Uniformity of Stably Integral Points on Elliptic Curves

Let X be a variety of logarithmic general type, defined over a number field K. Let S be a finite set of places in K and let OK,S be the ring of S-integers. Suppose that X is a model of X over Spec OK,S . As a natural generalizasion of theorems of Siegel and Faltings, It was conjectured by S. Lang and P. Vojta ([Vojta], conjecture 4.4) that the set of S-integral points X (OK,S) is not Zariski dense in X . In case X is projective, one may chose an arbitrary projective model X and then X (OK,S) is identified with X(K). In such a case, one often refers to this conjecture of Lang and Vojta as just Lang’s conjecture. L. Caporaso, J. Harris and B. Mazur [CHM] apply Lang’s conjecture in the following way: Let X → B be a smooth family of curves of genus g > 1. Let X B → B be the n-th fibered power of X over B. In [CHM] it is shown that for high enough n, the variety X B dominates a variety of general type. Assuming Lang’s conjecture, they deduce the following remarkable result: the number of rational points on a curve of genus g over a fixed number field is uniformly bounded.