A finiteness theorem for canonical heights attached to rational maps   over function fields
  Let K be a function field, let f be a rational function of degree d at least 2 defined over K, and suppose that f is not isotrivial. In this paper, we show that a point P in P^1(Kbar) has f-canonical height zero if and only if P is preperiodic for f. 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 P^1(K) with f-canonical height 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_v(x,y) attached to f at each place v of K. For example, we show that every conjugate of f has bad reduction at v if and only if g_v(x,x) > 0 for all x in the Berkovich projective line over the completion of the algebraic closure of K_v. In an appendix, we show how a similar method can be used to give a new proof of the Mordell-Weil theorem for elliptic curves over K. 
Matthew Baker