Autour de la conjecture de Zilber-Pink

Gaël Rémond[1]

  • [1] Institut Fourier, UMR 5582 BP 74 38402 Saint-Martin-d’Hères Cedex, France

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 2, page 405-414
  • ISSN: 1246-7405

Abstract

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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 .

How to cite

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Rémond, Gaël. "Autour de la conjecture de Zilber-Pink." Journal de Théorie des Nombres de Bordeaux 21.2 (2009): 405-414. <http://eudml.org/doc/10887>.

@article{Rémond2009,
abstract = {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 $\dim X+1$ n’est pas Zariski-dense dans $X$ dès que $X$ n’est pas contenue dans un sous-groupe algébrique strict de $A$.},
affiliation = {Institut Fourier, UMR 5582 BP 74 38402 Saint-Martin-d’Hères Cedex, France},
author = {Rémond, Gaël},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Zilber-Pink conjecture; semi-abelian variety; diophantine approximation},
language = {fre},
number = {2},
pages = {405-414},
publisher = {Université Bordeaux 1},
title = {Autour de la conjecture de Zilber-Pink},
url = {http://eudml.org/doc/10887},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Rémond, Gaël
TI - Autour de la conjecture de Zilber-Pink
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 2
SP - 405
EP - 414
AB - 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 $\dim X+1$ n’est pas Zariski-dense dans $X$ dès que $X$ n’est pas contenue dans un sous-groupe algébrique strict de $A$.
LA - fre
KW - Zilber-Pink conjecture; semi-abelian variety; diophantine approximation
UR - http://eudml.org/doc/10887
ER -

References

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  1. 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. Zbl1150.11021MR2075986
  2. J. Ax, Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups. Amer. J. Math. 94 (1972) 1195–1204. Zbl0258.14014MR435088
  3. E. Bombieri, D. Masser et U. Zannier, Intersecting a curve with algebraic subgroups of multiplicative groups. Internat. Math. Res. Notices 20 (1999) 1119–1140. Zbl0938.11031MR1728021
  4. E. Bombieri, D. Masser et U. Zannier, Finiteness results for multiplicatively dependent points on complex curves. Michigan Math. J. 51 (2003) 451–466. Zbl1048.11056MR2021000
  5. E. Bombieri, D. Masser et U. Zannier, Intersecting curves and algebraic subgroups : conjectures and more results. Trans. Amer. Math. Soc. 358 (2006) 2247–2257. Zbl1161.11025MR2197442
  6. E. Bombieri, D. Masser et U. Zannier, Anomalous subvarieties — structure theorems and applications. Internat. Math. Res. Notices 19 (2007) 1–33. Zbl1145.11049MR2359537
  7. 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. Zbl1150.11022MR2413672
  8. E. Bombieri, D. Masser et U. Zannier, On unlikely intersections of complex varieties with tori. Acta Arith. 133 (2008), 309–323. Zbl1162.11031MR2457263
  9. M. Carrizosa, Problème de Lehmer et variétés abéliennes CM. C. R. Acad. Sci. 346 (2008) 1219–1224. Zbl1204.11092MR2473296
  10. G. Faltings, Diophantine approximation on abelian varieties. Ann. of Math. 133 (1991) 549–576. Zbl0734.14007MR1109353
  11. 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. Zbl0823.14009MR1307396
  12. P. Habegger, Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension. Invent. math. 176 (2009), 405–447. Zbl1176.14008MR2495768
  13. M. Hindry, Autour d’une conjecture de Serge Lang. Invent. math. 94 (1988) 575–603. Zbl0638.14026MR969244
  14. J. Kirby, The theory of exponential differential equations. thèse. Oxford. (2006). 
  15. M. Laurent, Équations diophantiennes exponentielles. Invent. math. 78 (1984) 299–327. Zbl0554.10009MR767195
  16. M. McQuillan, Division points on semi-abelian varieties. Invent. math. 120 (1995) 143–159. Zbl0848.14022MR1323985
  17. G. Maurin, Courbes algébriques et équations multiplicatives. Math. Ann. 341 (2008) 789–824. Zbl1154.14017MR2407327
  18. R. Pink, A Common Generalization of the Conjectures of André-Oort, Manin-Mumford, and Mordell-Lang. prépublication (2005). Zbl1200.11041
  19. 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. Zbl0581.14031MR717600
  20. G. Rémond, Inégalité de Vojta généralisée. Bull. Soc. Math. France 133 (2005) 459–495. Zbl1136.11043MR2233693
  21. G. Rémond, Intersection de sous-groupes et de sous-variétés I. Math. Ann. 333 (2005) 525–548. Zbl1088.11047MR2198798
  22. G. Rémond, Intersection de sous-groupes et de sous-variétés II. J. Inst. Math. Jussieu 6 (2007) 317–348. Zbl1170.11014MR2311666
  23. G. Rémond, Intersection de sous-groupes et de sous-variétés III. Comm. Math. Helv. à paraître. 21 pages. 
  24. P. Vojta, Siegel’s theorem in the compact case. Ann. of Math. 133 (1991) 509–548. Zbl0774.14019MR1109352
  25. P. Vojta, Integral points on subvarieties of semiabelian varieties, I. Invent. math. 126 (1996) 133–181. Zbl1011.11040MR1408559
  26. B. Zilber, Exponential sums equations and the Schanuel conjecture. J. London Math. Soc. (2) 65 (2002) 27–44. Zbl1030.11073MR1875133

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