# O-minimality and diophantine geometry

Lead Research Organisation:
University of Manchester

Department Name: Mathematics

### Abstract

The investigators on this project are Alex Wilkie (Manchester, PI), Jonathan Pila (Oxford, co-I) and Gareth Jones (Manchester, co-I). It is a

joint proposal between the Universities of Manchester and Oxford with Manchester being the lead institution.

The aim of the project is to further the links between mathematical logic, specifically the branch of model theory known as o-minimality, and

diophantine geometry (i.e. the study of points on curves, surfaces etc with integer or rational coordinates). The o-minimality axiom applies to

collections of subsets of real euclidean spaces ("structures") and, when satisfied, implies a variety of topological, analytic and geometrical

finiteness properties that fail in general for the more classical classifications of sets that are studied in mainstream topology and analysis (such as those having a differentiable, or even analytic, manifold structure). Further, many interesting examples of o-minimal structures are known.

Wilkie was the first to notice that there are diophantine consequences for sets occurring in an o-minimal structure. Pila's work on diophantine

problems started with his influential 1989 paper with Bombieri and he proceeded to develop the diophantine theory of the so called real analytic

sets in several dimensions. This culminated in the Pila-Wilkie theorem establishing a general result in the broader setting of o-minimal structures. The application of this result, in work of Masser, Pila and Zannier, has opened up a new connection between mathematical logic and diophantine geometry with great potential, the most remarkable example to date being Pila's unconditional proof of a special case of the long standing Andre-Oort conjecture.

It is our intention to further such application as well as to advance the pure theory of o-minimal structures with this in mind. One step on the way is a conjecture of the PI concerning rational points on sets in one particular o-minimal structure known as the real exponential field. To carry this out one will also need more precise results on the model theory of the real Pfaffian field (another structure known to be o-minimal). Jones is an internationally recognised expert in both these areas and has obtained (jointly with several other researchers who are named as visitors in the proposal) the best results to date. All three aspects of our project, including details of visits and research meetings, are described at length in the following text.

joint proposal between the Universities of Manchester and Oxford with Manchester being the lead institution.

The aim of the project is to further the links between mathematical logic, specifically the branch of model theory known as o-minimality, and

diophantine geometry (i.e. the study of points on curves, surfaces etc with integer or rational coordinates). The o-minimality axiom applies to

collections of subsets of real euclidean spaces ("structures") and, when satisfied, implies a variety of topological, analytic and geometrical

finiteness properties that fail in general for the more classical classifications of sets that are studied in mainstream topology and analysis (such as those having a differentiable, or even analytic, manifold structure). Further, many interesting examples of o-minimal structures are known.

Wilkie was the first to notice that there are diophantine consequences for sets occurring in an o-minimal structure. Pila's work on diophantine

problems started with his influential 1989 paper with Bombieri and he proceeded to develop the diophantine theory of the so called real analytic

sets in several dimensions. This culminated in the Pila-Wilkie theorem establishing a general result in the broader setting of o-minimal structures. The application of this result, in work of Masser, Pila and Zannier, has opened up a new connection between mathematical logic and diophantine geometry with great potential, the most remarkable example to date being Pila's unconditional proof of a special case of the long standing Andre-Oort conjecture.

It is our intention to further such application as well as to advance the pure theory of o-minimal structures with this in mind. One step on the way is a conjecture of the PI concerning rational points on sets in one particular o-minimal structure known as the real exponential field. To carry this out one will also need more precise results on the model theory of the real Pfaffian field (another structure known to be o-minimal). Jones is an internationally recognised expert in both these areas and has obtained (jointly with several other researchers who are named as visitors in the proposal) the best results to date. All three aspects of our project, including details of visits and research meetings, are described at length in the following text.

### Planned Impact

Society and economy.

The proposed work is part of pure mathematics and we do not expect direct economic impact. This does not mean that we will not be alert to possibilities. If opportunities for non-academic exploitation of our work do arise, then the Manchester Institute for Mathematical Sciences (MIMS) is well-positioned to make the most of them. The presence of the CICADA project in MIMS means that many of the staff are used to an interdisciplinary outlook. There are also established interactions between some members of MIMS and commercial organizations. So, should commercial opportunities arise, there will be people in MIMS who are able to assist us in developing them.

Knowledge.

The main beneficiaries of the project will be other mathematicians working in model theory and number theory. To ensure rapid diffusion of our results we will hold small focused workshops around the visits of some of the visiting researchers. These workshops will differ from the standard format. There will be an emphasis on discussion and interaction among participants, rather than formal presentations. We anticipate several benefits of the ensuing knowledge exchange. For example, we expect that our visiting researchers will make a positive contribution to work being done by those workshop participants who are not part of the proposal. In the other direction, we hope that these participants will also make contributions that will impact on our work and that this will lead to further collaborations on the subject of the proposal. We have requested funds to cover the workshops, and resources to facilitate the collaborations that emerge.

Some of our work should also have impact in parts of the computer science community. Researchers in Bath and Manchester work on the complexity of various reachability problems in hybrid systems. They have developed a notion of Pfaffian hybrid system and have established algorithms for several problems on these systems. Our work on Pfaffian functions should lead to a wider and more robust class of systems to which their work would apply. We will meet the Manchester based researchers regularly to discuss our work. We will inform them of our progress and assist them as required in applying our results.

