A note on divisible points of curves
HTML articles powered by AMS MathViewer
- by M. Bays and P. Habegger PDF
- Trans. Amer. Math. Soc. 367 (2015), 1313-1328 Request permission
Abstract:
Let $C$ be an irreducible algebraic curve defined over a number field and inside an algebraic torus of dimension at least $3$. We partially answer a question posed by Levin on points on $C$ for which a non-trivial power lies again on $C$. Our results have connections to Zilber’s Conjecture on Intersections with Tori and yield to methods arising in transcendence theory and the theory of o-minimal structures.References
- James Ax, On Schanuel’s conjectures, Ann. of Math. (2) 93 (1971), 252–268. MR 277482, DOI 10.2307/1970774
- A. Baker and G. Wüstholz, Logarithmic forms and group varieties, J. Reine Angew. Math. 442 (1993), 19–62. MR 1234835, DOI 10.1515/crll.1993.442.19
- Enrico Bombieri and Walter Gubler, Heights in Diophantine geometry, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006. MR 2216774, DOI 10.1017/CBO9780511542879
- 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 1728021, DOI 10.1155/S1073792899000628
- Y. Bugeaud and M. Laurent, Minoration effective de la distance $p$-adique entre puissances de nombres algébriques, J. Number Theory 61 (1996), no. 2, 311–342 (French, with English summary). MR 1423057, DOI 10.1006/jnth.1996.0152
- Pietro Corvaja, David Masser, and Umberto Zannier, Sharpening ‘Manin-Mumford’ for certain algebraic groups of dimension 2, Enseign. Math. 59 (2013), no. 3-4, 225–269. MR 3189035, DOI 10.4171/LEM/59-3-2
- E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979), no. 4, 391–401. MR 543210, DOI 10.4064/aa-34-4-391-401
- Antonio J. Engler and Alexander Prestel, Valued fields, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. MR 2183496
- P. Habegger, On the bounded height conjecture, Int. Math. Res. Not. IMRN 5 (2009), 860–886. MR 2482128, DOI 10.1093/imrn/rnn149
- Marc Hindry, Autour d’une conjecture de Serge Lang, Invent. Math. 94 (1988), no. 3, 575–603 (French). MR 969244, DOI 10.1007/BF01394276
- Serge Lang, Division points on curves, Ann. Mat. Pura Appl. (4) 70 (1965), 229–234. MR 190146, DOI 10.1007/BF02410091
- Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, 1983.
- G. Maurin, Équations multiplicatives sur les sous-variétés des tores, Int. Math. Res. Not. IMRN 23 (2011), 5259–5366 (French). MR 2855071, DOI 10.1093/imrn/rnq248
- Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859, DOI 10.1007/978-3-662-03983-0
- J. Pila and A. J. Wilkie, The rational points of a definable set, Duke Math. J. 133 (2006), no. 3, 591–616. MR 2228464, DOI 10.1215/S0012-7094-06-13336-7
- Jonathan Pila and Umberto Zannier, Rational points in periodic analytic sets and the Manin-Mumford conjecture, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 19 (2008), no. 2, 149–162. MR 2411018, DOI 10.4171/RLM/514
- Gaël Rémond, Sur les sous-variétés des tores, Compositio Math. 134 (2002), no. 3, 337–366 (French, with English summary). MR 1943907, DOI 10.1023/A:1020982431028
- Lou van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR 1633348, DOI 10.1017/CBO9780511525919
- A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996), no. 4, 1051–1094. MR 1398816, DOI 10.1090/S0894-0347-96-00216-0
- Umberto Zannier, Some problems of unlikely intersections in arithmetic and geometry, Annals of Mathematics Studies, vol. 181, Princeton University Press, Princeton, NJ, 2012. With appendixes by David Masser. MR 2918151
- Shouwu Zhang, Positive line bundles on arithmetic varieties, J. Amer. Math. Soc. 8 (1995), no. 1, 187–221. MR 1254133, DOI 10.1090/S0894-0347-1995-1254133-7
- Boris Zilber, Exponential sums equations and the Schanuel conjecture, J. London Math. Soc. (2) 65 (2002), no. 1, 27–44. MR 1875133, DOI 10.1112/S0024610701002861
Additional Information
- M. Bays
- Affiliation: Department of Mathematics and Statistics, McMaster University, Ontario, Canada
- Email: mbays@sdf.org
- P. Habegger
- Affiliation: Fachbereich Mathematik, TU Darmstadt, Schlossgartenstrasse 7, Darmstadt, 64289 Germany
- MR Author ID: 774657
- Email: habegger@mathematik.tu-darmstadt.de
- Received by editor(s): May 8, 2013
- Published electronically: September 4, 2014
- Additional Notes: The first author was partially supported by the Agence Nationale de Recherche [MODIG, Project ANR-09-BLAN-0047]
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 1313-1328
- MSC (2010): Primary 14H25; Secondary 03C64, 11G50, 11J86, 11U09
- DOI: https://doi.org/10.1090/S0002-9947-2014-06494-5
- MathSciNet review: 3280045