Abstract
Using model-theoretic methods we prove:
Theorem A
If G is a Nash group over the real or p-adic field, then there is a Nash isomorphism between neighbourhoods of the identity of G and of the set of F-rational points of an algebraic group defined over F.
Theorem B
Let G be a connected affine Nash group over ℝ. Then G is Nash isogeneous with the (real) connected component of the set of real points of an algebraic group defined over ℝ.
Theorem C
Let G be a group definable in a pseudo-finite field F. Then G is definably virtually isogeneous with the set of F-rational points of an algebraic group defined over F.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
[A-M] M. Artin and B. Mazur,On periodic points, Ann. of Math.81 (1965), 82–99.
[BaL] J. T. Baldwin and A. H. Lachlan,On strongly minimal sets, J. Symbolic Logic36 (1971), 79–96.
[B1] E. Bouscaren,Model theoretic versions of Weil's theorem on pregroups, inModel Theory of Groups (A. Nesin and A. Pillay, eds.), Notre Dame Press, 1989, pp. 177–185.
[B2] E. Bouscaren,The group configuration—after E. Hrushovski, inModel Theory of Groups (A. Nesin and A. Pillay, eds.), Notre Dame Press, 1989, pp. 199–209.
[Bor] A. Borel,Linear Algebraic Groups, Springer-Verlag, Berlin, 1991.
[BCR] J. Bochnak, M. Coste and M-F. Roy,Geometrie algebrique reelle, Springer-Verlag, Berlin, 1987.
[Ch-v.d.D-Mac] Z. Chatzidakis, L. van den Dries and A. Macintyre,Definable sets over finite fields, J. Reine Angew427 (1992) 107–135.
[v.d.D] L. van den Dries,Algebraic theories with definable Skolem functions, J. Symbolic Logic49 (1984), 625–629.
[v.d.D-S] L. van den Dries and Ph. Scowcroft,On the structure of semialgebraic sets over p-adic fields, J. Symbolic Logic53 (1988), 1138–1164.
[Fr-J] M. Fried and M. Jarden,Field Arithmetic, Springer-Verlag, Heidelberg, 1986.
[Hr1] E. Hrushovski,Contributions to stable model theory, Ph.D. thesis, Berkeley, 1986.
[Ja] M. Jarden,Algebraic dimension over Frobenius fields, preprint, 1992.
[Mac] A. Macintyre,On definable sets of p-adic fields, J. Symbolic Logic41 (1976), 605–610.
[M-S] J. Madden and C. Stanton,One dimensional Nash groups, Pacific J. Math.154 (1992), 331–344.
[No] M. V. Nori,On subgroups of GL n (Fp), Inventiones Mathematicae88 (1987), 257–275.
[Pe] Perrin,Groupes henseliens, Publications d'Orsay, Universite Paris-Sud, 1975.
[P1] A. Pillay,Forking, normalisation and canonical bases, Ann. Pure Appl. Logic32 (1986), 61–81.
[P2] A. Pillay,Groups and fields definable in O-minimal structures, J. Pure Appl. Algebra53 (1988), 239–255.
[P3] A. Pillay, On fields definable in ℚ p , Arch. Math. Logic29 (1989), 1–7.
[P4] A. Pillay,An application of model theory to real and p-adic algebraic groups, J. Algebra126 (1989), 139–146.
[P5] A. Pillay,Model theory, stability theory and stable groups, inModel Theory of Groups (A. Nesin and A. Pillay, eds.), Notre Dame Press, 1989, pp. 1–40.
[P6] A. Pillay,Some remarks on definable equivalence relations in O-minimal structures, J. Symbolic Logic51 (1986), 709–714.
[P7] A. Pillay,Some remarks on modular regular types, J. Symbolic Logic56 (1991), 1003–1011.
[Po1] B. Poizat,Cours de Theorie des modeles, Nur al-Mantiq Wal-Marifah, Paris, 1985.
[Po2] B. Poizat,Groupes stables, Nur al-Mantiq Wal-Marifah, Paris, 1987.
[Po3] B. Poizat,An introduction to algebraically closed fields and varieties, inModel Theory of Groups (A. Nesin and A. Pillay, eds.), Notre Dame Press, 1989, pp. 41–67.
[S-W] A. Sagle and R. Warner,Introduction to Lie Groups, Academic Press, New York, 1973.
[Sh] M. Shiota,Nash manifolds, Lecture Notes in Math. 1269, Springer-Verlag, Berlin, 1987.
[We] A. Weil,On algebraic groups of transformations, Am. J. Math.77 (1955), 355–391.
Author information
Authors and Affiliations
Corresponding author
Additional information
Both authors supported by NSF grants.
Rights and permissions
About this article
Cite this article
Hrushovski, E., Pillay, A. Groups definable in local fields and pseudo-finite fields. Israel J. Math. 85, 203–262 (1994). https://doi.org/10.1007/BF02758643
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02758643