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Zariski dense orbits for regular self-maps on split semiabelian varieties

Part of:Arithmetic problems. Diophantine geometryAbelian varieties and schemes

Published online by Cambridge University Press:  05 March 2021

Dragos Ghioca and
Sina Saleh
Dragos Ghioca*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada e-mail: sinas@math.ubc.ca
*
e-mail: dghioca@math.ubc.ca
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Abstract

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We provide a direct proof of the Medvedev–Scanlon’s conjecture from Medvedev and Scanlon (Ann. Math. Second Series 179(2014), 81–177) regarding Zariski dense orbits under the action of regular self-maps on split semiabelian varieties defined over a field of characteristic $0$. Besides obtaining significantly easier proofs than the ones previously obtained in Ghioca and Scanlon (Trans. Am. Math. Soc. 369(2017), 447–466; for the case of abelian varieties) and Ghioca and Satriano (Trans. Am. Math. Soc. 371(2019), 6341–6358; for the case of semiabelian varieties), our method allows us to exhibit numerous starting points with Zariski dense orbits, which the methods from Ghioca and Scanlon (Trans. Am. Math. Soc. 369(2017), 447–466) and Ghioca and Satriano (Trans. Am. Math. Soc. 371(2019), 6341–6358) could not provide.

Keywords

Abelian varietiesZariski dense orbitsdominant endomorphisms

MSC classification

Primary:14K15: Arithmetic ground fields
Secondary:14G05: Rational points

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 1 , March 2022, pp. 116 - 122
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Copyright
© Canadian Mathematical Society 2021

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