Abstract
In this article, we show how to use the first and second Minkowski Theorems and some Diophantine geometry to bound explicitly the height of the points of rank N - 1 on transverse curves in E N , where E is an elliptic curve without Complex Multiplication (CM). We then apply our result to give a method for finding the rational points on such curves, when E has Q -rank ≤ N - 1 . We also give some explicit examples. This result generalises from rank 1 to rank N - 1 previous results of S. Checcoli, F. Veneziano and the author.
Highlights
A classical question in the context of Diophantine geometry is to determine the points of a certain shape, for instance the rational points, on an algebraic curve
A number field and V a variety defined over k, we denote by V (k ) the set of k-rational points on V
Veneziano, we prove that the points of rank one on a weak-transverse curve of E N have bounded height and we explicitly bound their height if E is non Complex Multiplication (CM)
Summary
A classical question in the context of Diophantine geometry is to determine the points of a certain shape, for instance the rational points, on an algebraic curve. A curve of genus at least 2 defined over a number field k has only finitely many k-rational points This is a very deep result, first conjectured by Mordell in [1] and known as Faltings Theorem after the ground-breaking proof in [2]. Kulesz, Matera and Schost [14] for some families of algebraic curves of genus 2 with Jacobian isogenous to a product of special elliptic curves of Q-rank one These methods do not give an explicit dependence of the height of the k-rational points neither in terms of the curve nor in terms of the ambient variety. These are just examples and many others can be created using the same ideas
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