Title:

Expansions of 1dimensional algebraic groups by a predicate for a subgroup

This thesis considers theories of expansions of the natural algebraic struc ture on the multiplicative group and on an elliptic curve by a predicate for a subgroup that are constructed by Hrushovski's predimension method. In the case of the multiplicative group, these are the theories of fields with green points constructed by Poizat. The convention of calling the elements of the distinguished subgroup green points is maintained throughout this work, also in the elliptic curve case, and we speak of theories of green points. In the first part of the thesis, we give a detailed account of the construction of the theories of green points. The work of Poizat is extended to the case of elliptic curves and an open question is answered in order to complete the construction in the cases where the distinguished subgroup is allowed to have torsion. Proofs of the main modeltheoretic properties of the theories, wstability and near modelcompleteness, are included, as well as rank calculations. In the second part, following ideas of Zilber, we find natural models of the constructed theories on the complex points of the corresponding algebraic group. In the case of elliptic curves, this is done under the assumption that the curve has no complex multiplication and is defined over the reals. In general, we also need to assume a consequence of the Schanuel Conjecture, in the multiplicative group case, and an analogous statement in the elliptic curve case. For the multiplicative group, the assumption is known to hold in generic cases by a theorem of Bays, Kirby and Wilkie; our result is therefore unconditional in these cases. Motivated by Zilber's work on connections between model theory and non commutative geometry, we prove similar results for variations of the above theories in which the distinguished subgroup is elementarily equivalent to the additive group of the integers, which we call theories of emerald points.
