Intersecting a plane with algebraic subgroups of multiplicative groups
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 1, pp. 51-80.

Consider an arbitrary algebraic curve defined over the field of all algebraic numbers and sitting in a multiplicative commutative algebraic group. In an earlier article from 1999 bearing almost the same title, we studied the intersection of the curve and the union of all algebraic subgroups of some fixed codimension. With codimension one the resulting set has bounded height properties, and with codimension two it has finiteness properties. The main aim of the present work is to make a start on such problems in higher dimension by proving the natural analogues for a linear surface (with codimensions two and three). These are in accordance with some general conjectures that we have recently proposed elsewhere.

Classification : 11G35, 11G50, 14G25, 14J20 
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Bombieri, Enrico; Masser, David; Zannier, Umberto. Intersecting a plane with algebraic subgroups of multiplicative groups. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 1, pp. 51-80. http://archive.numdam.org/item/ASNSP_2008_5_7_1_51_0/

[1] F. Amoroso and S. David, Le problème de Lehmer en dimension supérieure, J. Reine Angew. Math. 513 (1999), 145-179. | MR | Zbl

[2] F. Amoroso and S. David, Distribution des points de petite hauteur dans les groupes multiplicatifs, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 3 (2004), 325-348. | Numdam | MR | Zbl

[3] F. Amoroso and U. Zannier, A relative Dobrowolski lower bound over abelian extensions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), 711-727. | Numdam | MR | Zbl

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

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

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

[7] E. Bombieri, D. Masser and U. Zannier, Anomalous subvarieties - structure theorems and applications, Int. Math. Res. Not. IMRN 19 (2007), 33 pages. | MR | Zbl

[8] E. Bombieri and J. Vaaler, On Siegel's Lemma, Invent. Math. 73 (1983), 11-32. | MR | Zbl

[9] E. Bombieri and U. Zannier, Algebraic points on subvarieties of 𝔾 m n , Internat. Math. Res. Notices 7 (1995), 333-347. | MR | Zbl

[10] J. W. S. Cassels, “An Introduction to Diophantine Approximation”, Cambridge Tracts in Mathematics and Mathematical Physics, Vol. 45, Cambridge, 1965. | MR | Zbl

[11] T. Loher and D. Masser, Uniformly counting points of bounded height, Acta Arith. 111 (2004), 277-297. | MR | Zbl

[12] R. Pink, A common generalization of the conjectures of André-Oort, Manin-Mumford, and Mordell-Lang, manuscript dated 17th April 2005.

[13] A. Schinzel, “Polynomials with Special Regard to Reducibility”, Encyclopaedia of Mathematics and its Applications, Vol. 77, Cambridge, 2000. | MR | Zbl

[14] U. Zannier, Proof of Conjecture 1, Appendix to [13], 517-539.

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