On a dynamical Brauer–Manin obstruction
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, p. 235-250

Let ϕ:XX be a morphism of a variety defined over a number field K, let VX be a K-subvariety, and let 𝒪 ϕ (P)={ϕ n (P):n0} be the orbit of a point PX(K). We describe a local-global principle for the intersection V𝒪 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  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:XX un morphisme d’une variété définie sur un corps de nombres K, VX une sous-variété définie sur K et 𝒪 ϕ (P)={ϕ n (P):n0} l’orbite d’un point PX(K). Nous décrivons un principe local-global pour l’intersection V𝒪 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  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.

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}
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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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