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Intersecting curves and algebraic subgroups: Conjectures and more results


Authors: E. Bombieri, D. Masser and U. Zannier
Journal: Trans. Amer. Math. Soc. 358 (2006), 2247-2257
MSC (2000): Primary 11J95; Secondary 11G30, 11G50
DOI: https://doi.org/10.1090/S0002-9947-05-03810-9
Published electronically: October 31, 2005
MathSciNet review: 2197442
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper solves in the affirmative, up to dimension $ n=5$, a question raised in an earlier paper by the authors. The equivalence of the problem with a conjecture of Shou-Wu Zhang is proved in the Appendix.


References [Enhancements On Off] (What's this?)

  • [AD] F. Amoroso, S. David, Le problème de Lehmer en dimension supérieure, J. Reine Angew. Math. 513 (1999), 145-179. MR 1713323 (2001a:11116)
  • [AZ] F. Amoroso, U. Zannier, A relative Dobrowolski lower bound over abelian extensions, Ann. Scuola Norm. Sup. Pisa 29 (2000), 711-727.MR 1817715 (2003a:11078)
  • [BMZ] E. Bombieri, D. Masser and U. Zannier, Intersecting a curve with algebraic subgroups of multiplicative groups, International Math. Research Notices 20 (1999), 1119-1140. MR 1728021 (2001c:11081)
  • [BZ] E. Bombieri, U. Zannier, Algebraic points on subvarieties of $ \mathbf{G}_{m}^{n}$, International Math. Research Notices 7 (1995), 333-347.MR 1350686 (96h:11061)
  • [CZ] P.B. Cohen, U. Zannier, Multiplicative independence and bounded height, an example, Proc. Algebraic Number Theory and Dioph. Approx. Conference, Graz, 1998 (Walter de Gruyter, 2000), 93-101.MR 1770456 (2001f:11103)
  • [GKZ] I.M. Gelfand, M.M. Kapranov, A.V. Zelevinski, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, 1993.MR 1264417 (95e:14045)
  • [La1] S. Lang, Introduction to algebraic geometry, Addison-Wesley, 1973.MR 0344244 (49:8983)
  • [La2] S. Lang, Fundamentals of Diophantine Geometry, Springer Verlag, 1983.MR 0715605 (85j:11005)
  • [Li] P. Liardet, Sur une conjecture de Serge Lang, Astérisque 24-25 (1975), 187-210.MR 0376688 (51:12863)
  • [RV] G. Rémond, E. Viada, Problème de Mordell-Lang modulo certaines sous-variétés abéliennes, International Math. Research Notices 35 (2003), 1915-1931.MR 1995142 (2004h:11054)
  • [S] A. Schinzel, Polynomials with special regard to reducibility, Encyclopaedia of Mathematics and its Applications, vol. 77, Cambridge, 2000.MR 1770638 (2001h:11135)
  • [Zag] D. Zagier, Algebraic numbers close to both 0 and $ 1$, Math. Comp. 61 (1993), 485-491.MR 1197513 (94c:11104)
  • [Zan] U. Zannier, Proof of Conjecture 1, Appendix to [S].MR 1770638 (2001h:11135)
  • [Zh] S. Zhang, Positive line bundles on arithmetic surfaces, Annals of Math. 136 (1992), 569-587.MR 1189866 (93j:14024)

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Additional Information

E. Bombieri
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
Email: eb@math.ias.edu

D. Masser
Affiliation: Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland
Email: masser@math.unibas.ch

U. Zannier
Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri, 56100 Pisa, Italy
Email: u.zannier@sns.it

DOI: https://doi.org/10.1090/S0002-9947-05-03810-9
Received by editor(s): February 2, 2004
Received by editor(s) in revised form: July 14, 2004
Published electronically: October 31, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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