On a Dynamical Brauer-Manin Obstruction

Abstract

Let F : X --> X be a morphism of a variety defined over a number field K, let V be a K-subvariety of X, and let O_F(P)= {F^n(P) :n=0,1,2,...} be the orbit of a point P in X(K). We describe a local-global principle for the intersection of V and O_F(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.Comment: 17 page