Good reduction and canonical heights of subvarieties

TL;DR

This paper establishes bounds on the periodicity of preperiodic subvarieties in dynamical systems over number fields, using good reduction properties and canonical heights, with explicit results for hypersurfaces.

Contribution

It extends good reduction bounds from points to all projective varieties and provides explicit bounds for preperiodic subvarieties, including hypersurfaces.

Findings

Bound on the length of the periodic part of preperiodic subvarieties

Extension of good reduction bounds to all projective varieties

Explicit bounds for hypersurfaces

Abstract

We bound the length of the periodic part of the orbit of a preperiodic rational subvariety via good reduction information. This bound depends only on the degree of the map, the degree of the subvariety, the dimension of the projective space, the degree of the number field, and the prime of good reduction. As part of the proof, we extend the corresponding good reduction bound for points proven by the author for non-singular varieties to all projective varieties. Toward proving an absolute bound on the period for a given map, we study the canonical height of a subvariety via Chow forms and compute the bound between the height and canonical height of a subvariety. This gives the existence of a bound on the number of preperiodic rational subvarieties of bounded degree for a given map. An explicit bound is given for hypersurfaces.