The Manin-Mumford conjecture and the model theory of difference fields

Using methods of geometric stability (sometimes generalized to finite S1 rank), we determine the structure of Abelian groups definable in ACFA, the model companion of fields with an automorphism. We also give general bounds on sets definable in ACFA. We show that these tools can be used to study torsion points on Abelian varieties; among other results, we deduce a fairly general case of a conjecture of Tate and Voloch on p-adic distances of torsion points from subvarieties. (C) 2001 Elsevier Science B.V. All rights reserved.