Groups definable in local fields and pseudo-finite fields

Abstract

Using model-theoretic methods we prove: 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. 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. 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.