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.
@article{ASNSP_2008_5_7_1_51_0, author = {Bombieri, Enrico and Masser, David and Zannier, Umberto}, title = {Intersecting a plane with algebraic subgroups of multiplicative groups}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {51--80}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 7}, number = {1}, year = {2008}, zbl = {1150.11022}, mrnumber = {2413672}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2008_5_7_1_51_0/} }
TY - JOUR AU - Bombieri, Enrico AU - Masser, David AU - Zannier, Umberto TI - Intersecting a plane with algebraic subgroups of multiplicative groups JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2008 DA - 2008/// SP - 51 EP - 80 VL - Ser. 5, 7 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2008_5_7_1_51_0/ UR - https://zbmath.org/?q=an%3A1150.11022 UR - https://www.ams.org/mathscinet-getitem?mr=2413672 LA - en ID - ASNSP_2008_5_7_1_51_0 ER -
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] Le problème de Lehmer en dimension supérieure, J. Reine Angew. Math. 513 (1999), 145-179. | MR 1713323 | Zbl 1011.11045
and ,[2] Distribution des points de petite hauteur dans les groupes multiplicatifs, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 3 (2004), 325-348. | Numdam | MR 2075986 | Zbl 1150.11021
and ,[3] A relative Dobrowolski lower bound over abelian extensions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), 711-727. | Numdam | MR 1817715 | Zbl 1016.11026
and ,[4] Intersecting a curve with algebraic subgroups of multiplicative groups, Internat. Math. Res. Notices 20 (1999), 1119-1140. | MR 1728021 | Zbl 0938.11031
, and ,[5] Finiteness results for multiplicatively dependent points on complex curves, Michigan Math. J. 51 (2003), 451-466. | MR 2021000 | Zbl 1048.11056
, and ,[6] Intersecting curves and algebraic subgroups: conjectures and more results, Trans. Amer. Math. Soc. 358 (2006), 2247-2257. | MR 2197442 | Zbl 1161.11025
, and ,[7] Anomalous subvarieties - structure theorems and applications, Int. Math. Res. Not. IMRN 19 (2007), 33 pages. | MR 2359537 | Zbl 1145.11049
, and ,[8] On Siegel's Lemma, Invent. Math. 73 (1983), 11-32. | MR 707346 | Zbl 0533.10030
and ,[9] Algebraic points on subvarieties of , Internat. Math. Res. Notices 7 (1995), 333-347. | MR 1350686 | Zbl 0848.11030
and ,[10] “An Introduction to Diophantine Approximation”, Cambridge Tracts in Mathematics and Mathematical Physics, Vol. 45, Cambridge, 1965. | MR 120219 | Zbl 0077.04801
,[11] Uniformly counting points of bounded height, Acta Arith. 111 (2004), 277-297. | MR 2039627 | Zbl 1084.11034
and ,[12] A common generalization of the conjectures of André-Oort, Manin-Mumford, and Mordell-Lang, manuscript dated 17th April 2005.
,[13] “Polynomials with Special Regard to Reducibility”, Encyclopaedia of Mathematics and its Applications, Vol. 77, Cambridge, 2000. | MR 1770638 | Zbl 0956.12001
,[14] Proof of Conjecture , Appendix to [13], 517-539.
,[15] Exponential sums equations and the Schanuel conjecture, J. London Math. Soc. 65 (2002), 27-44. | MR 1875133 | Zbl 1030.11073
,