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Transactions of the American Mathematical Society

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A statistical view on the conjecture of Lang about the canonical height on elliptic curves


Author: Pierre Le Boudec
Journal: Trans. Amer. Math. Soc. 372 (2019), 8347-8361
MSC (2010): Primary 11D45, 11G05, 11G50, 14G05
DOI: https://doi.org/10.1090/tran/7912
Published electronically: September 25, 2019
MathSciNet review: 4029699
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Abstract | References | Similar Articles | Additional Information

Abstract: We adopt a statistical point of view on the conjecture of Lang which predicts a lower bound for the canonical height of nontorsion rational points on elliptic curves defined over $ \mathbb{Q}$. More specifically, we prove that among the family of all elliptic curves defined over $ \mathbb{Q}$ and having positive rank, there is a density one subfamily of curves which satisfy a strong form of Lang's conjecture.


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Additional Information

Pierre Le Boudec
Affiliation: Departement Mathematik und Informatik, Fachbereich Mathematik, Spiegelgasse 1, 4051 Basel, Switzerland
Email: pierre.leboudec@unibas.ch

DOI: https://doi.org/10.1090/tran/7912
Keywords: Elliptic curves, rational points, canonical height
Received by editor(s): February 2, 2018
Received by editor(s) in revised form: February 22, 2019
Published electronically: September 25, 2019
Additional Notes: The research of the author was integrally funded by the Swiss National Science Foundation through SNSF Professorship number $170565$ awarded to the project “Height of rational points on algebraic varieties”. Both the financial support of the SNSF and the perfect working conditions provided by the University of Basel are gratefully acknowledged.
This work was initiated while the author was working as an Instructor at the École Polytechnique Fédérale de Lausanne. The financial support and the wonderful working conditions that the author enjoyed during the four years he worked at this institution are gratefully acknowledged.
Article copyright: © Copyright 2019 American Mathematical Society