Distribution of integral division points on the algebraic torus
HTML articles powered by AMS MathViewer
- by Philipp Habegger and Su-ion Ih PDF
- Trans. Amer. Math. Soc. 371 (2019), 357-386 Request permission
Abstract:
Let $K$ be a number field with algebraic closure $\overline K$, and let $S$ be a finite set of places of $K$ containing all the infinite ones. Let $\Gamma _0$ be a finitely generated subgroup of $\mathbb {G}_{\mathrm {m}} (\overline K)$, and let $\Gamma \subset \mathbb {G}_{\mathrm {m}} (\overline K)$ be the division group attached to $\Gamma _0$. Here is an illustration of what we will prove in this article. Fix a proper closed subinterval $I$ of $[0, \infty )$ and a nonzero effective divisor $D$ on $\mathbb {G}_{\mathrm {m}}$ which is not the translate of any torsion divisor on the algebraic torus $\mathbb {G}_{\mathrm {m}}$ by any point of $\Gamma$ with height belonging to $I$.
Then we prove a statement which easily implies that the set of “integral division points on $\mathbb {G}_{\mathrm {m}}$ with height near $I$”, i.e., the set of points of $\Gamma$ with (standard absolute logarithmic Weil) height in $J$ which are $S$-integral on $\mathbb {G}_{\mathrm {m}}$ relative to $D,$ is finite for some fixed subinterval $J$ of $[0, \infty )$ properly containing $I$. We propose a conjecture on the nondensity of integral division points on semi-abelian varieties with prescribed height values, which generalizes some previously known conjectures as well as this finiteness result for $\mathbb {G}_{\mathrm {m}}$. Finally, we also propose an analogous version for a dynamical system on $\mathbb {P}^1$.
References
- Francesco Amoroso and Umberto Zannier, A relative Dobrowolski lower bound over abelian extensions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 3, 711–727. MR 1817715
- Pascal Autissier, Sur une question d’équirépartition de nombres algébriques, C. R. Math. Acad. Sci. Paris 342 (2006), no. 9, 639–641 (French, with English and French summaries). MR 2225867, DOI 10.1016/j.crma.2006.02.021
- 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
- Matthew Baker, Su-ion Ih, and Robert Rumely, A finiteness property of torsion points, Algebra Number Theory 2 (2008), no. 2, 217–248. MR 2377370, DOI 10.2140/ant.2008.2.217
- Yuri Bilu, Limit distribution of small points on algebraic tori, Duke Math. J. 89 (1997), no. 3, 465–476. MR 1470340, DOI 10.1215/S0012-7094-97-08921-3
- 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
- Antoine Chambert-Loir, Points de petite hauteur sur les variétés semi-abéliennes, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 6, 789–821 (French, with English and French summaries). MR 1832991, DOI 10.1016/S0012-9593(00)01053-3
- Sinnou David and Patrice Philippon, Sous-variétés de torsion des variétés semi-abéliennes, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 8, 587–592 (French, with English and French summaries). MR 1799094, DOI 10.1016/S0764-4442(00)01634-7
- 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
- David Grant and Su-Ion Ih, Integral division points on curves, Compos. Math. 149 (2013), no. 12, 2011–2035. MR 3143704, DOI 10.1112/S0010437X13007318
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 6th ed., Oxford University Press, Oxford, 2008. Revised by D. R. Heath-Brown and J. H. Silverman; With a foreword by Andrew Wiles. MR 2445243
- Glyn Harman, Metric number theory, London Mathematical Society Monographs. New Series, vol. 18, The Clarendon Press, Oxford University Press, New York, 1998. MR 1672558
- Marc Hindry and Joseph H. Silverman, Diophantine geometry, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. An introduction. MR 1745599, DOI 10.1007/978-1-4612-1210-2
- Su-Ion Ih, Integral points on the Chebyshev dynamical systems, J. Korean Math. Soc. 52 (2015), no. 5, 955–964. MR 3393112, DOI 10.4134/JKMS.2015.52.5.955
- 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
- Hans Peter Schlickewei, Lower bounds for heights on finitely generated groups, Monatsh. Math. 123 (1997), no. 2, 171–178. MR 1430503, DOI 10.1007/BF01305970
- H. P. Schlickewei and Wolfgang M. Schmidt, On polynomial-exponential equations, Math. Ann. 296 (1993), no. 2, 339–361. MR 1219906, DOI 10.1007/BF01445109
- Wolfgang M. Schmidt, Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, vol. 1467, Springer-Verlag, Berlin, 1991. MR 1176315, DOI 10.1007/BFb0098246
- Joseph H. Silverman, Integer points, Diophantine approximation, and iteration of rational maps, Duke Math. J. 71 (1993), no. 3, 793–829. MR 1240603, DOI 10.1215/S0012-7094-93-07129-3
- Joseph H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Mathematics, vol. 241, Springer, New York, 2007. MR 2316407, DOI 10.1007/978-0-387-69904-2
- Michel Waldschmidt, Diophantine approximation on linear algebraic groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 326, Springer-Verlag, Berlin, 2000. Transcendence properties of the exponential function in several variables. MR 1756786, DOI 10.1007/978-3-662-11569-5
- Kun Rui Yu, Linear forms in $p$-adic logarithms. III, Compositio Math. 91 (1994), no. 3, 241–276. MR 1273651
- Frank Zorzitto, Discretely normed abelian groups, Aequationes Math. 29 (1985), no. 2-3, 172–174. MR 819306, DOI 10.1007/BF02189825
Additional Information
- Philipp Habegger
- Affiliation: Departement Mathematik und Informatik, Fachbereich Mathematik, University of Basel, Spiegelgasse 1, 4051 Basel, Switzerland
- MR Author ID: 774657
- Email: philipp.habegger@unibas.ch
- Su-ion Ih
- Affiliation: Department of Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309-0395
- MR Author ID: 703039
- Email: ih@math.colorado.edu
- Received by editor(s): August 22, 2015
- Received by editor(s) in revised form: October 7, 2016, and February 20, 2017
- Published electronically: April 25, 2018
- Additional Notes: The work of the first author was partially supported by the National Science Foundation, grant number DMS-1128155.
The work of the second author was partially supported by the Simons Foundation, grant number 267613.
The opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of those foundations. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 357-386
- MSC (2010): Primary 11G50, 11J61, 11J71, 11J86, 11L15, 14G25, 14G40, 20G30, 37P05, 37P35
- DOI: https://doi.org/10.1090/tran/7238
- MathSciNet review: 3885147