Interpretable groups in Mann pairs

Abstract

In this paper, we study an algebraically closed field $\Omega$ expanded by two unary predicates denoting an algebraically closed proper subfield k and a multiplicative subgroup $\Gamma$. This will be a proper expansion of algebraically closed field with a group satisfying the Mann property, and also pairs of algebraically closed fields. We first characterize the independence in the triple $(\Omega , k, \Gamma )$. This enables us to characterize the interpretable groups when $\Gamma$ is divisible. Every interpretable group H in $(\Omega ,k, \Gamma )$ is, up to isogeny, an extension of a direct sum of k-rational points of an algebraic group defined over k and an interpretable abelian group in $\Gamma$ by an interpretable group N, which is the quotient of an algebraic group by a subgroup $N_1$, which in turn is isogenous to a cartesian product of k-rational points of an algebraic group defined over k and an interpretable abelian group in $\Gamma$.