## 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^{1} () 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 *ε* > 0 such that the set of points *P* ∈ ℙ^{1} (*K*) with *φ*-canonical height at most *ε* is finite. Our proof is essentially analytic, making use of potential theory on Berkovich spaces to study the dynamical Green's functions *g*
_{φ, υ} (*x, y*) attached to *φ* at each place *υ* of *K*. For example, we show that every conjugate of *φ* has bad reduction at *υ* if and only if *g*
_{φ, υ} (*x, x*) > 0 for all , where denotes the Berkovich projective line over the completion of
_{υ}. In an appendix, we use a similar method to give a new proof of the Mordell-Weil theorem for elliptic curves over *K*.

Received: 2006-05-31Revised: 2007-10-22Published Online: 2009-01-08Published in Print: 2009-01-01Citation Information:Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2009, Issue 626, Pages 205–233, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/CRELLE.2009.008, January 2009