Autour de la conjecture de Zilber-Pink

Rémond, Gaël

Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 2, pp. 405-414.

Nous dressons un rapide panorama de résultats allant dans le sens de la conjecture suivante : l’intersection d’une sous-variété $X$ d’une variété semi-abélienne $A$ et de l’union de tous les sous-groupes algébriques de $A$ de codimension au moins $dimX+1$ n’est pas Zariski-dense dans $X$ dès que $X$ n’est pas contenue dans un sous-groupe algébrique strict de $A$.

We describe some results toward the following conjecture: if $X$ is an irreducible subvariety of a semi-abelian variety $A$, its intersection with the union of all algebraic subgroups $A$ of codimension greater than the dimension of $X$ is not Zariski-dense in $X$, unless $X$ is contained in a proper algebraic subgroup of $A$.

[AD] F. Amoroso et S. David, Distribution des points de petite hauteur dans les groupes multiplicatifs. Ann. Scuola Norm. Sup. Pisa Série V 3 (2004) 325–348. | Numdam | MR | Zbl

[Ax] J. Ax, Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups. Amer. J. Math. 94 (1972) 1195–1204. | MR | Zbl

[BMZ1] E. Bombieri, D. Masser et U. Zannier, Intersecting a curve with algebraic subgroups of multiplicative groups. Internat. Math. Res. Notices 20 (1999) 1119–1140. | MR | Zbl

[BMZ2] E. Bombieri, D. Masser et U. Zannier, Finiteness results for multiplicatively dependent points on complex curves. Michigan Math. J. 51 (2003) 451–466. | MR | Zbl

[BMZ3] E. Bombieri, D. Masser et U. Zannier, Intersecting curves and algebraic subgroups : conjectures and more results. Trans. Amer. Math. Soc. 358 (2006) 2247–2257. | MR | Zbl

[BMZ4] E. Bombieri, D. Masser et U. Zannier, Anomalous subvarieties — structure theorems and applications. Internat. Math. Res. Notices 19 (2007) 1–33. | MR | Zbl

[BMZ5] E. Bombieri, D. Masser et U. Zannier, Intersecting a plane with algebraic subgroups of multiplicative groups. Ann. Scuola Norm. Sup. Pisa Série V 7 (2008) 51–80. | Numdam | MR | Zbl

[BMZ6] E. Bombieri, D. Masser et U. Zannier, On unlikely intersections of complex varieties with tori. Acta Arith. 133 (2008), 309–323. | MR | Zbl

[C] M. Carrizosa, Problème de Lehmer et variétés abéliennes CM. C. R. Acad. Sci. 346 (2008) 1219–1224. | MR

[F1] G. Faltings, Diophantine approximation on abelian varieties. Ann. of Math. 133 (1991) 549–576. | MR | Zbl

[F2] G. Faltings, The general case of S. Lang’s conjecture. Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991). Perspect. Math. 15. Academic Press. San Diego. (1994) 175–182. | MR | Zbl

[Ha] P. Habegger, Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension. Invent. math. 176 (2009), 405–447. | MR | Zbl

[Hi] M. Hindry, Autour d’une conjecture de Serge Lang. Invent. math. 94 (1988) 575–603. | EuDML | MR | Zbl

[K] J. Kirby, The theory of exponential differential equations. thèse. Oxford. (2006).

[La] M. Laurent, Équations diophantiennes exponentielles. Invent. math. 78 (1984) 299–327. | EuDML | MR | Zbl

[McQ] M. McQuillan, Division points on semi-abelian varieties. Invent. math. 120 (1995) 143–159. | EuDML | MR | Zbl

[Mau] G. Maurin, Courbes algébriques et équations multiplicatives. Math. Ann. 341 (2008) 789–824. | MR | Zbl

[P] R. Pink, A Common Generalization of the Conjectures of André-Oort, Manin-Mumford, and Mordell-Lang. prépublication (2005).

[Ray] M. Raynaud, Sous-variétés d’une variété abélienne et points de torsion. Arithmetic and geometry, Vol. I. Progr. Math. 35. Birkhäuser. Boston. (1983) 327–352. | MR | Zbl

[R1] G. Rémond, Inégalité de Vojta généralisée. Bull. Soc. Math. France 133 (2005) 459–495. | EuDML | Numdam | MR | Zbl

[R2] G. Rémond, Intersection de sous-groupes et de sous-variétés I. Math. Ann. 333 (2005) 525–548. | MR | Zbl

[R3] G. Rémond, Intersection de sous-groupes et de sous-variétés II. J. Inst. Math. Jussieu 6 (2007) 317–348. | MR | Zbl

[R4] G. Rémond, Intersection de sous-groupes et de sous-variétés III. Comm. Math. Helv. à paraître. 21 pages. | Zbl

[V1] P. Vojta, Siegel’s theorem in the compact case. Ann. of Math. 133 (1991) 509–548. | MR | Zbl

[V2] P. Vojta, Integral points on subvarieties of semiabelian varieties, I. Invent. math. 126 (1996) 133–181. | MR | Zbl

[Z] B. Zilber, Exponential sums equations and the Schanuel conjecture. J. London Math. Soc. (2) 65 (2002) 27–44. | MR | Zbl