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/
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