Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-17T19:52:54.070Z Has data issue: false hasContentIssue false
  • English
  • Français

Abelian Quadratic Forms

Published online by Cambridge University Press:  20 November 2018

K. G. Ramanathan
K. G. Ramanathan*
Affiliation:
Tata Institute for Fundamental Research
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.

Hermite [4] in the course of his investigations on the transformation theory of abelian functions, introduced the notion of abelian quadratic forms. They are quadratic forms whose matrices of orders 2n, satisfy

where k ≠ 0 is a real number, and is the unit matrix of order n. Laguerre [7] and, more systematically, Cotty [3] developed an arithmetical theory of abelian forms in four variables.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 4 , 1952 , pp. 352 - 368
Copyright
Copyright © Canadian Mathematical Society 1952

References

1. Braun, H., Hermitian modular functions III, Ann. Math,, vol. 53 (1951), 143–160.Google Scholar
2. Chevalley, C., Theory of Lie groups I (Princeton, 1946).Google Scholar
3. Cotty, G., Les fonctions abéliennes et la théorie des nombres, Ann. Fac. Sci. Toulouse, ser. 3, vol. 3 (1911), 209–376.Google Scholar
4. Hermite, C., Sur la théorie de la transformation des fonctions abéliennes, Oeuvres, vol. 1 (Paris, 1905), 444–478.Google Scholar
5. Humbert, P., Réduction de formes quadratiques dans un corps algébrique fini, Comment. Math. Helv., vol. 23 (1949), 50–63.Google Scholar
6. Klein, F. and Fricke, R., Vorlesungen Über die Theorie der Modul-Funktionen, vol. 1 (Leipzig, 1890), 243–269.Google Scholar
7. Laguerre, E., Sur le calcul des systèmes linéaires, Oeuvres, vol. 1 (Paris, 1898), 221–267.Google Scholar
8. Ramanathan, K. G., The theory of units of quadratic and hermitian forms, Amer. J. Math., vol. 73 (1951), 233–255.Google Scholar
9. Siegel, C. L., Einheiten quadratischer Formen, Abh. Sem. Hansischen Univ., vol. 13 (1939), 209–239.Google Scholar
10. Siegel, C. L., Symplectic geometry, Amer. J. Math., vol. 65 (1943), 1–86.Google Scholar
11. Siegel, C. L. Some remarks on discontinuous groups, Ann. Math., vol. 46 (1945), 708–718.Google Scholar
12. van der Waerden, B. L., Moderne Algebra, vol. 1 (Berlin, 1930).Google Scholar