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

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