The model-theoretic content of Lang’s conjecture

Abstract

The rest of this volume is dedicated to explaining Hrushovski's model-theoretic approach [Hr 96] to the geometric case of a conjecture of Lang. See [Hi] for a presentation of the conjecture and [Bous] for the proof. The purpose of this note is to point out that the use of model-theoretic and stability-theoretic methods should not be so surprising, as the full Lang conjecture itself is equivalent to a purely model-theoretic statement. The structure (ℚ, +,.) is wild (undecidable, definable sets have no ``structure'' etc.), as is the structure (ℂ, +,.) with a predicate for the rationals. What comes out of the diophantine-type conjectures on the other hand is that certain enrichments of the structure (ℂ, +,.) (more specifically expansions obtained by adding a predicate, not for ℚ itself, but rather for the ℚ-points of certain algebraic groups) are not wild, in particular are stable.