On a dynamical Brauer–Manin obstruction
Liang-Chung Hsia1 ; Joseph Silverman2
1 Department of Mathematics National Central University Chung-Li, 32054 Taiwan, R. O. C.
2 Mathematics Department Box 1917 Brown University Providence, RI 02912 USA
Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 1, pp. 235-250.

Soient ϕ:X→X un morphisme d’une variété définie sur un corps de nombres K, V⊂X une sous-variété définie sur K et 𝒪 ϕ (P)={ϕ n (P):n≥0} l’orbite d’un point P∈X(K). Nous décrivons un principe local-global pour l’intersection V∩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.

Let ϕ:X→X be a morphism of a variety defined over a number field K, let V⊂X be a K-subvariety, and let 𝒪 ϕ (P)={ϕ n (P):n≥0} be the orbit of a point P∈X(K). We describe a local-global principle for the intersection V∩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.

DOI : 10.5802/jtnb.668
Mots clés : arithmetic dynamical systems, local-global principle, Brauer–Manin obstruction
Liang-Chung Hsia 1 ; Joseph Silverman 2

1 Department of Mathematics National Central University Chung-Li, 32054 Taiwan, R. O. C.
2 Mathematics Department Box 1917 Brown University Providence, RI 02912 USA 
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Liang-Chung Hsia; Joseph Silverman. On a dynamical Brauer–Manin obstruction. Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 1, pp. 235-250. doi : 10.5802/jtnb.668. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.668/

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