Uniformity of stably integral points on elliptic

Abstract

0. Introduction Let X be a variety of logarithmic general type, deened over a number eld K. Let S be a nite set of places in K and let O K;S be the ring of S-integers. Suppose that X is a model of X over Spec O K;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(O K;S) is not Zariski dense in X. In case X is projective, one may chose an arbitrary projective model X and then X(O K;S) is identiied with X(K). In such a case, one often refers to this conjecture of Lang and Vojta as just Lang's conjecture. apply Lang's conjecture in the following way: Let X ! B be a smooth family of curves of genus g > 1. Let X n B ! B be the n-th bered power of X over B. In CHM] it is shown that for high enough n, the variety X n 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 xed number eld is uniformly bounded. In this note we study an analogous implication for elliptic curves. Let E=K be an elliptic curve over a number eld, and let P 2 E(K). We say that P is stably S-integral, denoted P 2 E(K; S), if P is S-integral after semistable reduction (see x4). Our main theorem states (see x5): Theorem 1. (Main theorem in terms of points) Assume that the Lang-Vojta conjecture holds. Then for any number eld K and a nite set of places S, there is an integer N such that for any elliptic curve E=K we have #E(K; S) < N. Since the moduli space of elliptic curves is only one-dimensional, the computations and the proofs are a bit simpler than the higher genus cases. One can view the results in this paper as a simple application of the methods of CHM]. 0.1. Overview. In section 1 we prove a basic lemma analogous to lemma 1.1 CHM] on uniformity of correlated points. In section 2 we study a particular pencil of elliptic curves which is the main building block for proving theorem 1. In section 3 we look at quadratic twists of an elliptic …

Cite this paper

@inproceedings{Abramovich1995UniformityOS, title={Uniformity of stably integral points on elliptic}, author={Dan Abramovich}, year={1995} }