A finiteness theorem for canonical heights attached to rational maps over function fields

Abstract

Let K be a function field, let φ ∈ K(T ) be a rational map of degree d ≥ 2 defined over K, and suppose that φ is not isotrivial. In this paper, we show that a point P ∈ P(K) has φ-canonical height zero if and only if P is preperiodic for φ. This answers affirmatively a question of Szpiro and Tucker, and generalizes a recent result of Benedetto from polynomials to rational functions. We actually prove the following stronger result, which is a variant of the Northcott finiteness principle: there exists e > 0 such that the set of points P ∈ P(K) with φ-canonical height at most e is finite. Our proof is essentially analytic, making use of potential theory on Berkovich spaces to study the dynamical Green’s functions gφ,v(x, y) attached to φ at each place v of K. For example, we show that every conjugate of φ has bad reduction at v if and only if gφ,v(x, x) > 0 for all x ∈ P1Berk,v, where P 1 Berk,v denotes the Berkovich projective line over the completion of Kv. In an appendix, we use a similar method to give a new proof of the Mordell-Weil theorem for elliptic curves over K.