On a Dynamical Brauer-Manin Obstruction

TL;DR

This paper develops a dynamical analog of the Brauer-Manin obstruction to understand the intersection of orbits and subvarieties over number fields, with applications to power maps and abelian varieties.

Contribution

It introduces a local-global principle for dynamical intersections and proves cases where rational points are Brauer--Manin unobstructed for specific dynamical systems.

Findings

Rational points are Brauer--Manin unobstructed for power maps on P^2 in certain cases.

The principle applies to subvarieties that are translates of tori or lines with preperiodic points.

Analogous results are established for endomorphisms of abelian varieties.

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.