Applications of Model Theory

Recent years have witnessed some very exciting applications of model theory in various areas of mathematics; among which is the (nth) proof of Manin-Mumford Conjecture (I will refer to it as MMC) by Pila and Zannier ([7]). I will not get into why MMC is important as it is the subject of an article in a number theory blog. I will, however, get into details of how o-minimality helps proving MMC. In vague terms, MMC states that there aren’t too many torsion points on a subvariety of an abelian variety, unless that subvariety itself is close to being an abelian (sub)variety.

Recall that an abelian variety $A$ is a complex torus $\mathbb C^n/\Lambda$ that can be embedded into a projective space $\mathbb P^N(\mathbb C)$ (hence it has the structure of a projective complex variety). With this definition, an abelian variety is obviously an algebraic group which, in turn, is abelian.

There are various ways to state MMC with various strengths. Here is the version I’d like to discuss as it appears in [7].

Theorem 1. Let $A$ be an abelian variety and $X$ a subvariety, both defined over a number field. Suppose that $X$ does not contain a translate of an infinite abelian subvariety of A. Then $X$ contains only finitely many torsions of $A$ .

This has been proven first by Reynaud, and later came the proofs many others like Hindry, Buium and Hrushovski (this last one is a model theoretic proof as well). The proof I’ll discuss is due to Pila and Zannier and is quite different in vein, as uses o-minimality.

Without further ado, I’ll start explaining the main tool in that proof.

Pila-Wilkie Theorem

In a nutshell, Pila-Wilkie Theorem ([6]) gives an upper bound on the number of rational points of a set definable in an o-minimal expansion $\mathcal R$ of the real field, in terms of their heights. Well, this is very vague; anything this vague has to be correct. So where is the hardness? Let’s make things more precise.

First of all, the height of a rational number $\frac{a}{b}$ is the maximum of $|a|$ or $|b|$ , provided that $\gcd(a,b)=1$ and the height of a tuple of rational numbers is the maximum of the heights of its components. This is the most naive notion of height, but it works fine in the case in hand.

We wish to state that given a definable set $Y\subseteq\mathbb R^m$ , the number of rational points in $Y$ of height less than $T$ is bounded by a linear function of $T$ . But of course this cannot be the case, taking $Y$ to be the whole $\mathbb R^m$ ; besides such a result would finish a huge part of number theory. For this reason we need to take the (semi)algebraic part $Y^{alg}$ of $Y$ out; which is defined to be the union of all connected infinite semialgebraic sets contained in $Y$ .

With all this notation in hand, here is the precise statement of Pila-Wilkie Theorem (I’ll refer to it as PW).

Theorem 2. Let $Y\subseteq\mathbb R^n$ be definable and $\epsilon>0$ . There exists a constant $c=c(Y,\epsilon)>0$ such that for any $T>0$ we have

$|\{\alpha\in (Y\setminus Y^{alg})\cap\mathbb Q^n: H(\alpha)<T\}|\leq c T^{\epsilon}.$

The proof of this result is quite involved, but is a nice combination of the earlier individual works of the authors. Some years ago, Pila has proven a similar results in dimensions one and two ([1] (with Bombieri) and [5]) and Wilkie has a result on the integer points of one dimensional sets definable in o-minimal expansions of the real field ([8]).

Proof of MMC

The main difficulty in getting MMC from PW is to determine the algebraic part of a definable set. In general, we don’t really have any way to do that, but in our situation we deal with special kind of definable sets, definable in certain o-minimal structures.

Remember that we want to understand the torsions in a subvariety $X$ of an abelian variety $A=\mathbb C^n/\Lambda$ . In this case we have the universal covering map:

$\pi: \mathbb C^n\to A.$

Let $\lambda_1,\dots,\lambda_{2n}\in\mathbb C^n$ be a basis of the lattice $\Lambda$ . If we represent $\mathbb C^n$ with respect this basis, we see that the torsion points of $A$ correspond to the rational points of $\mathbb C^n$ .

