The perfect power problem for elliptic curves over function fields

Abstract

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 jE ∉ Kp. 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.