Proceedings of the Japan Academy, Series A, Mathematical Sciences

On isomorphism classes of Zariski dense subgroups of semisimple algebraic groups with isomorphic $p$-adic closures

Nguyêñ Quoc Th\v{a}ńg

Full-text: Open access


We prove under certain natural conditions the finiteness of the number of isomorphism classes of Zariski dense subgroups in semisimple groups with isomorphic $p$-adic closures.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 78, Number 5 (2002), 60-62.

First available in Project Euclid: 23 May 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]
Secondary: 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05}

Arithmetic subgroups $p$-adic groups


Th\v{a}ńg, Nguyêñ Quoc. On isomorphism classes of Zariski dense subgroups of semisimple algebraic groups with isomorphic $p$-adic closures. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 5, 60--62. doi:10.3792/pjaa.78.60.

Export citation


  • Borel, A.: Some finiteness properties of adèle groups over number fields. Pub. Math. I. H. E. S., 16, 101–126 (1963).
  • Borel, A., et Serre, J.-P.: Théorèmes de finitude en cohomologie galoisienne. Comm. Math. Helv., 39, 111–164 (1964).
  • Bruhat, F., et Tits, J.: Groupes réductifs sur un corps local. Pub. Math., 41, 5–251 (1971).
  • Hochschild, G., and Mostow, G.: Automorphisms of algebraic groups. J. Algebra, 23, 435–443 (1969).
  • Matthews, C. R., Vasserstein, L. N., and Weisfeiler, B.: Congruence properties of Zariski-dense subgroups I. Proc. London. Math. Soc., 48, 514–532 (1984).
  • Mazur, B.: On the passage from local to global in number theory. Bull. A. M. S., 29, 14–50 (1993).
  • Nori, M.: On subgroups of $GL_n(\mathbf{F}_p)$. Inv. Math., 88, 257–275 (1987).
  • Pink, R.: Compact subgroups of linear algebraic groups. J. Algebra, 206, 438–504 (1998).
  • Serre, J.-P.: Lie algebras and Lie groups. Lecture Notes in Mathematics, vol. 1500, Springer-Verlag, Berlin-Heidelberg-New York (1992).
  • Segal, D.: Some remarks on $p$-adic analytic groups. Bull. London Math. Soc., 31, 149–153 (1999).