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Density of orbits of endomorphisms of abelian varieties

Authors: Dragos Ghioca and Thomas Scanlon
Journal: Trans. Amer. Math. Soc. 369 (2017), 447-466
MSC (2010): Primary 11G10, 14G25
Published electronically: April 14, 2016
MathSciNet review: 3557780
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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

Thomas Scanlon
Affiliation: Department of Mathematics, Evans Hall, University of California Berkeley, Berkeley, California 94720-3840

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
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