On a conjecture of Buium and Poonen
[Sur une conjecture de Buium et Poonen]
Annales de l'Institut Fourier, Tome 70 (2020) no. 2, pp. 457-477.

Étant donnée une correspondance entre une courbe modulaire S et une courbe elliptique A, nous prouvons que l’intersection de tout sous-groupe de rang fini de A avec l’ensemble de points de A d’une classe isogénie sur S est fini. Le question a été posée par A. Buium et B. Poonen en 2009. Nous suivons la stratégie proposée par les auteurs, utilisant un résultat sur l’équidistribution des points de Hecke sur les variétés de Shimura et le théorème de l’image ouverte de Serre. Le résultat est un cas particulier de la conjecture de Zilber–Pink.

Given a correspondence between a modular curve S and an elliptic curve A, we prove that the intersection of any finite-rank subgroup of A with the set of points on A corresponding to an isogeny class on S is finite. The question was proposed by A. Buium and B. Poonen in 2009. We follow the strategy proposed by the authors, using a result about the equidistribution of Hecke points on Shimura varieties and Serre’s open image theorem. The result is an instance of the Zilber–Pink conjecture.

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DOI : https://doi.org/10.5802/aif.3317
Classification : 11G18,  11G05,  14G35
Mots clés : courbe modulaire, courbe de Shimura, classes d’isogénie, intersections atypiques et conjecture de Zilber–Pink.
@article{AIF_2020__70_2_457_0,
     author = {Baldi, Gregorio},
     title = {On a conjecture of {Buium} and {Poonen}},
     journal = {Annales de l'Institut Fourier},
     pages = {457--477},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {70},
     number = {2},
     year = {2020},
     doi = {10.5802/aif.3317},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3317/}
}
Baldi, Gregorio. On a conjecture of Buium and Poonen. Annales de l'Institut Fourier, Tome 70 (2020) no. 2, pp. 457-477. doi : 10.5802/aif.3317. https://aif.centre-mersenne.org/articles/10.5802/aif.3317/

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