Amos Turchet: Uniformity in Diophantine Geometry

The seminal paper of Caporaso, Harris and Mazur proved that the Lang Conjecture, i.e., rational points in general type varieties are not Zariski dense, has a striking implication on the distribution of rational points on curves; they proved that given a number field K and an integer g, assuming the Lang Conjecture, there exists a constant N, depending only on K and g, such that every curve of genus g defined over K has at most N rational points. This result has been extended to higher dimensional varieties by work of Hassett, Abramovich and Voloch.

In this talk we will describe the ideas behind their proof and present new results about the distribution of integral points obtained in joint work with Kenneth Ascher (Brown University). The techniques used come from the theory of moduli space of curves, or more general of general type varieties and stable pairs à la Kollár.