OMinimal Ingredients in Proofs of Arithmetical Conjectures Such as ManinMumford and AndreOort
Introductory Workshop: Model Theory, Arithmetic Geometry and Number Theory February 03, 2014  February 07, 2014
Location: MSRI: Simons Auditorium
The goal of these talks is to describe a method which was rst proposed by Pila and Zannier in their new proof of the ManinMumford conjecture ([4]), and since then implemented to various other arithmetical questions. Most notably, it was used by Pila to prove the AndreOort conjecture for Cn, [3]. A common feature to the problems which t this method is their general form: Name a collection of \special" complex algebraic varieties (e.g. Abelian varieties, Shimura varieties, mixed Shimura varieties). Within one such special variety V one isolates a set of \special points" (e.g. the set of torsion points in the group of an abelian variety). Consider an algebraic subvariety X of V (apriori not necessarily special itself) and assume that X contains \many" (e.g. A Zariski dense set of) special points. The conjectures claim that under these assumptions X itself must be a special variety, or at least contain a nontrivial special subvariety. The novelty in the method of Pila and Zannier is the application of a Theorem of Pila and Wilkie, [8] about rational points on denable sets in ominimal structures. That theorem itself has a similar form to the above conjectures: Assume that X Rn is a denable set in an ominimal structure. If X \ Qn is \large" then X must contain an innite semialgebraic set. More precisely, if one removes from X all innite semialgebraic sets then the remaining intersection with Qn is \small". The application of PilaWilkie to the arithmetic problems goes via three steps: 1. Find a map , denable in an ominimal structure M which transforms the variety V to an Mdenable set eV Cn. Let (X) = eX . These maps are usually dened via classical transcendental maps such as the complex exponentials or the Jinvariant. In all cases thus far it is sucient to consider the structure = Ran;exp. 2. Show that if X contains \many" special points in the arithmetical sense then eX contains a \large" set of rational or algebraic points, in the sense of PilaWilkie. This step is number theoretic in nature. It requires rst to show that sends the special points in X to rational (or more generally algebraic) points in eX . Mainly, in order to apply PilaWilkie, one is required to obtain precise lower bounds on the number of the rational (or algebraic) points, in terms of their height. 3. After applying PilaWilkie one obtains a nontrivial semialgebraic subset eS eX . Replacing S by such subset of maximal dimension, one tries to show that
Peterzil notes

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