On a dynamical Brauer–Manin obstruction

Hsia, Liang-Chung; Silverman, Joseph

doi:10.5802/jtnb.668

Hsia, Liang-Chung; Silverman, Joseph

Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, p. 235-250

Abstract
Résumé

Let $ϕ : X \to X$ be a morphism of a variety defined over a number field $K$ , let $V \subset X$ be a $K$ -subvariety, and let $𝒪_{ϕ} (P) = {ϕ^{n} (P) : n \geq 0}$ be the orbit of a point $P \in X (K)$ . We describe a local-global principle for the intersection $V \cap 𝒪_{ϕ} (P)$ . This principle may be viewed as a dynamical analog of the Brauer–Manin obstruction. We show that the rational points of $V (K)$ are Brauer–Manin unobstructed for power maps on $ℙ^{2}$ in two cases: (1) $V$ is a translate of a torus. (2) $V$ is a line and $P$ has a preperiodic coordinate. A key tool in the proofs is the classical Bang–Zsigmondy theorem on primitive divisors in sequences. We also prove analogous local-global results for dynamical systems associated to endomoprhisms of abelian varieties.

Soient $ϕ : X \to X$ un morphisme d’une variété définie sur un corps de nombres $K$ , $V \subset X$ une sous-variété définie sur $K$ et $𝒪_{ϕ} (P) = {ϕ^{n} (P) : n \geq 0}$ l’orbite d’un point $P \in X (K)$ . Nous décrivons un principe local-global pour l’intersection $V \cap 𝒪_{ϕ} (P)$ . Ce principe peut être vu comme l’analogue dynamique de l’obstruction de Brauer–Manin. Nous prouvons que les points rationnels de $V (K)$ ne sont pas soumis à l’obstruction de Brauer–Manin pour l’application puissance sur $ℙ^{2}$ dans deux cas : (1) $V$ est la translatée d’un tore. (2) $V$ est une droite and $P$ a une coordonnée prépériodique. Un outil principal des preuves est le théorème classique de Bang–Zsigmondy sur les diviseurs primitifs dans les suites. Nous prouvons également des résultats local-globaux analogues pour les systèmes dynamiques associés aux endomorphismes de variétés abéliennes.

MR 2537714 | Zbl pre05620679

DOI : https://doi.org/10.5802/jtnb.668
Keywords: arithmetic dynamical systems, local-global principle, Brauer–Manin obstruction

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     author = {Hsia, Liang-Chung and Silverman, Joseph},
     title = {On a dynamical Brauer--Manin obstruction},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {1},
     year = {2009},
     pages = {235-250},
     doi = {10.5802/jtnb.668},
     mrnumber = {2537714},
     zbl = {pre05620679},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2009__21_1_235_0}
}

Hsia, Liang-Chung; Silverman, Joseph. On a dynamical Brauer–Manin obstruction. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, pp. 235-250. doi : 10.5802/jtnb.668. http://www.numdam.org/item/JTNB_2009__21_1_235_0/

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