Let be a morphism of a variety defined over a number field , let be a -subvariety, and let be the orbit of a point . We describe a local-global principle for the intersection . This principle may be viewed as a dynamical analog of the Brauer–Manin obstruction. We show that the rational points of are Brauer–Manin unobstructed for power maps on in two cases: (1) is a translate of a torus. (2) is a line and 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 un morphisme d’une variété définie sur un corps de nombres , une sous-variété définie sur et l’orbite d’un point . Nous décrivons un principe local-global pour l’intersection . Ce principe peut être vu comme l’analogue dynamique de l’obstruction de Brauer–Manin. Nous prouvons que les points rationnels de ne sont pas soumis à l’obstruction de Brauer–Manin pour l’application puissance sur dans deux cas : (1) est la translatée d’un tore. (2) est une droite and 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.
Keywords: arithmetic dynamical systems, local-global principle, Brauer–Manin obstruction
@article{JTNB_2009__21_1_235_0, 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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