The proposed work is part of pure mathematics and we do not expect direct economic impact. This does not mean that we will not be alert to possibilities. If opportunities for non-academic exploitation of our work do arise, then the Manchester Institute for Mathematical Sciences (MIMS) is well-positioned to make the most of them. The presence of the CICADA project in MIMS means that many of the staff are used to an interdisciplinary outlook. There are also established interactions between some members of MIMS and commercial organizations. So, should commercial opportunities arise, there will be people in MIMS who are able to assist us in developing them.

Knowledge.

The main beneficiaries of the project will be other mathematicians working in model theory and number theory. To ensure rapid diffusion of our results we will hold small focused workshops around the visits of some of the visiting researchers. These workshops will differ from the standard format. There will be an emphasis on discussion and interaction among participants, rather than formal presentations. We anticipate several benefits of the ensuing knowledge exchange. For example, we expect that our visiting researchers will make a positive contribution to work being done by those workshop participants who are not part of the proposal. In the other direction, we hope that these participants will also make contributions that will impact on our work and that this will lead to further collaborations on the subject of the proposal. We have requested funds to cover the workshops, and resources to facilitate the collaborations that emerge.

Some of our work should also have impact in parts of the computer science community. Researchers in Bath and Manchester work on the complexity of various reachability problems in hybrid systems. They have developed a notion of Pfaffian hybrid system and have established algorithms for several problems on these systems. Our work on Pfaffian functions should lead to a wider and more robust class of systems to which their work would apply. We will meet the Manchester based researchers regularly to discuss our work. We will inform them of our progress and assist them as required in applying our results.

### Publications

Boxall G
(2015)

*Rational Values of Entire Functions of Finite Order*in International Mathematics Research Notices
Boxall G
(2015)

*Algebraic Values of Certain Analytic Functions*in International Mathematics Research Notices
Cluckers, R.
(2020)

*Uniform parameterization of subanalytic sets and diophantine applications*in Annales scientifiques de l'ENS
HABEGGER P
(2016)

*Six unlikely intersection problems in search of effectivity*in Mathematical Proceedings of the Cambridge Philosophical Society
Jones G
(2019)

*On Local definability of holomorphic functions*in The Quarterly Journal of Mathematics
Jones G
(2020)

*Effective Pila-Wilkie bounds for unrestricted Pfaffian surfaces*in Mathematische Annalen
Jones G
(2015)

*O-Minimality and Diophantine Geometry*
Jones G
(2014)

*LOCAL INTERDEFINABILITY OF WEIERSTRASS ELLIPTIC FUNCTIONS*in Journal of the Institute of Mathematics of Jussieu
Jones Go

*Rational values of Weierstrass zeta functions*in Proceedings of the Edinburgh Mathematical SocietyDescription | One objective of the project was the study of certain uniformization maps which generalize the exponential function. These uniformizations relate two different objects each of quite simple algebraic type, but the map represents a quotient by the action of an infinite discrete group action, and is not algebraic: it is transcendental. A classic theorem of Ax, about the exponential function, asserts that it is roughly speaking as transcendental as possible, avoiding interacting with algebraic subvarieties except when this is necessitated by its (well-known) functional equation: exp(x+y)=\exp(x).\exp(y). A major desideratum was to prove analogues of this result for more general maps of a similar type, and a key finding of the project was to prove such an analogue for the elliptic modular function. This result appears in a paper by Pila and Jacob Tsimerman (Toronto). A further objective was to use such results to generalize to further cases the approach of using o-minimality and point-counting to prove cases of the Zilber-Pink conjecture. A key finding in this direction is contained in a paper written Pila and Philipp Habegger (Basel), which gives, first, a result in the setting of abelian varieties which is the analogue of a well-known result of Bombieri-Masser-Zannier in the multiplicative case, and, second, a full conditional proof of Zilber-Pink in products of modular curves based on the above mentioned theorem for the elliptic modular function, though conditional on some conjectures for Galois orbits (which seem to be very difficult). Concerning the objectives around pfaffian functions, Jones and Patrick Speissegger (McMaster) have made progress, but have not yet achieved complete results. This kind of long term work is made much easier by the support of EPSRC. On the topic of improved bounds on the distribution of rational points on graphs of transcendental functions, there has been progress made by Gareth Boxall (Stellenbosch) and Jones. In particular, they proved strong bounds for various classical function such as the gamma function, various L-functions, and the Weierstrass sigma-function. The methods used were then adapted by Habegger, Jones and David Masser (Basel) to prove new effective instances of the Zilber-Pink conjecture. Habegger, Jones and Masser also used a variety of other methods, distinct from the Pila-Zannier approach, to obtain new effective instances of Zilber-Pink. Some work begun under the project has only recently been completed. In particular, Jones and Margaret Thomas (McMaster) have proved an effective form of the Pila-Wilkie result for Pfaffian surfaces. This will have applications to obtain further effective results around Zilber-Pink. |

Exploitation Route | As mentioned in our proposal the findings are of most interest to other researchers in the area. This is a very active area and our findings have been developed by several other groups. |

Sectors | Other |