Some Definability Results in Abstract Kummer Theory

Let S be a semiabelian variety over an algebraically closed field, and let X be an irreducible subvariety not contained in a translate of a proper algebraic subgroup of S. We show that the number of irreducible components of [n]⁻¹(X) is bounded uniformly in n, and moreover that the bound is uniform in families Xt. We prove this by Galois-theoretic methods. This proof can be formulated purely model theoretically, and applies in the more general context of divisible abelian groups of finite Morley rank. In this latter context, we deduce a definability result under the assumption of the definable multiplicity property (DMP). We give sufficient conditions for finite Morley rank groups to have the DMP, and hence give examples where our definability result holds.