 

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 


Abstract
We generalise the SiegelVoloch theorem about Sintegral 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 jinvariant 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.


Acknowledgements
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
g.cornelissen@uu.nl
Jonathan Reynolds:
INTO University of East Anglia, Norwich Research Park, Norwich, Norfolk, NR4 7TJ, United Kingdom
jonathan.reynolds@uea.ac.uk

