Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-28T15:02:05.708Z Has data issue: false hasContentIssue false
  • English
  • Français

On S-class number relations of algebraic tori in Galois extensions of global fields

Published online by Cambridge University Press:  22 January 2016

Masanori Morishita
Masanori Morishita*
Affiliation:
Department of Mathematics, The Johns Hopkins University, Baltimore, MD 21218, U.S.A.
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.

As an interpretation and a generalization of Gauss’ genus theory on binary quadratic forms in the language of arithmetic of algebraic tori, Ono [02] established an equality between a kind of “Euler number E(K/k)” for a finite Galois extension K/k of algebraic number fields and other arithmetical invariants associated to K/k. His proof depended on his Tamagawa number formula [01] and Shyr’s formula [Sh] which follows from the analytic class number formula of a torus. Later, two direct proofs were given by Katayama [K] and Sasaki [Sa].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 124 , December 1991 , pp. 133 - 144
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1991

References

[E.A] Artin, E., Quadratishe Körper im Gebiete der höheren Kongruenzen. I, Math. Z., 19 (1924), 153–206.Google Scholar
[A-T] Artin, E. and Tate, J., Class Field Theory, Harvard Univ., 1961.Google Scholar
[M.A] Artin, M., Grothendieck Topologies, Lecture Notes, Harvard Univ., 1962.Google Scholar
[D-G] Demazure, M. and Grothendieck, A., Schémas en Groupes II, Lect. Notes in Math., 152, Springer, 1971.Google Scholar
[G-R] Galovich, S. and Rosen, M., The class number of cyclotomic function fields, J. Number Theory, 13 (1981), 363–375.Google Scholar
[G] Greenberg, M., Rational points in henselian discrete valuation rings, Publ. Math. I.H.E.S., 31 (1966), 59–64.CrossRefGoogle Scholar
[H] Hörlimann, W., On algebraic tori of norm type. Comment. Math. Helv., 59 (1984), 539–549.CrossRefGoogle Scholar
[K] Katayama, S., E(K/k) and other arithmetical invariants for finite Galois extensions, Nagoya Math. J., 114 (1989), 135–142.Google Scholar
[Nl] Nisnevich, Ye. A., Arithmetic and cohomology invariants of semisimple group schemes and compactifications of locally symmetric spaces, Funct. Anal. Appl., 14 (1980), no. 1, 61–62.Google Scholar
[N2] Nisnevich, Ye. A., On certain arithmetic and cohomological invariants of semisimple groups, preprint, 1989.Google Scholar
[01] Ono, T., On the Tamagawa number of algebraic tori. Ann. of Math., 78 (1963), 47–73.Google Scholar
[02] Ono, T., On some class number relations for Galois extensions. Nagoya Math. J., 107 (1987),121–133.Google Scholar
[Sa] Sasaki, R., Some remarks to Ono’s theorem on a generalization of Gauss’ genus theory, Nagoya Math. J., 111 (1988), 131–142.Google Scholar
[Se] Serre, J.-P., Cohomologie Galoisienne. Lect. Notes in Math., 5, Springer, 1965.Google Scholar
[Sh] Shyr, J.-M., Class number formulas of algebraic tori with applications to relative class numbers of certain relative quadratic extensions of algebraic number fields, Thesis, The Johns Hopkins Univ., 1974.Google Scholar