Proceedings of the Japan Academy, Series A, Mathematical Sciences

On the topology of relative orbits for actions of algebraic groups over complete fields

Dao Phuong Bac and Nguyen Quoc Thang

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We investigate the problem of equipping a topology on cohomology groups (sets) in its relation with the problem of closedness of (relative) orbits for the action of algebraic groups on affine varieties defined over complete, especially $\mathfrak{p}$-adic fields and give some applications.

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Proc. Japan Acad. Ser. A Math. Sci., Volume 86, Number 8 (2010), 133-138.

First available in Project Euclid: 4 October 2010

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Zentralblatt MATH identifier

Primary: 14L24: Geometric invariant theory [See also 13A50]
Secondary: 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 20G15: Linear algebraic groups over arbitrary fields

Closed orbits local fields algebraic group actions


Bac, Dao Phuong; Thang, Nguyen Quoc. On the topology of relative orbits for actions of algebraic groups over complete fields. Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 8, 133--138. doi:10.3792/pjaa.86.133.

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  • D. Birkes, Orbits of linear algebraic groups, Ann. of Math. (2) 93 (1971), 459–475.
  • A. Borel, Introduction aux groupes arithmétiques, Hermann, Paris, 1969.
  • A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535.
  • R. J. Bremigan, Quotients for algebraic group actions over non-algebraically closed fields, J. Reine Angew. Math. 453 (1994), 21–47.
  • M. Demazure and P. Gabriel, Groupes algébriques. Tome I, Masson & Cie, Éditeur, Paris, 1970.
  • P. Gille and L. Moret-Bailly, Action algébriques des groupes arithmétiques. Appendice to the article by Ullmo-Yafaev “Galois orbits and equidistribution of special subvarieties: towards the André-Oort conjecture”. (Preprint).
  • T. Kambayashi, M. Miyanishi and M. Takeuchi, Unipotent algebraic groups, Lecture Notes in Math., 414, Springer, Berlin, 1974.
  • G. R. Kempf, Instability in invariant theory, Ann. of Math. (2) 108 (1978), no. 2, 299–316.
  • L. Lifschitz, Superrigidity theorems in positive characteristic, J. Algebra 229 (2000), no. 1, 375–404.
  • G. A. Margulis, Discrete subgroups of semisimple Lie groups, Springer, Berlin, 1991.
  • J. S. Milne, Arithmetic duality theorems, Second edition, BookSurge, LLC, Charleston, SC, 2006.
  • D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory, Third edition, Springer, Berlin, 1994.
  • J. Oesterlé, Nombres de Tamagawa et groupes unipotents en caractéristique $p$, Invent. Math. 78 (1984), no. 1, 13–88.
  • M. S. Raghunathan, A note on orbits of reductive groups, J. Indian Math. Soc. (N.S.) 38 (1974), no. 1-4, 65–70 (1975).
  • S. Ramanan and A. Ramanathan, Some remarks on the instability flag, Tohoku Math. J. (2) 36 (1984), no. 2, 269–291.
  • J.-P. Serre, Cohomologie galoisienne, Fifth edition, Springer, Berlin, 1994. MR1324577 (96b:12010)
  • R. Steinberg, Conjugacy classes in algebraic groups, Springer, Berlin, 1974.
  • N. Q. Thǎńg and N. D. Tan, On the Galois and flat cohomology of unipotent algebraic groups over local and global function fields. I, J. Algebra 319 (2008), no. 10, 4288–4324.
  • J. Tits, Lectures on algebraic groups, Yale Univ., 1967.
  • T. N. Venkataramana, On superrigidity and arithmeticity of lattices in semisimple groups over local fields of arbitrary characteristic, Invent. Math. 92 (1988), no. 2, 255–306.