New York Journal of Mathematics
Volume 22 (2016) 95-114


Gunther Cornelissen and Jonathan Reynolds

The perfect power problem for elliptic curves over function fields

view    print

Published: February 3, 2016
Keywords: Elliptic divisibility sequences, Siegel's theorem, perfect powers
Subject: 11G05, 11D41

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 \notin 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.


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.

Author information

Gunther Cornelissen:
Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, Nederland

Jonathan Reynolds:
INTO University of East Anglia, Norwich Research Park, Norwich, Norfolk, NR4 7TJ, United Kingdom