On a dynamical Brauer–Manin obstruction

Hsia, Liang-Chung, and Silverman, Joseph. "On a dynamical Brauer–Manin obstruction." Journal de Théorie des Nombres de Bordeaux 21.1 (2009): 235-250. <http://eudml.org/doc/10874>.

@article{Hsia2009,
abstract = {Let $\varphi :X\rightarrow X$ be a morphism of a variety defined over a number field $K$, let $V\subset X$ be a $K$-subvariety, and let $\{\mathcal\{O\}\}_\varphi (P)=\lbrace \varphi ^n(P):n\ge 0\rbrace $ be the orbit of a point $P\in X(K)$. We describe a local-global principle for the intersection $V\cap \{\mathcal\{O\}\}_\varphi (P)$. 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 $\mathbb\{P\}^2$ 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.},
affiliation = {Department of Mathematics National Central University Chung-Li, 32054 Taiwan, R. O. C.; Mathematics Department Box 1917 Brown University Providence, RI 02912 USA},
author = {Hsia, Liang-Chung, Silverman, Joseph},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {arithmetic dynamical systems; local-global principle; Brauer–Manin obstruction; Arithmetic dynamics; Brauer-Manin obstruction},
language = {eng},
number = {1},
pages = {235-250},
publisher = {Université Bordeaux 1},
title = {On a dynamical Brauer–Manin obstruction},
url = {http://eudml.org/doc/10874},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Hsia, Liang-Chung
AU - Silverman, Joseph
TI - On a dynamical Brauer–Manin obstruction
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 1
SP - 235
EP - 250
AB - Let $\varphi :X\rightarrow X$ be a morphism of a variety defined over a number field $K$, let $V\subset X$ be a $K$-subvariety, and let ${\mathcal{O}}_\varphi (P)=\lbrace \varphi ^n(P):n\ge 0\rbrace $ be the orbit of a point $P\in X(K)$. We describe a local-global principle for the intersection $V\cap {\mathcal{O}}_\varphi (P)$. 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 $\mathbb{P}^2$ 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.
LA - eng
KW - arithmetic dynamical systems; local-global principle; Brauer–Manin obstruction; Arithmetic dynamics; Brauer-Manin obstruction
UR - http://eudml.org/doc/10874
ER -