# Moduli Spaces and Rational Points

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

One of the central questions in number theory is the solution of diophantine equation: to determine the set of all rational solutions of a system of polynomial equations with rational coefficients. The name diophantine is derived from name of Diophantus of Alexandria, of the third century AD, whose influential books "Arithmetica" shaped the development of number theory. From the point of view of algebraic geometry, the equations that Diophantus studied mostly define curves and his goal was to determine the set of integral (or rational) solutions.

By virtue of many celebrated results, the case of rational points on curves is theoretically well-understood, and the motto "Geometry Determines Arithmetic" is fully justified. Indeed, an algebraic curve has a unique discrete invariant, its genus, taking non-negative integral values. If a curve has genus zero, then the question of determining whether it has rational points or not is completely algorithmic, and the set of all its rational points can be efficiently determined. If a curve has genus one, then the question of determining whether it has a rational point or not is typically feasible in concrete cases. There is a procedure to decide whether a curve of genus one has a point, but, if the curve does not have points, it is not known whether this procedure necessarily terminates. The finiteness of this procedure essentially relies on the finiteness of the Tate-Shafarevich groups. If a curve of genus one admits a point, then the set of its rational points can be endowed with a very natural structure of an abelian group. This group is finitely generated over number fields by the Mordell-Weil Theorem; explicit generators can again be found subject essentially to the Birch--Swinnerton-Dyer Conjecture. Finally, curves of genus at least two only have finitely many rational points, by Faltings' celebrated proof of the Mordell Conjecture.

The situation is entirely different in higher dimensions. Bombieri and Lang formulated a conjecture implying that the distribution of rational points on varieties shares many similarities with the case of curves.

Conjecture (Bombieri-Lang). The set of rational points of a smooth projective variety of general type over a number field is not Zariski dense.

While this conjecture is very appealing, already in the case of surfaces, there is very little supporting evidence for it.

The overall goal of this project is to study algebraic surfaces, mostly of general type, of special arithmetic interested, with the aim of gathering evidence for the Bombieri-Lang Conjecture. For this purpose we will compute the Picard groups and automorphism groups of various surfaces. We will use this information to look for curves of genus at most one on the surfaces, and determine the rational points on such curves. All this data will provide clues on possible modular interpretations of the surfaces: we will try to establish the modularity of these surfaces, trying first among moduli spaces of Abelian varieties and moduli spaces of vector bundles.

By virtue of many celebrated results, the case of rational points on curves is theoretically well-understood, and the motto "Geometry Determines Arithmetic" is fully justified. Indeed, an algebraic curve has a unique discrete invariant, its genus, taking non-negative integral values. If a curve has genus zero, then the question of determining whether it has rational points or not is completely algorithmic, and the set of all its rational points can be efficiently determined. If a curve has genus one, then the question of determining whether it has a rational point or not is typically feasible in concrete cases. There is a procedure to decide whether a curve of genus one has a point, but, if the curve does not have points, it is not known whether this procedure necessarily terminates. The finiteness of this procedure essentially relies on the finiteness of the Tate-Shafarevich groups. If a curve of genus one admits a point, then the set of its rational points can be endowed with a very natural structure of an abelian group. This group is finitely generated over number fields by the Mordell-Weil Theorem; explicit generators can again be found subject essentially to the Birch--Swinnerton-Dyer Conjecture. Finally, curves of genus at least two only have finitely many rational points, by Faltings' celebrated proof of the Mordell Conjecture.

The situation is entirely different in higher dimensions. Bombieri and Lang formulated a conjecture implying that the distribution of rational points on varieties shares many similarities with the case of curves.

Conjecture (Bombieri-Lang). The set of rational points of a smooth projective variety of general type over a number field is not Zariski dense.

While this conjecture is very appealing, already in the case of surfaces, there is very little supporting evidence for it.

The overall goal of this project is to study algebraic surfaces, mostly of general type, of special arithmetic interested, with the aim of gathering evidence for the Bombieri-Lang Conjecture. For this purpose we will compute the Picard groups and automorphism groups of various surfaces. We will use this information to look for curves of genus at most one on the surfaces, and determine the rational points on such curves. All this data will provide clues on possible modular interpretations of the surfaces: we will try to establish the modularity of these surfaces, trying first among moduli spaces of Abelian varieties and moduli spaces of vector bundles.

### Planned Impact

As a result of the proposed research on the various objectives outlined above there will be several research papers that will be published in relevant research journals. Preprint versions of the articles will be made available via the arXiv. Also, the PI will maintain an up-to-date account of the progress on his webpage, by also posting there the relevant preprints and publications.

At the same time, throughout the duration of the project, the PI will give seminar talks at other universities, at international conferences, at specialized workshops, and will use these opportunities to disseminate the results of his research. For instance, the PI intends to participate in the yearly conference ALGA at IMPA, Rio de Janeiro in 2014, as well as the workshop "Rational Points" in Bayreuth, expected for 2015.

