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Tensor products of positive definite quadratic forms IV

Published online by Cambridge University Press:  22 January 2016

Yoshiyuki Kitaoka
Yoshiyuki Kitaoka*
Affiliation:
Department of Mathematics, Nagoya University
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Let L, M, N be positive definite quadratic lattices over Z. We treated the following problem in [5], [6]:

If L ⊗ M is isometric to L ⊗ N, then is M isometric to N?

We gave a condition (**) in [6] such that the answer is affirmative for an indecomposable lattice L satisfying (**), and we gave some examples. In this paper we introduce a certain apparently weaker condition (A) than the condition (**), and we show that the condition (A) implies the condition (**) and more on integral orthogonal groups than a result in [6].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 73 , March 1979 , pp. 149 - 156
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1979

References

[1] Bartels, H.-J., Zur Galoiskohomologie definiter arithmetischer Gruppen, J. reine angew. Math. 298 (1978), 89–97.Google Scholar
[2] Bartels, H.-J., Definite arithmetische Gruppen, ibid. 301 (1978), 27–29.Google Scholar
[3] Kitaoka, Y., Scalar extension of quadratic lattices, Nagoya Math. J. 66 (1977), 139–149.Google Scholar
[4] Kitaoka, Y., Scalar extension of quadratic lattices II, ibid. 67 (1977), 159–164.Google Scholar
[5] Kitaoka, Y., Tensor products of positive definite quadratic forms, Göttingen Nachr. Nr. 4 (1977).Google Scholar
[6] Kitaoka, Y., Tensor products of positive definite quadratic forms II, J. reine angew. Math. 299/300 (1978), 161–170.Google Scholar
[7] Kitaoka, Y., Tensor products of positive definite quadratic forms III, Nagoya Math. J. 70 (1978), 173–181.CrossRefGoogle Scholar
[8] O’Meara, 0. T., Introduction to quadratic forms, Berlin, Heidelberg, New York (1971).Google Scholar
[9] Yang, J. H., Positive definite quadratic forms under some field extensions, A. M. S. Notices 23 (1976), A–424.Google Scholar