Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-02T22:46:53.322Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimal discriminants of indefinite ternary quadratic forms having specified class number

Published online by Cambridge University Press:  26 February 2010

A. G Earnest
A. G Earnest
Affiliation:
Department of Mathematics, Southern Illinois UniversityCarbondale, Illinois, USA. 62901-4408
Get access

Extract

In order for an indefinite integral ternary quadratic form to have class number exceeding one, its discriminant must be divisible by the cube of at least one odd prime, or by a sufficiently large power of 2 (see [4], [1]). More generally, for such a form to have class number 2t, t> 1, it is necessary not only that the discriminant be divisible by at least t distinct primes, but also that these primes interact with each other in rather specific ways. Consequently, the minimal absolute value ∆(t) of the discriminant of an indefinite integral ternary quadratic form of class number 2' increases rapidly as a function of the natural number t.

Type
Research Article
Information
Mathematika , Volume 35 , Issue 1 , June 1988 , pp. 95 - 100
Copyright
Copyright © University College London 1988

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.Earnest, A. G. and Hsia, J. S.. Spinor norms of local integral rotations, II. Pacific J. Math., 61 (1975), 71–86.CrossRefGoogle Scholar
2.Hsia, J. S.. Spinor norms of local integral rotations, I. Pacific J. Math., 57 (1975), 199–206.CrossRefGoogle Scholar
3.Jones, B. W.. The Arithmetic Theory of Quadratic Forms. (Carus Math. Monographs, Buffalo, 1950).CrossRefGoogle Scholar
4.Kneser, M.. Klassenzahlen indefiniter quadratischer Formen in drei oder mehr Veranderlichen. Arch Math., 7 (1956), 323–332.CrossRefGoogle Scholar
5.O'Meara, O. T.. Introduction to Quadratic Forms (Springer, Berlin, 1963).CrossRefGoogle Scholar
6.O'Meara, O. T.. Hilbert's eleventh problem: the arithmetic theory of quadratic forms. Mathematical developments arising from Hilbert problems (ed. Browder, F. E.). Proc. Symposia Pure Math., 28 (1976), 379–400.CrossRefGoogle Scholar
7.Watson, G. L.. Integral Quadratic Forms (Cambridge University Press, 1960).Google Scholar