I will also devote some of my time with the Warwick in Africa program. There are two combined programs involving organising discussions and giving lectures to either students or teachers in African schools. This involves usually some preparatory meetings and then approximately 2-3 weeks of summer teaching in Africa.

Finally, the workshop planned for July 2015 near the end of the project will also serve the purpose of circulating the ideas and results obtained while working on the objectives mentioned above.

At the same time, throughout the duration of the project, the PI will give seminar talks at other universities, at international conferences, at specialized workshops, and will use these opportunities to disseminate the results of his research. For instance, the PI intends to participate in the yearly conference ALGA at IMPA, Rio de Janeiro in 2014, as well as the workshop "Rational Points" in Bayreuth, expected for 2015.

I will also devote some of my time with the Warwick in Africa program. There are two combined programs involving organising discussions and giving lectures to either students or teachers in African schools. This involves usually some preparatory meetings and then approximately 2-3 weeks of summer teaching in Africa.

Finally, the workshop planned for July 2015 near the end of the project will also serve the purpose of circulating the ideas and results obtained while working on the objectives mentioned above.

## People |
## ORCID iD |

Damiano Testa (Principal Investigator) |

### Publications

Artebani M
(2014)

*On Büchi's K3 surface*in Mathematische Zeitschrift
Flynn E
(2014)

*Finite Weil restriction of curves*in Monatshefte für Mathematik
Pacini M
(2015)

*Plane quartics with at least 8 hyperinflection points*in Bulletin of the Brazilian Mathematical Society, New SeriesDescription | The area of pure mathematics under the scope of the grant is influenced by two different subjects. On the one hand, there is algebraic geometry, dealing mostly with shapes and intrinsic structures: for instance, understanding which types of curves can be found on a surface. On the other hand, there is number theory, dealing mostly with symmetries of geometrical shapes and their fixed points. In several instances, we were able to successfully combine a deep geometric understanding of shapes that were relevant to arithmetic questions to fully analyse all the special curves. This lead to important steps towards the solution of several old problems. |

Exploitation Route | Our findings have already been used by others. There are research papers that have been written building on top of our results (e.g. http://arxiv.org/abs/1408.0269). The authors are a professor at a Dutch university and one of his a postgraduate students pursuing the findings of the funded research. On a larger scale, the results of our research open the way to a systematic study of special curves on surfaces that will be developed further in our own future work, as well as that of our collaborators and colleagues. |

Sectors | Digital/Communication/Information Technologies (including Software),Security and Diplomacy |

URL | http://homepages.warwick.ac.uk/~maskal/MSRP/ |

Description | Research Support Fund Committee |

Amount | £997 (GBP) |

Organisation | Edinburgh Mathematical Society |

Sector | Academic/University |

Country | United Kingdom |

Start | 06/2014 |

End | 06/2014 |

Description | Research Workshop Grant |

Amount | £1,893 (GBP) |

Funding ID | Ws-1314-05 |

Organisation | London Mathematical Society |

Sector | Academic/University |

Country | United Kingdom |

Start | 07/2014 |

End | 07/2014 |

Description | Magic squares of squares |

Organisation | Rice University |

Department | Department of Mathematics |

Country | United States |

Sector | Academic/University |

PI Contribution | My collaborator and I worked on an arithmetic problem: the existence of magic squares of squares. |

Collaborator Contribution | My collaborator and I worked on an arithmetic problem: the existence of magic squares of squares. |

Impact | We are in the process of writing up a paper based on the outcomes of our collaboration. |

Start Year | 2014 |

Description | Magic squares of squares |

Organisation | Rice University |

Department | Department of Mathematics |

Country | United States |

Sector | Academic/University |

PI Contribution | My collaborator and I worked on an arithmetic problem: the existence of magic squares of squares. |

Collaborator Contribution | My collaborator and I worked on an arithmetic problem: the existence of magic squares of squares. |

Impact | We are in the process of writing up a paper based on the outcomes of our collaboration. |

Start Year | 2014 |

Description | Moduli spaces of K3 surfaces and del Pezzo surfaces |

Organisation | Rice University |

Department | Department of Mathematics |

Country | United States |

Sector | Academic/University |

PI Contribution | My collaborators and I worked on extending known relations between moduli spaces of K3 surfaces of small degree and del Pezzo surfaces. |

Collaborator Contribution | My collaborators and I worked on extending known relations between moduli spaces of K3 surfaces of small degree and del Pezzo surfaces. |

Impact | We are in the process of writing up a paper based on our findings. |

Start Year | 2014 |

Description | Moduli spaces of K3 surfaces and del Pezzo surfaces |

Organisation | University of the Andes, Chile |

Country | Chile |

Sector | Academic/University |

PI Contribution | My collaborators and I worked on extending known relations between moduli spaces of K3 surfaces of small degree and del Pezzo surfaces. |

Collaborator Contribution | |

Impact | We are in the process of writing up a paper based on our findings. |

Start Year | 2014 |