Model Theory and Diophantine Geometry Lectures 3 , 4 and 5

These notes continue the notes of Anand Pillay on model theory and diophantine geometry. In my lectures I describe a model theoretic approach to some analogues of the Mordell-Lang conjecture for Drinfeld modules. Many questions remain open and algebraic proofs along the lines of the proof of the Manin-Mumford conjecture described by Pillay may be possible. We discuss these questions and potential alternate approaches to these problems throughout these notes. These notes are organized as follows. We begin in Section 1 with a discussion of the Mordell-Lang conjecture in its original form and some of its generalization. A discussion of Drinfeld modules and the Drinfeld module analogues of the the Mordell-Lang conjectures raised by Laurent Denis follows in Section 2. In Section 3 we discuss the general technique for proving Mordell-Lang type theorems by working with locally modular groups in enriched fields. In Section 5 we outline a weak solution to the the Drinfeld module Mordell-Lang conjecture proved using the model theory of separably closed fields. In Section 4 we prove the the Drinfeld module version of the Manin-Mumford conjecture using the model theory of difference fields. In Section 6 we end these notes with several open questions.