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Transactions of the American Mathematical Society

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ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Distribution of integral division points on the algebraic torus
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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 $\mathbb {P}^1$.

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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