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NOMBRE DE PETITS POINTS SUR UNE VARIÉTÉ ABÉLIENNE

Part of: Diophantine approximation, transcendental number theory Arithmetic algebraic geometry

Published online by Cambridge University Press:  18 March 2025

Éric Gaudron Open the ORCID record for Éric Gaudron [Opens in a new window]  and
Gaël Rémond
Éric Gaudron*
Affiliation:
CNRS, LMBP, Université Clermont Auvergne, F-63000 Clermont-Ferrand, France
Gaël Rémond
Affiliation:
Institut Fourier, UMR 5582 CS 40700, 38058 Grenoble Cedex 9, France (Gael.Remond@univ-grenoble-alpes.fr)
*
(Eric.Gaudron@uca.fr)
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Abstract

Given a polarised abelian variety over a number field, we provide totally explicit upper bounds for the cardinality of the rational points whose Néron-Tate height is less than a small threshold. These imply new estimates for the number of torsion points as well as the minimal height of a non-torsion point. Our bounds involve the Faltings height and dimension of the abelian variety together with the degrees of the polarisation and the number field but we also get a stronger statement where we use certain successive minima associated to the period lattice at a fixed archimedean place, in the spirit of a result of David for elliptic curves.

MSC classification

Primary: 11G10: Abelian varieties of dimension $> 1$
Secondary: 11G50: Heights 11J95: Results involving abelian varieties
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , First View , pp. 1 - 57
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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