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NOMBRE DE PETITS POINTS SUR UNE VARIÉTÉ ABÉLIENNE

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

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 ([email protected])
*
([email protected])
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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 , Volume 24 , Issue 3 , May 2025 , pp. 705 - 761
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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References

Références

Anderson, M et Masser, D (1980) Lower bounds for heights on elliptic curves. Math. Z. 174, 23–34.CrossRefGoogle Scholar
Ballaÿ, F (2021) Successive minima and asymptotic slopes in Arakelov geometry. Compositio Math. 157, 1302–1339.CrossRefGoogle Scholar
Bantegnie, R (1970) Sur une propriété de transfert concernant les boules de ${\mathbb{R}}^n$ . Monatsh. Math. 74, 1–5.CrossRefGoogle Scholar
Blichfeldt, HF (1914) A new principle in the geometry of numbers, with some applications. Trans. Amer. Math. Soc. 15, 227–235.CrossRefGoogle Scholar
Bosser, V et Gaudron, É (2019) Logarithmes des points rationnels des variétés abéliennes. Canadian J. Math. 71, 247–298.CrossRefGoogle Scholar
Bost, JB (1996) Intrinsic heights of stable varieties and abelian varieties. Duke Math. J. 82, 21–70.CrossRefGoogle Scholar
Carrizosa, M (2009) Petits points et multiplication complexe. Int. Math. Res. Not. IMRN 16, 3016–3097.Google Scholar
Clark, P et Pollack, P (2015) The truth about torsion in the CM case. C. R. Math. Acad. Sci. Paris 353, 683–688.CrossRefGoogle Scholar
Cohn, H et Elkies, N (2003) New upper bounds on sphere packings I. Ann. Math. 157, 689–714.CrossRefGoogle Scholar
David, S (1991) Fonctions thêta et points de torsion des variétés abéliennes. Compositio Math. 78, 121–160.Google Scholar
David, S (1993) Minorations de hauteurs sur les variétés abéliennes. Bull. Soc. Math. France 121, 509–544.CrossRefGoogle Scholar
David, S (1997) Points de petite hauteur sur les courbes elliptiques. J. Number Theory 64, 104–129.CrossRefGoogle Scholar
Gaudron, É (2014) Minorations simultanées de formes linéaires de logarithmes de nombres algébriques. Bull. Soc. Math. France 142, 1–62.CrossRefGoogle Scholar
Gaudron, É (2019) Some explicit computations in Arakelov geometry of abelian varieties. J. Ramanujan Math. Soc. 34, 433–447.Google Scholar
Gaudron, É (2021) Minima and slopes of rigid adelic spaces. In Arakelov Geometry and Diophantine Applications, édité par Peyre, E. et Rémond., G. Lecture Notes in Math. 2276. Cham : Springer, 37–76.Google Scholar
Gaudron, É et Rémond, G (2014) Théorème des périodes et degrés minimaux d’isogénies. Comment. Math. Helv. 89, 343–403.CrossRefGoogle Scholar
Gaudron, É et Rémond, G (2014) Polarisations et isogénies. Duke Math. J. 163, 2057–2108.CrossRefGoogle Scholar
Gaudron, É et Rémond, G (2018) Torsion des variétés abéliennes CM. Proc. Amer. Math. Soc. 146, 2741–2747.CrossRefGoogle Scholar
Gaudron, É et Rémond, G (2023) Nouveaux théorèmes d’isogénie. Mém. Soc. Math. France 176, vi+123 p.Google Scholar
Lang, S (1975) Division points of elliptic curves and Abelian functions over number fields. Amer. J. Math. 97, 124–132.CrossRefGoogle Scholar
Masser, D (1977) Division fields of elliptic functions. Bull. London Math. Soc. 9, 49–53.CrossRefGoogle Scholar
Masser, D (1981) Small values of the quadratic part of the Néron-Tate height. Sémin. Delange-Pisot-Poitou, Paris 1979–80 , Prog. Math. 12, 213–222.Google Scholar
Masser, D (1984) Small values of the quadratic part of the Néron-Tate height on an abelian variety. Compositio Math. 53, 153–170.Google Scholar
Masser, D (1987) Small values of heights on families of abelian varieties. In Diophantine Approximation and Transcendence Theory (Springer, Bonn, 1985) , Lecture Notes in Math. 1290. 109–148.CrossRefGoogle Scholar
Masser, D (1986) Lettre à Daniel Bertrand. 17 novembre 1986. 6 pages.CrossRefGoogle Scholar
Masser, D (1989) Counting points of small height on elliptic curves. Bull. Soc. Math. France 117, 247–265.CrossRefGoogle Scholar
Masser, D (1993) Large period matrices and a conjecture of Lang. Séminaire de Théorie des Nombres, Paris 1991–92 , Progr. Math. 116, 153–177.CrossRefGoogle Scholar
Merel, L (1996) Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124, 437–449.CrossRefGoogle Scholar
Moret-Bailly, L (1985) Pinceaux de variétés abéliennes. Astérisque 129.Google Scholar
Nakamaye, M (2007) Multiplicity estimates on commutative algebraic groups. J. reine angew. Math. 607, 217–235.Google Scholar
Parent, P (1999) Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres. J. reine angew. Math. 506, 85–116.CrossRefGoogle Scholar
Rémond, G (2018) Conjectures uniformes sur les variétés abéliennes. Quart. J. Math. 69, 459–486.CrossRefGoogle Scholar
Rémond, G (2022) Propriétés de la hauteur de Faltings. Ann. Scuola Norm. Sup. Pisa 23, 1589–1596.Google Scholar
Serre, JP (2013) Résumé des cours au Collège de France 1985–1986. In Œuvres $/$ Collected papers. IV. 1985–1998. Springer, 33–37.Google Scholar
Silverberg, A (1988) Torsion points on abelian varieties of CM-type. Compositio Math. 68, 241–249.Google Scholar