (1959), 57–79: and II, The construction of perfect and extreme forms I Acta Arith., 205–222.CrossRefGoogle Scholar
5.Coxeter, H. S. M.. “Extreme forms”, Canad. J. Math., 3 (1951), 391–441.CrossRefGoogle Scholar
6.Coxeter, H. S. M. and Todd, J. A.. “An extreme duodenary form”, An extreme duodenary form, 5 (1953), 384–392.Google Scholar
7.Craig, M.. “A characterization of certain extreme forms”, Illinois J. Math., 20 (1976), 706–717.CrossRefGoogle Scholar
8.Craig, M.. “A cyclotomic construction for Leech's lattice”, to appear in Mathematika.Google Scholar
9.Janusz, G. J.. Algebraic number fields, Pure and Applied Mathematics Series, vol. 55 (Academic Press, 1973).Google Scholar
10.Kneser, M.. “Two remarks on extreme forms”, Canad. J. Math., 7 (1955), 145–149.CrossRefGoogle Scholar
11.Ko, Chao. “On the positive definite quadratic forms with determinant unity”, Acta Arith., 3 (1939), 79–85.CrossRefGoogle Scholar
12.O'Connor, R. E. and Pall, G.. “The construction of integral quadratic forms of determinant 1 “. Duke Math. J., 11 (1944), 319–331.Google Scholar
13.Stacey, K. C.. “The enumeration of perfect septenary forms”, J. Lond. Math. Soc, (2), 10 (1975), 97–104.CrossRefGoogle Scholar
14.Weiss, E.. Algebraic number theory, International Series in Pure and Applied Mathematics (McGraw-Hill, 1963).Google Scholar