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Lattices and rational points. (English) Zbl 1434.11121

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 S. Checcoli et al. [Trans. Am. Math. Soc. 369, No. 9, 6465–6491 (2017; Zbl 1428.11124)].

MSC:

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11G05 Elliptic curves over global fields
14G05 Rational points
14G40 Arithmetic varieties and schemes; Arakelov theory; heights

Citations:

Zbl 1428.11124
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References:

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