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Logarithmes des points rationnels des variétés abéliennes

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

Published online by Cambridge University Press:  09 January 2019

Vincent Bosser  and
Éric Gaudron
Vincent Bosser
Affiliation:
Laboratoire de mathématiques Nicolas Oresme, Université Caen Normandie, Boulevard Maréchal Juin, 14032 Caen, France Email: vincent.bosser@unicaen.fr
Éric Gaudron
Affiliation:
Université Clermont Auvergne, CNRS, LMBP, F-63000, Clermont-Ferrand, France Email: eric.gaudron@uca.fr
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Abstract

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Keywords

théorème des périodesvariété abéliennelogarithmeméthode de Gel’fond-Bakerthéorie des penteshauteur de Néron-Tatelemme d’interpolation

MSC classification

Primary: 11J86: Linear forms in logarithms; Baker's method
Secondary: 11J95: Results involving abelian varieties 11G10: Abelian varieties of dimension $> 1$ 11G50: Heights 14G40: Arithmetic varieties and schemes; Arakelov theory; heights
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 2 , April 2019 , pp. 247 - 298
Copyright
© Société mathématique du Canada 2018 

Footnotes

Les auteurs ont bénéficié du soutien du projet ANR Gardio 14-CE25-0015.

References

Références

Autissier, P., Un lemme matriciel effectif . Math. Z. 273(2013), 355–361. https://doi.org/10.1007/s00209-012-1008-x.Google Scholar
Baker, A. and Wüstholz, G., Logarithmic forms and group varieties . J. Reine Angew. Math. 442(1993), 19–62. https://doi.org/10.1515/crll.1993.442.19.Google Scholar
Banaszczyk, W., New bounds in some transference theorems in the geometry of numbers . Math. Ann. 296(1993), 625–635. https://doi.org/10.1007/BF01445125.Google Scholar
Bertrand, D., Minimal heights and polarizations on group varieties . Duke Math. J. 80(1995), 223–250. https://doi.org/10.1215/S0012-7094-95-08009-0.Google Scholar
Bertrand, D. and Philippon, P., Sous-groupes algébriques de groupes algébriques commutatifs . Illinois J. Math. 32(1988), 263–280. https://projecteuclid.org/euclid.ijm/1255989130.Google Scholar
Bost, J.-B., Périodes et isogénies des variétés abéliennes sur les corps de nombres (d’après D. Masser et G. Wüstholz). Astérisque (1996), no. 237, 115–161. http://www.numdam.org/item?id=SB_1994-1995_37_115_0.Google Scholar
Bost, J.-B., Intrinsic heights of stable varieties and abelian varieties . Duke Math. J. 82(1996), 21–70. https://doi.org/10.1215/S0012-7094-96-08202-2.Google Scholar
Cijsouw, P. and Waldschmidt, M., Linear forms and simultaneous approximations . Compositio Math. 34(1977), 173–197. http://www.numdam.org/item?id=CM_1977_34_2_173_0.Google Scholar
David, S., Approximation diophantienne sur les variétés abéliennes. École doctorale de Géométrie diophantienne. Rennes (France), 15–26 juin 2009.Google Scholar
Gaudron, É., Formes linéaires de logarithmes effectives sur les variétés abéliennes . Ann. Sci. École Norm. Sup. 39(2006), 699–773. https://doi.org/10.1016/j.ansens.2006.09.001.Google Scholar
Gaudron, É., Pentes des fibrés vectoriels adéliques sur un corps global . Rend. Sem. Mat. Univ. Padova 119(2008), 21–95. https://doi.org/10.4171/RSMUP/119-2.Google Scholar
Gaudron, É., Minorations simultanées de formes linéaires de logarithmes de nombres algébriques . Bull. Soc. Math. Fr. 142(2014), 1–62. https://doi.org/10.24033/bsmf.2658.Google Scholar
Gaudron, É., Lower bound for the Néron–Tate height (with V. Bosser). In: Oberwolfach Reports, 21. 2016, p. 22–24.Google Scholar
Gaudron, É. and Rémond, G., Minima, pentes et algèbre tensorielle . Israel J. Math. 195(2013), 565–591. https://doi.org/10.1007/s11856-012-0109-x.Google Scholar
Gaudron, É. and Rémond, G., Théorème des périodes et degrés minimaux disogénies . Commen. Math. Helv. 89(2014), 343–403. https://doi.org/10.4171/CMH/322.Google Scholar
Gaudron, É. and Rémond, G., Polarisations et isogénies . Duke Math. J. 163(2014), 2057–2108. https://doi.org/10.1215/00127094-2782528.Google Scholar
Gillet, H. and Soulé, C., An arithmetic Riemann-Roch theorem . Invent. Math. 110(1992), 473–543. https://doi.org/10.1007/BF01231343.Google Scholar
Masser, D., Small values of heights on families of abelian varieties. In: Diophantine approximation and transcendence theory. Lecture Notes in Math., 1290, Springer, Berlin, 1987, pp. 109–148. https://doi.org/10.1007/BFb0078706.Google Scholar
Masser, D. and Wüstholz, G., Periods and minimal abelian subvarieties . Ann. of Math. 137(1993), 407–458. https://doi.org/10.2307/2946542.Google Scholar
Mignotte, M. and Waldschmidt, M., Linear forms in two logarithms and Schneider’s method. II . Acta Arith. 53(1989), 251–287. https://doi.org/10.4064/aa-53-3-251-287.Google Scholar
Milnor, J. and Husemoller, D., Symmetric bilinear forms. volume 73 de Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 1973.Google Scholar
Nakamaye, M., Multiplicity estimates on commutative algebraic groups . J. Reine Angew. Math. 607(2007), 217–235. https://doi.org/10.1515/CRELLE.2007.049.Google Scholar
Pellarin, F., Sur la distance d’un point algébrique à l’origine dans les variétés abéliennes . J. Number Theory 88(2001), 241–262. https://doi.org/10.1006/jnth.2000.2612.Google Scholar
Philippon, P. and Waldschmidt, M., Formes linéaires de logarithmes sur les groupes algébriques commutatifs . Illinois J. Math. 32(1988), 281–314. https://projecteuclid.org/euclid.ijm/1255989131.Google Scholar
Rémond, G., Conjectures uniformes sur les variétés abéliennes . Quarterly J. Math. 69(2018), 459–486. https://doi.org/10.1093/qmath/hax042.Google Scholar
Rémond, G., Degré de définition des endomorphismes d’une variété abélienne. Prépublication 2017. https://www-fourier.ujf-grenoble.fr/∼remond/4441.pdf.Google Scholar
Silverberg, A., Fields of definition for homomorphisms of abelian varieties . J. Pure Appl. Algebra 77(1992), 253–262. https://doi.org/10.1016/0022-4049(92)90141-2.Google Scholar
Winckler, B., Problème de Lehmer sur les courbes elliptiques à multiplications complexes . Acta Arith. 182(2018), 347–396. https://doi.org/10.4064/aa170404-5-11.Google Scholar