Moduli Spaces and Rational Points

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.

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.