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 ( ¯ 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 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 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 ∈ P 1 Berk,v, where P1 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.

Abstract. In this paper we prove several Lehmer type inequalities for Drinfeld modules which will enable us to prove certain Mordell-Weil type structure theorems for Drinfeld modules. Key words: Lehmer conjecture, Mordell-Weil theorem, Drinfeld modules, Heights. 1.

Abstract. We present the details of a model theoretic proof of an analogue of the Manin-Mumford conjecture for semiabelian varieties in positive characteristic. 1.