ABSTRACT
Introduction We give an exposition of several results on definable groups in dif
ferentially closed fields, and applications thereof. We tend to work in a naive context, replacing algebraic-geometric notions and constructions by model-theoretic ones whenever possible. There is a certain resulting clum siness, but at least the material should be accessible to somebody with a good knowledge of stability/stable groups, but without too much algebraicgeometric background. Among other things we give a proof of the result [HS] that there are continuum many countable differentially closed fields of characteristic 0. The theory DCFn (of differentially closed fields of charac teristic 0) is complete and ■stable and thus by fSHMl has either or
countable models. But until recently it was not known which. Rather surprisingly it turns out that classical mathematical objects, specifically el liptic curves, lie behind the existence of continuum many countable models (or at least behind the present proof). One of the essential points is to find some strongly regular nonisolated type which is orthogonal to the empty set. The required type p is found inside a suitable definable (in DCFq) subgroup G (of finite Morley rank) of an elliptic curve E(a) with differen tially transcendental j -invariant a. So it turns out that there are “exotic” groups of finite Morley rank definable in differentially closed fields. In any case in section 2 of this paper we prove the existence of countable dif ferentially closed fields. The argument we present was sketched for us by E. Hrushovski, although we have a few additional simplifications. In fact, given an example due to Manin [M], showing that for any elliptic curve E there is differential rational homomorphism from E onto Ga (the additive group), the existence of the required type p turns out to rather a direct matter, requiring neither the deep Zariski-geometry interpretation, nor the properties of “jet group” of algebraic groups.