In particular, the torsion points of $X$ correspond to the rational points of the set $Y'=\pi^{-1}(X)$ . The main virtue of this set $Y'$ is that it is periodic with respect to $\Lambda.$ Therefore we could restrict our attention to its rational points in $[0,1]^n$ . Now this new set $Y:=Y'\cap [0,1]^n$ is definable in the expansion $\mathbb R_{an}$ of the real field by restricted analytic functions. From now on we play with this set. Before forgetting $X$ , note that the assumption that it doesn’t contain any translate of any infinite abelian subvariety of $A$ translates to $Y$ as it doesn’t contain any translate of any $\mathbb C$ -linear subspace $V$ of $\mathbb C^n$ such that $dim V= rank (V\cap \Lambda)$ . Such translates are called torus cosets.

A large part of the paper ([7]) is devoted to the following.

Theorem 3. Let $Z\subseteq\mathbb C^n$ be an analytic set which is periodic with respect to a full-lattice. Suppose that $Z$ does not contain any torus coset. Then $Z$ does not contain any infinite connected semialgebraic set. This is to say that $Z^{alg}=\emptyset$ .

Note that this result holds for any complex torus, rather than just the abelian varieties. In short, the proof of this is just local (and tedious) analysis.

Now using this we get that $(Y')^{alg}=\emptyset$ and hence $Y^{alg}=\emptyset$ . So applying Theorem 2, we get that for a given $\epsilon>0$ , there is a constant $c=c(Y,\epsilon)>0$ such that there are at most $cT^{\epsilon}$ many rational points in $Y$ with common denominator $T$ , hence at most that many torsions in $X$ of exponent at most $T$ . This only gives an upper bound and the following result by Masser ([3]) gives the lower bound, finishing the proof of MMC.

Theorem 4. Let $A$ be an abelian variety defined over a number field $K$ . Then there is $d>0$ and $\rho$ depending only on the dimension of $A$ such that for every torsion point $P\in A$ of order $T$ we have

$[K(P):\mathbb Q]\geq d T^{\rho}.$

Final remarks

Recently this strategy played the role of a template for the proofs of even more complicated results such as the proof of certain cases of Andre-Oort conjecture by Habbegger and Pila ([2]). Other than finding the right setting, the main obstacles in such proofs are determining the algebraic part of the appropriate set and finding the substitute for Theorem 4. Of course, finding the right kind of o-minimal structure is another important step; as done by Peterzil and Starchenko in ([4]).

References

E. Bombieri, J. Pila. The number of integral points on arcs and ovals. Duke Math. J. 59 (1989), no. 2, 337–357.
P. Habegger, J. Pila. Some Unlikely Intersections Beyond André-Oort. Compositio Math. 148 (2012).
D. W. Masser. Small values of the quadratic part of the Néron-Tate height on an abelian variety. Compositio Math., 53(2):153–170, 1984.
Y. Peterzil and S. Starchenko. Definablity of restricted theta functions and families of abelian varieties. arXiv:1103.3110.
J. Pila. Rational points on a subanalytic surface. Ann. Inst. Fourier (Grenoble) 55 (2005), no. 5, 1501–1516.
J. Pila and A. Wilkie. The rational points of a deﬁnable set. Duke Math. J. 133 (2006), no. 3, 591–616.
Jonathan Pila and Umberto Zannier. Rational points in periodic analytic sets and the Manin-Mumford conjecture. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19(2):149–162, 2008.
A. J. Wilkie. Diophantine properties of sets deﬁnable in o-minimal structures. J. Symbolic Logic 69 (2004), no. 3, 851–861.

Tag Archives: Applications of Model Theory

“Model theory and applications to geometry”, Satellite workshop associated with LC 2013, Lisbon, 18th-19th of July 2013

Pila-Wilkie Theorem and Its Consequences

Pila-Wilkie Theorem

Proof of MMC

Final remarks

References

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