Groups Definable in ACFA

Abstract

These notes should be read in conjunction with Anand Pillay’s notes [7]. We present some of the intermediate results given by E. Hrushovski [3] in his proof of the Manin-Mumford conjecture. We work in an existentially closed difference field. One of the main results is the existence in characteristic 0 of a simple criterion for the stability and 1-basedness (LMS) of a definable subgroup of an abelian variety A. A consequence of this criterion is:

Let A be an abelian variety defined over the field k, and assume that the automorphism σ fixes k. Let $p\left( T \right) = \sum\nolimits_{i = 0}^m {{a_i}{T^i} \in \mathbb{Z}\left[ T \right]}$ and let B = $\ker \left( {p\left( \sigma \right)} \right) = {}_{def}\left\{ {A|\sum\nolimits_{i = 0}^m {\left[ {{a_i}} \right]{\sigma ^2}\left( b \right) = 0} } \right\}$.