Last update: 11/08/07

Abstract

Let K be a function field in one variable over an arbitrary field. Let &phi be a rational function of degree at least 2 defined over K, and suppose that &phi is not isotrivial. In this paper, we show that a point P in P¹(K) has canonical height zero with respect to &phi if and only if P is preperiodic for &phi. 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 &epsilon > 0 such that the set of points P in P¹(K) whose canonical height with respect to &phi is at most &epsilon is finite. Our proof is essentially analytic, making use of potential theory on Berkovich spaces to prove some new results about the dynamical Green's functions g_&phi,v(x,y) attached to &phi at each place v of K. For example, we show that every conjugate of &phi has bad reduction at v if and only if g_&phi,v(x,x) > 0 for all x in the Berkovich projective line over the completion of the algebraic closure of K_v. As an application of our results, we give a new proof of the Mordell-Weil theorem for elliptic curves over K (under some simplifying hypotheses).