Abstract
We introduce an algebraicity criterion. It has the following form: Consider an analytic subvariety of some algebraic variety X over a global field K. Under certain conditions, if X contains many K-points, then X is algebraic over K. This gives a way to show the transcendence of points via the transcendence of analytic subvarieties. Such a situation often appears when we have a dynamical system, because we can often produce infinitely many points from one point via iterates. Combining this criterion and the study of invariant subvarieties, we get some results on the transcendence in arithmetic dynamics. We get a characterization for products of Böttcher coordinates or products of multiplicative canonical heights for polynomial dynamical pairs to be algebraic. For this, we study the invariant subvarieties for products of endomorphisms. In particular, we partially generalize Medvedev–Scanlon’s classification of invariant subvarieties of split polynomial maps to separable endomorphisms on \(({\mathbb P}^1)^N\) in any characteristic. We also get some high dimensional partial generalization via introducing a notion of independence. We then study dominant endomorphisms f on \({\mathbb A}^N\) over a number field of algebraic degree \(d\ge 2\). We show that in most cases (e.g. when such an endomorphism extends to an endomorphism on \({\mathbb P}^N\)), there are many analytic curves centered at infinity which are periodic. We show that for most of them, it is algebraic if and only if it contains at least one algebraic point. We also study the periodic curves. We show that for most f, all periodic curves have degree at most 2. When \(N=2\), we get a more precise classification result. We show that under a condition which is satisfied for a general f, if f has infinitely many periodic curves, then f is homogenous up to change of origin.
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Notes
We may define the distance function \(d_v\) on \(X(K_v)\) via an embedding of \(X(K_v)\) to a projective space \({\mathbb P}^N(K_v)\). The equivalent class of \(d_v\) does not depend on the choice of the embedding.
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Acknowledgements
I would like to thank Xinyi Yuan for his help for the proof of Lemma 2.3. I thank Charles Favre for telling me Lemma 5.3. I also thank Zhiyu Tian for helpful discussions. I thank Thomas Scanlon for explaining to me the notions “almost orthogonal” and “orthogonal” in model theory. I thank Jason Bell and Khoa D. Nguyen for kindly telling me their independent work in the same results as Theorem 1.13 and Proposition 1.14. I also thank Khoa D. Nguyen for reminding me that my notion “independent” relates to the notions “almost orthogonal” and “orthogonal” in model theory. I thank Serge Cantat and Joseph Silverman for helpful comments of the first version of this paper. I thank the referee for his helpful suggestions.
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Xie, J. Algebraicity Criteria, Invariant Subvarieties and Transcendence Problems from Arithmetic Dynamics. Peking Math J 7, 345–398 (2024). https://doi.org/10.1007/s42543-022-00059-9
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DOI: https://doi.org/10.1007/s42543-022-00059-9