O-minimality and  certain atypical intersections

Habegger, Philipp; Pila, Jonathan

doi:10.24033/asens.2296

O-minimality and certain atypical intersections
[o-minimalité et certaines intersections atypiques]

Habegger, Philipp ; Pila, Jonathan

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 4, pp. 813-858.

Résumé
Abstract

On démontre que la stratégie de comptage dans des structures o-minimales est suffisante pour traiter plusieurs problèmes qui vont au-delà des conjectures de Manin-Mumford et André-Oort. On vérifie la conjecture de Zilber-Pink pour un produit de courbes modulaires en supposant une minoration assez forte pour la taille de l'orbite de Galois et en supposant une version modulaire du théorème de Ax-Schanuel. Dans le cas des variétés abéliennes, on démontre la conjecture de Zilber-Pink pour les courbes si tous les objets sont définis sur un corps de nombres. Pour les sous-variétés de dimension supérieure, on obtient quelques résultats plus faibles et quelques résultats conditionnels.

We show that the strategy of point counting in o-minimal structures can be applied to various problems on unlikely intersections that go beyond the conjectures of Manin-Mumford and André-Oort. We verify the so-called Zilber-Pink Conjecture in a product of modular curves on assuming a lower bound for Galois orbits and a sufficiently strong modular Ax-Schanuel Conjecture. In the context of abelian varieties we obtain the Zilber-Pink Conjecture for curves unconditionally when everything is defined over a number field. For higher dimensional subvarieties of abelian varieties we obtain some weaker results and some conditional results.

Publié le : 2016-11-12

MR Zbl

DOI : 10.24033/asens.2296

Classification : 11G10, 11G18; 03C64, 11G50, 14K12, 14G35, 14G40.
Keywords: Zilber-Pink conjecture, unlikely intersections, o-minimality.
Mot clés : Conjecture de Zilber-Pink, intersections exceptionnelles, o-minimalité.

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     title = {O-minimality and  certain atypical intersections},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {813--858},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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     mrnumber = {3552014},
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Habegger, Philipp; Pila, Jonathan. O-minimality and  certain atypical intersections. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 4, pp. 813-858. doi : 10.24033/asens.2296. http://www.numdam.org/articles/10.24033/asens.2296/

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