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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
Posted:
October 31, 2005
MathSciNet review:
2197442
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: This paper solves in the affirmative, up to dimension  , 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.
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Zannier, A relative Dobrowolski lower bound over abelian
extensions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)
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(96h:11061), http://dx.doi.org/10.1155/S1073792895000250
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Schinzel, Polynomials with special regard to reducibility,
Encyclopedia of Mathematics and its Applications, vol. 77, Cambridge
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Math. (2) 136 (1992), no. 3, 569–587. MR 1189866
(93j:14024), http://dx.doi.org/10.2307/2946601
- [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
 , 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
 , 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:
http://dx.doi.org/10.1090/S0002-9947-05-03810-9
PII:
S 0002-9947(05)03810-9
Received by editor(s):
February 2, 2004
Received by editor(s) in revised form:
July 14, 2004
Posted:
October 31, 2005
Article copyright:
© Copyright 2005 American Mathematical Society
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