The dynamical Mordell-Lang conjecture in positive characteristic
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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.References
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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
- 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.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 1151-1167
- MSC (2010): Primary 11G10; Secondary 37P55
- DOI: https://doi.org/10.1090/tran/7261
- MathSciNet review: 3885174