Evidence for the Dynamical Brauer–Manin Criterion

Abstract

Let ϕ: X → X be a morphism of a variety over a number field K. We consider local conditions and a “Brauer–Manin” condition, defined by Hsia and Silverman, for the orbit of a point P ∈ X(K) to be disjoint from a subvariety V⊆X, i.e., for We provide evidence that the dynamical Brauer–Manin condition is sufficient to explain the lack of points in the intersection ; this evidence stems from a probabilistic argument as well as unconditional results in the case of étale maps.

2000 AMS Subject Classification::

• 37P15
• 11S82

Keywords::

• arithmetic dynamics
• Brauer-Manin obstruction

Notes

¹Since , the intersection is contained in a finite union of geometrically irreducible closed -subvarieties V_i⊂V. Therefore, there is no loss of generality in restricting to geometrically irreducible subvarieties V.