The semester, we are following a paper of Peterzil and Starchenko in the model theory seminar. Dave Marker gave the first lecture, which was a general interest to the topics the paper will consider. This post is going to give rough notes of Dave’s talk. We begin with a result of Ax:

**Theorem:** Let -linearly independent over . Then

**Geometric version:** Let be the group of . Let be convergent power series on . Let (note: I will use to denote the graph of a function). Then the rank of the matrix is equal to The dimension of the Zariski closure of denoted is at least unless the projection of $U$ to its first $n$ coordinates is contained in a linear subspace: , in which case, .

The statement which is usually called the **Ax-Lindemann-Weierstrass Theorem** (hence ALW) is a more detailed statement of the algebraic content of the geometric version of Ax’s result:

**ALW** If are irreducible algebraic varieties, with , then there is a coset of an algebraic subgroup of so that .

Now, we will consider statements of a similar nature in a more general setting. Let be an abelian variety of dimension (again we are working over the complex numbers throughout). Then there are so that we can realize as the quotient of by in an analytic manner: that is, there is a surjective analytic map , with . We will be considering this map Much like the complex exponential function, this global map is poorly behaved when we consider it in any expansion of the real field (because it is periodic on , with this map and the field structure, it is easy to see that you can interpret the integers).

But, we are still going to use tame model theory (o-minimality) to prove things about this map (eventually) – we are just going to have to be careful about restricting to bounded subsets of the domain. In this setting, we have another ALW-type result (this one was proved by Pila and some cases by others, I think – if you can give a definitive account of this, please comment below and I will modify the post).

**ALW:** Let be as above. Suppose that is an irreducible algebraic variety, then is a coset of a subabelian variety.

Of course, to many of us, these statements about transcendence and restrictions on the algebraic closures of very transcendental maps are interesting in their own right, but there are other reasons that these results have proven useful. We will spend the rest of the time describing the Pila-Zannier approach to diophantine problems in one specific interesting case.

**Manin-Mumford (special case):** If is a simple abelian variety over a number field and is a curve, then is finite.

We will describe the Pila-Zannier proof. First, we will think about the following maps: , where . Then , and But, we only need to understand the map on , by periodicity.

So, let be a curve, and let and let now we will think about , which is definable in the o-minimal structure $\mathbb R_{an}$ (the real field augmented by maps of any analytic function of any number of variables restricted to a compact subset). Now, a powerful theorem will come onto the scene (actually, it already did in Pila’s proof of ALW, but this seminar is about proving ALW in another manner, so lets disregard that…).

**Pila-Wilkie:** Let be definable in an o-minimal structure and let be the union of connected semialgebraic curves. Then for any , let with in lowest terms, and for all ,

Note that in general can be a countable union of algebraic curves (for instance, think about the graph of$latex x^y=z$ in . So, one point of the result is that one can characterize a certain a priori analytic curve as algebraic just by finding a lot of points on the curve. Then if you know a transcendence result (e.g. ALW above), these two points can be played off of each other.

Under the assumptions for our MM-type problem, is empty, because one can prove that if contains a semi-algebraic curve, then contains an algebraic curve (we won’t prove this). Of course, the only way this happens is for to be the coset of an abelian subvariety, but there are none, by simplicity.

Now, there is a result of Masser: such that if is a torsion point of order $n \geq N $, then has at least many conjugates over $k$. Now of course, we have a problem – if we do have infinitely many torsion points on our curve, then add some torsion point of order at least . One can, bound the height of the preimage of the torsion points in as a linear function of their order. But, now we get at least many of these points, which violates Pila-Wilkie. So, we must have had only finitely many torsion points on our curve.

This was the end of Dave’s seminar. Several comments: as mentioned, Pila-Wilkie is used in two places – once to prove the functional transcendence results (this is quite involved) and onces to play off of a result like Masser’s (in other contexts, there are other results of a similar spirit). This seminar is about proving the functional transcendence results in a more straightforward manner without appealing to Pila-Wilkie. Pila-Wilkie is “ineffective” in the sense that we know that the growth rate of the points on the certain definable sets must be small, but we don’t know the constants or at what point the growth rates kick in – so there are fewer than many points happens eventually, for large enough , but we don’t know what large enough means. So, effectivizing these finiteness statements with specific bounds is of quite a bit of interest. It has been done in certain contexts, but that is another story for another time.