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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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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Density of orbits of endomorphisms of abelian varieties
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by Dragos Ghioca and Thomas Scanlon PDF
Trans. Amer. Math. Soc. 369 (2017), 447-466 Request permission

Abstract:

Let $A$ be an abelian variety defined over $\bar {\mathbb {Q}}$, and let $\varphi$ be a dominant endomorphism of $A$ as an algebraic variety. We prove that either there exists a non-constant rational fibration preserved by $\varphi$ or there exists a point $x\in A(\bar {\mathbb {Q}})$ whose $\varphi$-orbit is Zariski dense in $A$. This provides a positive answer for abelian varieties of a question raised by Medvedev and the second author. We also prove a stronger statement of this result in which $\varphi$ is replaced by any commutative finitely generated monoid of dominant endomorphisms of $A$.
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Additional Information
  • Dragos Ghioca
  • Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2 Canada
  • MR Author ID: 776484
  • Email: dghioca@math.ubc.ca
  • Thomas Scanlon
  • Affiliation: Department of Mathematics, Evans Hall, University of California Berkeley, Berkeley, California 94720-3840
  • MR Author ID: 626736
  • ORCID: 0000-0003-2501-679X
  • Email: scanlon@math.berkeley.edu
  • Received by editor(s): December 5, 2014
  • Received by editor(s) in revised form: January 6, 2015
  • Published electronically: April 14, 2016
  • Additional Notes: The first author was partially supported by an NSERC grant. The second author was partially supported by NSF Grant DMS-1363372. This material is based upon work supported by the National Science Foundation under Grant No. 0932078 000 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the spring 2014 semester
  • © Copyright 2016 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 447-466
  • MSC (2010): Primary 11G10, 14G25
  • DOI: https://doi.org/10.1090/tran6648
  • MathSciNet review: 3557780