We generalise the Siegel-Voloch theorem about S-integral points on elliptic curves as follows: let K/F denote a global function field over a finite field F of characteristic p≧ 5, let S denote a finite set of places of K and let E/K denote an elliptic curve over K with j-invariant j_E \notin K^p. Fix a function f ∈ K(E) with a pole of order N>0 at the zero of E. We prove that there are only finitely many rational points P ∈ E(K) such that for any valuation outside S for which f(P) is negative, that valuation of f(P) is divisible by some integer not dividing N. We also present some effective bounds for certain elliptic curves over rational function fields.

This work was completed whilst the authors enjoyed the hospitality of the University of Warwick (special thanks to Richard Sharp for making it possible) and during a visit of the first author to the Hausdorff Institute in Bonn.