Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T10:41:27.796Z Has data issue: false hasContentIssue false
  • English
  • Français

XXV.—On the Groups of Units of Ternary Quadratic Forms with Rational Coefficients

Published online by Cambridge University Press:  14 February 2012

J. Mennicke
J. Mennicke
Affiliation:
Mathematisches Institut A, Technische Hochschule, Braunschweig, Germany.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Fuchsian groups that are unit groups of ternary quadratic forms with rational integer coefficients are studied. By means of the well-known Nielsen classification of finitely generated Fuchsian groups, a complete survey of the unit groups is given. For this, we have to use the arithmetical methods of B. W. Jones. In the second part, the relations between Fuchsian groups arising from different quadratic forms are studied. It turns out that, with a finite number of exceptions, all these Fuchsian groups are subgroups of a particular one.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 67 , Issue 4 , 1967 , pp. 309 - 352
Copyright
Copyright © Royal Society of Edinburgh 1967

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

References to Literature

Bachmann, F., 1959. Aufbau der Geometrie aus dem Spiegelungsbegriff. Heidelberg: Springer Verlag.CrossRefGoogle Scholar
Coxeter, H. S. M., 1961. Introduction to geometry. New York: Wiley.Google Scholar
Coxeter, H. S. M., and Moser, W. O. J., 1957. ”Generators and relations for discrete groups”, Ergebn. Math., 14, 1–155.Google Scholar
Dickson, L. E., 1950. Modern elementary theory of numbers. University of Chicago Press.Google Scholar
Dickson, L. E., 1961. Linear groups, with an exposition of the Galois field theory. New York: Dover Publications Inc.Google Scholar
Dieudonn é, J., 1948. Sur les groupes classiques. Paris: Hermann.Google Scholar
Eichler, M., 1952. Quadratische Formen und orthogonale Gruppen. Heidelberg: Springer Verlag.CrossRefGoogle Scholar
Greenberg, L., 1960. “Discrete groups of motions”, Canad. J. Math., 12, 415–426.CrossRefGoogle Scholar
Heegner, K., 1938. “Transformierbare automorphe Funktionen und quadratische Formen, I, II”, Math. Z., 43, 161–204, 321–352.CrossRefGoogle Scholar
Hull, R., 1939. “On the units of indefinite quaternion algebras”, Amer. J. Math., 61, 365–374.CrossRefGoogle Scholar
Jones, B. W., 1950. The arithemtic theory of quadratic forms. New York: Mathl. Ass. Am.CrossRefGoogle Scholar
Kneser, M., 1956. “Klassenzahlen indefiniter quadratischer Formen in drei oder mehr Veranderlichen”, Arch. Math., Karlsruhe, 7, 323–332.CrossRefGoogle Scholar
Mennicke, J., 1961. “A note on regular coverings of closed orientable surfaces”, Proc. Glasg. Math. Ass., 5, 49–66.CrossRefGoogle Scholar
Nielsen, J., 1940. “Uber Gruppen linearer Transformationen”, Mitt. Math. Ges., Hamburg, 8, 82–104.Google Scholar
Nielsen, J., 1952. “Some fundamental concepts concerning discontinuous groups of linear substitutions in a complex variable”, Proc. Skand. Math. Kongr., Trondheim, 1949, 61–70.Google Scholar
Nielsen, J. and Bundgaard, S., 1951. “On normal subgroups of finite index in F-groups”, Mat. Tidsskr., 1951 B, 56–58.Google Scholar
Siegel, C. L., 1935, 1936. “Uber die analytische Theorie der quadratischen Formen, I, II”, Ann. Math., Princeton, 36, 527–606; 37, 230–263.CrossRefGoogle Scholar
Siegel, C. L., 1939. “Einheiten quadratishcher Formen”, Abh. Math. Sem. Hamburg Univ., 13, 209–239.CrossRefGoogle Scholar
Scholz, A. and Schoeneberg, B., 1955. Einführung in die Zahlentheorie, in Sammlung Goschen, 1131. Berlin: de Gruyter.CrossRefGoogle Scholar
Tatuzawa, T., 1951. “On a theorem of Siegel”, Jap. J. Math., 21, 163–178.CrossRefGoogle Scholar
Threlfall, W., 1932. “Gruppenbilder”, Abh. Sächs. Ges. (Akad.) Wiss., Math. Phys. Kl, 41, 1–59.Google Scholar
Watson, G. L., 1960. Integral quadratic forms. Cambridge University Press.Google Scholar
Van Der Waerden, B. L., 1948. Gruppen von linearen Transformationen. New York: Chelsea Publishing Co.Google Scholar
Zassenhaus, H., 1958. The theory of groups. New York: Chelsea Publishing Co.Google Scholar