Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-08T05:34:56.212Z Has data issue: false hasContentIssue false
  • English
  • Français

Quaternary even positive definite quadratic forms of prime discriminant

Published online by Cambridge University Press:  22 January 2016

Yoshiyuki Kitaoka
Yoshiyuki Kitaoka*
Affiliation:
Department of Mathematics, Nagoya University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this note we study quaternary even positive definite quadratic forms of prime discriminant. In § 1 we classify quaternary even positive definite quadratic forms of prime discriminant p = 1 mod 4 (called simply nice quaternary lattices in this note) which represent two. We note that the class number of such forms is closely related to the dimension of the space of certain automorphic forms. (Remark 4 in the text). By using the classification in § 1 and the theory of integral representations of cyclic groups we show that the orthogonal group of a nice quaternary lattice is generated by ±1 and symmetries (of the lattice). In § 3, we calculate the class number of nice quaternary lattices. Notations and terminologies will generally be those of O’Meara [5]. Any exceptions to this convention will be stated explicitly.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 52 , December 1973 , pp. 147 - 161
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1973

References

[1] Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras, Interscience Pub., 1962.Google Scholar
[2] Eisenstein, G., Tabelle der reducirten positive ternären quadratischen Formen, J. für Math., 41 (1851), 141–190.Google Scholar
[3] Kitaoka, Y., Two theorems on the class number of positive definite quadratic forms, Nagoya Math. J., 51 (1973), 79–89.CrossRefGoogle Scholar
[4] Mordell, L. J., On the class number for definite ternary quadratics, Messenger of Math., 47 (1918), 65–78.Google Scholar
[5] O’Meara, O. T., Introduction to quadratic forms, Springer-Verlag, 1963.Google Scholar
[6] Pfeuffer, H., Einklassige Geschlechter totalpositiver quadratischer Formen in totalreellen algebraischen Zahlkörpern, J. number theory, 3 (1971), 371–411.Google Scholar
[7] Ponomarev, P., Class number of definite quaternary forms with nonsquare discriminant, Bull. Amer. Math. Soc, 79 (1973), 594–598.Google Scholar
[8] Siegel, C. L., Über die analytische Theorie der quadratischen Formen, Ann. Math., 36 (1935), 527–606.Google Scholar