Lattices and rational points (Q2413122)

Summary: 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 \(\mathbb Q\)-rank \(\leq N -1\). We also give some explicit examples. This result generalises from rank 1 to rank \(N-1\) previous results of \textit{S. Checcoli} et al. [Trans. Am. Math. Soc. 369, No. 9, 6465--6491 (2017; Zbl 1428.11124)].