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Chapter V: Beyond Heights: Slopes and Distribution of Rational Points

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Arakelov Geometry and Diophantine Applications

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2276))

Abstract

The distribution of rational points of bounded height on algebraic varieties is far from uniform. Indeed the points tend to accumulate on thin subsets which are images of non-trivial finite morphisms. The problem is to find a way to characterise the points in these thin subsets. The slopes introduced by Jean-Benoît Bost are a useful tool for this problem. These notes will present several cases in which this approach is fruitful. We shall also describe the notion of locally accumulating subvarieties which arises when one considers rational points of bounded height near a fixed rational point.

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Acknowledgements

The author was supported by the ANR Grant Gardio 14-CE25-0015

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  1. Institut Fourier, Université Grenoble Alpes, Grenoble, France

    Emmanuel Peyre

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  1. Emmanuel Peyre
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Correspondence to Emmanuel Peyre .

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  1. Institut Fourier, Université Grenoble Alpes, Grenoble, France

    Emmanuel Peyre

  2. Institut Fourier, Université Grenoble Alpes, Grenoble, France

    Gaël Rémond

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Peyre, E. (2021). Chapter V: Beyond Heights: Slopes and Distribution of Rational Points. In: Peyre, E., Rémond, G. (eds) Arakelov Geometry and Diophantine Applications. Lecture Notes in Mathematics, vol 2276. Springer, Cham. https://doi.org/10.1007/978-3-030-57559-5_6

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