# O-minimality, Diophantine geometry, and functional transcendence

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

This project will apply techniques from mathematical logic to problems of functional transcendence and Diophantine geometry. Diophantine geometry is the study, using techniques from algebraic geometry, of the solutions to systems of polynomial equations in either the integers or the rational numbers. Typical questions studied are showing that certain systems of equations have a set of solutions which is either finite or can be described finitely in some particular way. A paradigmatic result of this kind is Faltings' proof of the Mordell conjecture, that curves of genus greater than $1$ have only finitely many rational points. This project will investigate conjectures related to this. While Mordell's conjecture has been proved, conjectures arising from it, such as the Andr\'{e}--Oort and Zilber--Pink conjectures remain open. Numerous partial results in the direction of these conjectures have already been obtained. The project would look to extend such partial results, perhaps by considering analogues of these results in different settings.

Transcendence theory investigates the algebraic nature of naturally defined numbers, particularly the values of certain classical functions such as the exponential function. The underlying conjecture in this subject is Schanuel's Conjecture, which captures the expected transcendence properties of the exponential function. Schanuel's Conjecture is that, given $z_1, \ldots, z_n \in \mathbb{C}$ linearly independent over $\mathbb{Q}$, the transcendence degree of the field $\mathbb{Q}(z_1, \ldots, z_n, e^{z_1}, \ldots, e^{z_n})$ is at least $n$. This project may investigate cognates of Schanuel's Conjecture in different settings, to see if the natural translations of the conjecture hold in such settings and what the consequences of such cognates are. An example of the kind of setting to be considered is functional transcendence theory, in which the algebraic independence of functions is studied. A typical result in this area is the Ax--Schanuel theorem, whereby Ax proved the relevant cognate of Schanuel's Conjecture holds in the setting of a differential field.

The logical techniques to be used in the project are from model theory. There are numerous connections between model theory and the problems discussed above. Many of the conjectures like Zilber--Pink have a model-theoretic provenance. Zilber's own formulation of the Zilber--Pink conjecture arose from his investigation of the model theory of complex exponentiation. Further, the study of o-minimal structures in model theory has provided useful approaches to the kinds of problems described above. The defining characteristic of o-minimal structures is that every definable set is a finite union of intervals and points. This characterisation imbues o-minimal structures with remarkable tameness properties, for example the Cell Decomposition Theorem. It is these tameness properties of o-minimal structures which are required in addressing problems in Diophantine geometry and functional transcendence. Recent applications of o-minimality to investigating such problems have been carried out by Jonathan Pila, Umberto Zannier, and others. This project will look to build on these results.

The project falls within the EPSRC research areas Logic & Combinatorics and Number Theory.

Transcendence theory investigates the algebraic nature of naturally defined numbers, particularly the values of certain classical functions such as the exponential function. The underlying conjecture in this subject is Schanuel's Conjecture, which captures the expected transcendence properties of the exponential function. Schanuel's Conjecture is that, given $z_1, \ldots, z_n \in \mathbb{C}$ linearly independent over $\mathbb{Q}$, the transcendence degree of the field $\mathbb{Q}(z_1, \ldots, z_n, e^{z_1}, \ldots, e^{z_n})$ is at least $n$. This project may investigate cognates of Schanuel's Conjecture in different settings, to see if the natural translations of the conjecture hold in such settings and what the consequences of such cognates are. An example of the kind of setting to be considered is functional transcendence theory, in which the algebraic independence of functions is studied. A typical result in this area is the Ax--Schanuel theorem, whereby Ax proved the relevant cognate of Schanuel's Conjecture holds in the setting of a differential field.

The logical techniques to be used in the project are from model theory. There are numerous connections between model theory and the problems discussed above. Many of the conjectures like Zilber--Pink have a model-theoretic provenance. Zilber's own formulation of the Zilber--Pink conjecture arose from his investigation of the model theory of complex exponentiation. Further, the study of o-minimal structures in model theory has provided useful approaches to the kinds of problems described above. The defining characteristic of o-minimal structures is that every definable set is a finite union of intervals and points. This characterisation imbues o-minimal structures with remarkable tameness properties, for example the Cell Decomposition Theorem. It is these tameness properties of o-minimal structures which are required in addressing problems in Diophantine geometry and functional transcendence. Recent applications of o-minimality to investigating such problems have been carried out by Jonathan Pila, Umberto Zannier, and others. This project will look to build on these results.

The project falls within the EPSRC research areas Logic & Combinatorics and Number Theory.

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/N509711/1 | 01/10/2016 | 30/09/2021 | |||

1941915 | Studentship | EP/N509711/1 | 01/10/2017 | 31/03/2021 | Guy Fowler |

Description | Proved some novel functional independence statements for modular functions. Used such results to establish some new results on multiplicative properties of the images of CM points under these functions. Proved some new effective versions of these results in low dimensional cases. |

Exploitation Route | I am currently in the process of writing up the results obtained so far, in order that they be disemminated to other researchers in the field. I have also spoken about them at conferences in the UK, and will speak about them at other conferences in the UK and Europe. |

Sectors | Other |