Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

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
DOI: https://doi.org/10.1090/tran6648
Published electronically: April 14, 2016
MathSciNet review: 3557780
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11G10, 14G25

Retrieve articles in all journals with MSC (2010): 11G10, 14G25


Additional Information

Dragos Ghioca
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2 Canada
Email: dghioca@math.ubc.ca

Thomas Scanlon
Affiliation: Department of Mathematics, Evans Hall, University of California Berkeley, Berkeley, California 94720-3840
Email: scanlon@math.berkeley.edu

DOI: https://doi.org/10.1090/tran6648
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
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society