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Dependence relations and exceptional intersections. (Relations de dépendance et intersections exceptionnelles.) (French) Zbl 1273.14068

Séminaire Bourbaki. Volume 2010/2011. Exposés 1027–1042. Avec table par noms d’auteurs de 1948/49 à 2009/10. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-351-5/pbk). Astérisque 348, 149-188 (2012).
In this survey paper, the author outlines the state of art of the problem of the so-called “unlikely intersections” in Diophantine geometry.
This problem arises from a number of conjectures, made more and more precise with time (Manin-Mumford, Mordell-Lang, André-Oort, Zilber-Pink), all related with the problem of classifying the situations where an unusual number of points with some diophantine property lies in the intersection of two algebraic varieties (e.g, rational points, or, in the case of algebraic groups, torsion points and, more generally, subgroups of finite rank).
After presenting the problem in general, the survey presents some of the major achievements in this context, essentially the theorems contained in the following papers:
[E. Bombieri, D. Masser and U. Zannier, Int. Math. Res. Not. 20, 1119–1140 (1999; Zbl 0938.11031)];
[G. Maurin, Math. Ann. 341, No. 4, 789–824 (2008; Zbl 1154.14017)];
[E. Bombieri, P. Habegger, D. Masser and U. Zannier, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 21, No. 3, 251–260 (2010; Zbl 1209.11057)];
[P. Habegger, Bull. Soc. Math. Fr. 137, No. 1, 93–125 (2009; Zbl 1270.11063)];
[P. Habegger, Int. Math. Res. Not. 2009, No. 5, 860–886 (2009; Zbl 1239.11070)];
[G. Rémond, J. Théor. Nombres Bordx. 21, No. 2, 405–414 (2009; Zbl 1196.11083)];
[G. Maurin, Int. Math. Res. Not. 2011, No. 23, 5259–5366 (2011; Zbl 1239.14020)].
For the entire collection see [Zbl 1257.00012].

MSC:

14H52 Elliptic curves
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
11G10 Abelian varieties of dimension \(> 1\)
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