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Transactions of the American Mathematical Society

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

 
 

 

The dynamical Mordell-Lang conjecture in positive characteristic


Author: Dragos Ghioca
Journal: Trans. Amer. Math. Soc. 371 (2019), 1151-1167
MSC (2010): Primary 11G10; Secondary 37P55
DOI: https://doi.org/10.1090/tran/7261
Published electronically: July 20, 2018
MathSciNet review: 3885174
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Abstract: Let $ K$ be an algebraically closed field of prime characteristic $ p$, let $ N\in \mathbb{N}$, let $ \Phi :\mathbb{G}_m^N\longrightarrow \mathbb{G}_m^N$ be a self-map defined over $ K$, let $ V\subset \mathbb{G}_m^N$ be a curve defined over $ K$, and let $ \alpha \in \mathbb{G}_m^N(K)$. We show that the set $ S=\{n\in \mathbb{N}\colon \Phi ^n(\alpha )\in V\}$ is a union of finitely many arithmetic progressions, along with a finite set and finitely many $ p$-arithmetic sequences, which are sets of the form $ \{a+bp^{kn}\colon n\in \mathbb{N}\}$ for some $ a,b\in \mathbb{Q}$ and some $ k\in \mathbb{N}$. We also prove that our result is sharp in the sense that $ S$ may be infinite without containing an arithmetic progression. Our result addresses a positive characteristic version of the dynamical Mordell-Lang conjecture, and it is the first known instance when a structure theorem is proven for the set $ S$ which includes $ p$-arithmetic sequences.


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Additional Information

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

DOI: https://doi.org/10.1090/tran/7261
Keywords: Dynamical Mordell-Lang problem, endomorphisms of algebraic tori over fields of characteristic $p$
Received by editor(s): October 13, 2016
Received by editor(s) in revised form: October 16, 2016, and April 20, 2017
Published electronically: July 20, 2018
Additional Notes: The author has been partially supported by a Discovery Grant from the National Science and Engineering Board of Canada.
Article copyright: © Copyright 2018 American Mathematical Society