Good reduction and canonical heights of subvarieties

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 the bound between the height and canonical height of a subvariety via Chow forms. 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.Theorem. (Theorem 3.8) Let K be a number field. Let f : P N → P N be a morphism of degree d defined over K. Let X ⊂ P N defined over K be an irreducible periodic subvariety of degree D and codimension t for f with minimal period n. Let p ∈ K be a prime of good reduction for f . If deg(f ℓ (X)) = deg(f ℓ (X)) for 0 ≤ ℓ ≤ n, then there exists a constant C depending only on d, D, N, t, p such that n ≤ C (d, D, N, t, p).The basic idea of the method is to move the problem to an endomorphism of a component of the Chow variety and appeal to the similar bound for points from the author's previous work [16]. The restriction on the degree can be thought of as primes of good reduction for the subvariety and can only not be satisfied for finitely many primes for a given periodic subvariety. However, this is probably not quite enough for

On Dynamical Cancellation

“…We summarize the basic properties of such heights in the next subsection. Standard references for canonical heights of subvarieties are [7,13].…”