[Algebraic points of bounded height on the projective line]
We consider an absolute adelic height on the set of algebraic points of the projective line , associate to an ample line bundle. We give an asymptotic formula for the number of algebraic points of fixed degree and of height lower than , when tends to infinity. The case of the standard height on has been studied by Masser and Vaaler. We generalize this result for any adelic height using a geometric point of view and one of he known cases of the Batyrev-Manin conjecture.
On considère une hauteur adélique absolue sur l’ensemble des points algébriques de la droite projective , relative à un fibré en droites ample. Nous donnons une formule asymptotique pour le nombre de points algébriques de de degré fixé et de hauteur inférieure à B, lorsque tend vers l’infini. Le cas où la hauteur considérée est la hauteur absolue usuelle a été traité par Masser et Vaaler. Nous généralisons ce résultat pour les hauteurs adéliques quelconques, en adoptant un point de vue géométrique faisant appel à l’un des résultats connus de la conjecture de Batyrev et Manin.
Accepted:
Published online:
DOI: 10.5802/jtnb.888
Author's affiliations:
@article{JTNB_2014__26_3_789_0,
author = {C\'ecile Le Rudulier},
title = {Points alg\'ebriques de hauteur born\'ee sur la droite projective},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {789--812},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {26},
number = {3},
year = {2014},
doi = {10.5802/jtnb.888},
zbl = {1338.14028},
mrnumber = {3320501},
language = {fr},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.888/}
}
TY - JOUR TI - Points algébriques de hauteur bornée sur la droite projective JO - Journal de Théorie des Nombres de Bordeaux PY - 2014 DA - 2014/// SP - 789 EP - 812 VL - 26 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.888/ UR - https://zbmath.org/?q=an%3A1338.14028 UR - https://www.ams.org/mathscinet-getitem?mr=3320501 UR - https://doi.org/10.5802/jtnb.888 DO - 10.5802/jtnb.888 LA - fr ID - JTNB_2014__26_3_789_0 ER -
Cécile Le Rudulier. Points algébriques de hauteur bornée sur la droite projective. Journal de Théorie des Nombres de Bordeaux, Volume 26 (2014) no. 3, pp. 789-812. doi : 10.5802/jtnb.888. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.888/
[1] E. Bombieri, W. Gubler, Heights in Diophantine geometry, New Mathematical Monographs, 4, Cambridge University Press, Cambridge, (2006). | MR: 2216774 | Zbl: 1115.11034
[2] V.V. Batyrev, Y. I. Manin, Sur le nombre des points rationnels de hauteur borné des variétés algébriques, Math. Ann., 286, (1990), 27–43. | MR: 1032922 | Zbl: 0679.14008
[3] F. Barroero, Counting algebraic integers of fixed degree and bounded height, arXiv :1305.0482 (2013). | MR: 3249885
[4] A. Chambert-Loir, Y. Tschinkel, Integral points of bounded height on partial equivariant compactifications of vector groups, Duke Math. J., 161, (2012), 2799–2836. | MR: 2999313
[5] S. J. Chern, J. Vaaler, The distribution of values of Mahler’s measure, J. reine angew. Math., 540 (2001), 1–47. | MR: 1868596 | Zbl: 0986.11017
[6] C. Christensen, W. Gubler, Der relative Satz von Schanuel, Manuscripta Math., 126, (2008), 505–525. | MR: 2425438 | Zbl: 1155.11034
[7] I. Dolgachev, Lectures on invariant theory, Cambridge University Press, (2003). | MR: 2004511 | Zbl: 1023.13006
[8] X. Gao, On Northcott’s theorem, Thesis (Ph.D.)–University of Colorado, (1995). | MR: 2693933
[9] D. Masser, J. Vaaler, Counting algebraic numbers with large height. II, Trans. Amer. Math. Soc., 359, (2007), 427–445. | MR: 2247898 | Zbl: 1215.11100
[10] L. Moret-Baily, Un théorème de l’application ouverte sur les corps valués algébriquement clos, Math. Scand., 111, (2012), 161–168. | MR: 3023520 | Zbl: 1276.14004
[11] D. G. Northoctt, An inequality in the theory of arithmetic on algebraic varieties, Proc. Cambridge Philos. Soc., 45, (1949), 502–509 et 510–518. | MR: 33094 | Zbl: 0035.30701
[12] E. Peyre, Hauteurs et mesures de Tamagawa sur les variétés de Fano, Duke Math. J., 79, (1995), 101–218. | MR: 1340296 | Zbl: 0901.14025
[13] R. Rumely, C. Lau, R. Varley, Existence of the sectional capacity, Mem. Amer. Math. Soc., 145, (2000). | MR: 1677934 | Zbl: 0987.14018
[14] S. H. Schanuel, Heights in number fields, Bull. Soc. Math. France, 4, (1979), 433–449. | Numdam | MR: 557080 | Zbl: 0428.12009
[15] W. M. Schmidt, Northcott’s theorem on heights. II. The quadratic case, Acta Arith., 70, (1995), 343–375. | MR: 1330740 | Zbl: 0784.11055
[16] J. P. Serre, Lectures on the Mordell-Weil theorem, Aspects of Mathematics, Third edition (1997). | MR: 1757192 | Zbl: 0863.14013
[17] M. Widmer, Counting points of fixed degree and bounded height, Acta Arith., 140, (2009), 145–168. | MR: 2558450 | Zbl: 1252.11054
[18] M. Widmer, Counting points of fixed degree and bounded height on linear varieties, J. Number Theory, 130, (2010), 145–168. | MR: 2651154 | Zbl: 1213.11140
[19] S. Zhang, Small points and adelic metrics, J. Algebraic Geom., 4, (1995), 1763–1784. | MR: 1311351 | Zbl: 0861.14019
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