Additive reducts of algebraically closed valued fields

Abstract

Abstract In this paper we prove a form of the Zilber's trichotomy conjecture for reducts of algebraically closed valued fields of characteristic 0 which are expansions of the valued vector space structure. We prove first that a non-modular reduct of a nontrivially valued algebraically closed field containing the valued vector space structure defines a non-semilinear curve. Then we show that the expansion of such a reduct by a non-semilinear curve defines multiplication on a nonempty open set.