Abstract
We compute the Shafarevich-Tate group, the kernel of the weak approximation and the Manin groups of three-dimensional algebraic tori defined over an algebraic number field. A minimal example of a torus with fractional Tamagawa number is constructed. A criterion for the validity of the Hasse norm principle for extensions of degree four of an algebraic number field is given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
Yu. I. Manin, Cubic Forms; Algebra, Geometry, Arithmetic, North-Holland, Amsterdam (1974).
V. E. Voskresenskii, Algebraic Tori [in Russian], Nauka, Moscow (1977).
B. É. Kunyavskii, “Birational classification of tori of small dimension,” in: Seminar on Arithmetic of Algebraic Varieties, Saratov State Univ. (1979), pp. 37–42.
T. Ono, “On the Tamagawa number of algebraic tori,” Ann. Math.,78, No. 1, 47–73 (1963).
S. Lang, Algebraic Numbers, Addison-Wesley, Reading, Mass. (1964).
A. F. Kryuchkov, “Finite Galois modules,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,86, 125–134 (1979).
S. Gurak, “The Hasse norm principle in non-Abelian extensions,” J. Reine Angew. Math., 303–304, 1978.
J.-L. Colliot-Thélène and J.-J. Sansuc, “La R-équivalence sur les tores,” Ann. Sci. Ecole Norm. Sup., 4ème Série,10, No. 2, 175–229 (1977).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 102–107, 1982.
The author expresses his gratitude to V. E. Voskresenskii and A. A. Klyachko for valuable discussions.
Rights and permissions
About this article
Cite this article
Kunyavskii, B.É. Arithmetic properties of three-dimensional algebraic tori. J Math Sci 26, 1898–1901 (1984). https://doi.org/10.1007/BF01670577
Issue Date:
DOI: https://doi.org/10.1007/BF01670577