Geodesic
O-minimality and certain atypical intersections
[o-minimalité et certaines intersections atypiques]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 4, pp. 813-858.

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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.

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.

Publié le :
DOI : 10.24033/asens.2296
Classification : 11G10, 11G18; 03C64, 11G50, 14K12, 14G35, 14G40.
Keywords: Zilber-Pink conjecture, unlikely intersections, o-minimality.
Mots-clés : Conjecture de Zilber-Pink, intersections exceptionnelles, o-minimalité. 
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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://geodesic.mathdoc.fr/articles/10.24033/asens.2296/

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