Expansions of algebraically closed fields and abstract Kummer theory

Using Hrushovski's amalgamation method, one may construct `nice' expansions of algebraically closed fields by new (non-algebraic) subgroups of the multiplicative group (green fields of Poizat, bad fields) or of a semi-abelian variety, with the help of an appropriate predimension function.

In these constructions, one is confronted with two uniformity issues: the first, more difficult one, is to get a control of the predimension; the second one deals with `multiplicities' in the division sequence attached to a generic point of a variety which may move in an algebraic family. The first difficulty is usually overcome using Ax-Schanuel type results (week CIT), the latter one is related to Kummer theory.

In this talk, we will first give a glance of the construction of the green and bad fields. We will then focus on the second issue, namely how one may uniformly control ' Kummer genericity' of subvarieties of algebraic tori. In fact, the results extend from semi-abelian varieties to the context of divisible abelian groups of finite Morley rank, since the relevant portions of Kummer theory may be developped in this abstract model-theoretic context. (Joint work with Martin Bays and Misha Gavrilovich.)