The Mordell-Lang Theorem from the Zilber Dichotomy

Abstract

We present a largely self-contained exposition of Ehud Hrushovski's proof of the function field Mordell-Lang conjecture beginning from the Zilber Dichotomy for differentially closed fields and separably closed fields. Our account is based on notes from a series of lectures given by Rahim Moosa at a MODNET workshop at Humboldt Universitat in Berlin in September 2007. We treat the characteristic 0 and characteristic p cases uniformly as far as is possible, then specialize to characteristic p in the final stages of the proof. We also take this opportunity to work out the extension of Hrushovski's ``Socle Theorem'' from the finite Morley rank setting to the finite U-rank setting, as is in fact required for Hrushovski's proof of Mordell-Lang to go through in positive characteristic.