The perfect power problem for elliptic curves over function fields

Large classes of linear recurrent sequences contain only finitely many S-perfect powers >2 (i.e., numbers of the form sx^q for q>2 and s only divisible by primes from a finite set S). We study the analogue of this problem for rational points on elliptic curves over global function fields. More specifically, we prove that if E/K is a non-isotrivial elliptic curve over a global function field K of characteristic p>3 with j-invariant a p^s-power in K, and f a non-constant function on E with a pole of order N at the zero element of E, then there are only finitely many points P in E(K) such that for any valuation v not in S with v(f(P))<0, v(f(P)) is divisible by some integer not dividing p^sN.  This strengthens the function field analogue of Siegel’s theorem on S-integral points, due to Voloch (joint work with Jonathan Reynolds).