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One-class genera of positive quadratic forms in nine and ten variables

Published online by Cambridge University Press:  26 February 2010

G. L. Watson
G. L. Watson
Affiliation:
University College, London.
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Let f be a positive-definite quadratic form with integer coefficients, and denote by c(f) (≥ 1) the class-number of f, that is, the number of classes in the genus of f. I showed in [4] that c(f) ≥ 2 for every f in n ≥ 11 variables; the transformations of [3] were used to make the problem easier. I have since sought to find all the one-class n-ary genera with 3 ≤ n ≤ 10 (the case n = 1 is trivial, and n = 2 is very difficult).

Type
Research Article
Information
Mathematika , Volume 25 , Issue 1 , June 1978 , pp. 57 - 67
Copyright
Copyright © University College London 1978

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References

1.Kneser, M.. “Zur Theorie der Kristallgitter”, Math. Annalen, 127 (1954), 105–106.CrossRefGoogle Scholar
2.Watson, G. L.. Integral quadratic forms, Cambridge Tracts in Mathematics and Mathematical Physics, No. 51 (Cambridge, 1960).Google Scholar
3.Watson, G. L.. “Transformations of a quadratic form which do not increase the class-number”, Proc. London Math. Soc, (3), 12 (1962), 577–587.CrossRefGoogle Scholar
4.Watson, G. L.. “The class-number of a positive quadratic form”, Proc. London Math. Soc, (3), 13 (1963), 549–576.Google Scholar
5.Watson, G. L.. “One-class genera of positive ternary quadratic forms”, Mathematika, 19 (1972), 96–104.Google Scholar
6.Watson, G. L.. “One-class genera of positive quaternary quadratic forms”, Ada Arithmetica, 24 (1973), 461–475.Google Scholar
7.Watson, G. L. “One-class genera of positive quadratic forms in at least five variables”, Acta Arithmetica, 26 (1975), 309–327.Google Scholar
8.Watson, G. L. “One-class genera of positive ternary quadratic forms—II”, Mathematika, 22 (1975), 1–11.Google Scholar
9.Watson, G. L. “Transformations of a quadratic form which do not increase the class-number (II)”, Acta Arithmetica, 27 (1975), 171–189.CrossRefGoogle Scholar
10.Watson, G. L. “The 2-adic density of a quadratic form”, Mathematika, 23 (1976), 94–106.CrossRefGoogle Scholar
11.Watson, G. L. “One-class genera of positive quadratic forms”, Journal London Math. Soc, 38 (1963), 387–392.Google Scholar