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- Date de réalisation : 3 Juillet 2019
- Durée du programme : 60 min
- Classification Dewey : Mathématiques
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- Catégorie : Conférences
- Niveau : niveau Doctorat (LMD), Recherche
- Disciplines : Mathématiques
- Collections : Ecoles d'été, 2019
- ficheLom : Voir la fiche LOM
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- Auteur(s) : BINYAMINI Gal
- Réalisateur(s) : Bastien Fanny, HUMPHRIES Donovan
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- Langue : Anglais
- Mots-clés : Grenoble, eem2019, Foliations and algebraic geometry, Feuilletages et géométrie algébrique, point counting, number fields
- Conditions d’utilisation / Copyright : CC BY-NC-ND 4.0
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G. Binyamini - Point counting for foliations over number fields
We consider an algebraic $V$ variety and its foliation, both defined over a number field. Given a (compact piece of a) leaf $L$ of the foliation, and a subvariety $W$ of complementary codimension, we give an upper bound for the number of intersections between $L$ and $W$. The bound depends polynomially on the degree of $W$, the logarithmic height of $W$, and the logarithmic distance between $L$ and the locus of points where leafs of the foliation intersect $W$ improperly.
Using this theory we prove the Wilkie conjecture for sets defined using leafs of foliations under a certain assumption about the algebraicity locus. For example, we prove the if none of the leafs contain algebraic curves then the number of algebraic points of degree $d$ and log-height $h$ on a (compact piece of a) leaf grows polynomially with $d$ and $h$. This statement and its generalizations have many applications in diophantine geometry following the Pila-Zannier strategy.
I will focus mostly on the proof of the main statement, which uses a combination of differential-algebraic methods related to foliations with some ideas from complex geometry and value distribution theory. If time permits I will briefly discuss the applications to counting algebraic points and diophantine geometry at the end.
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