Abstract
Let \({X\subset G_m^n}\) be an irreducible closed subvariety defined over \({\bar{Q}}\). We bound the height of algebraic points on X that are in a certain sense close to the union of all algebraic subgroup of \({G_m^n}\) of dimension m < n/dim X. The bound we obtain is effective and will be expressed as a function of the height of X, the degree of X, and n. We then apply this bound to derive certain finiteness results if m is also strictly less than n − dim X.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amoroso F., David S.: Distribution des points de petite hauteur dans les groupes multiplicatifs. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3(2), 325–348 (2004)
Bombieri E., Masser D., Zannier U.: Intersecting a curve with algebraic subgroups of multiplicative groups. Int. Math. Res. Not. 20, 1119–1140 (1999)
Bombieri, E., Masser, D., Zannier, U.: Anomalous subvarieties—structure theorems and applications. Int. Math. Res. Not. IMRN (19), 1–33 (2007)
Bombieri, E., Masser, D., Zannier, U.: Intersecting a plane with algebraic subgroups of multiplicative groups. Ann. Sc. Norm. Super. Pisa (to appear)
Bombieri E., Gubler W.: Heights in Diophantine Geometry. Cambridge University Press, London (2006)
Bost J.-B., Gillet H., Soulé C.: Heights of projective varieties and positive Green forms. J. Am. Math. Soc. 7(2), 903–1027 (1994)
Danilov V.I.: Algebraic varieties and schemes. In: Shafarevich, I.R.(eds) Algebraic geometry I. Encyclopaedia of Mathematical Sciences vol. 23, Springer, Heidelberg (1994)
Evertse, J.-H.: Points on Subvarieties of Tori, pp.214–230. Cambridge University Press, London (2002)
Fulton W.: Intersection Theory. Springer, Heidelberg (1984)
Habegger, P.: Heights and multiplicative relations on algebraic varieties, PhD Thesis University of Basel (2007)
Maurin, G.: Courbes algébriques et équations multiplicatives. Math. Ann. doi:10.1007/s00208-008-0212-9 (in press)
Philippon P.: Sur des hauteurs alternatives III. J. Math. Pures Appl. (9) 74(4), 345–365 (1995)
Pink, R.: A Common Generalization of the Conjectures of André–Oort, Manin–Mumford, and Mordell–Lang, 17 April 2005 (preprint)
Poonen B.: Mordell–Lang plus Bogomolov. Invent. Math. 137(2), 413–425 (1999)
Rémond G.: Approximation diophantienne sur les variétés semi-abéliennes. Ann. Sci. école Norm. Sup. (4) 36(2), 191–212 (2003)
Schinzel, A.: Polynomials with special regard to reducibility. With an appendix by Umberto Zannier. Encyclopedia of Mathematics and its Applications, vol. 77. Cambridge University Press, London (2000)
Schlickewei H.P.: Lower bounds for heights on finitely generated groups. Monatsh. Math. 123(2), 171–178 (1997)
Viada E.: The intersection of a curve with algebraic subgroups in a product of elliptic curves. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2(1), 47–75 (2003)
Zannier, U.: Appendix by Umberto Zannier in [16], pp. 517–539 (2000)
Zhang S.: Positive line bundles on arithmetic varieties. J. Am. Math. Soc. 8(1), 187–221 (1995)
Zilber B.: Exponential sums equations and the Schanuel conjecture. J. Lond. Math. Soc. (2) 65(1), 27–44 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Habegger, P. Intersecting subvarieties of \({G_m^n}\) with algebraic subgroups. Math. Ann. 342, 449–466 (2008). https://doi.org/10.1007/s00208-008-0242-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-008-0242-3