 

Jonathan Reynolds
On the preimage of a point under an isogeny and Siegel's theorem view print


Published: 
March 3, 2011 
Keywords: 
Isogeny; elliptic curve; Siegel's theorem 
Subject: 
11G05, 11A51 


Abstract
Consider a rational point on an elliptic curve under an isogeny. Suppose that the action of Galois partitions the set of its preimages into n orbits. It is shown that all but finitely many such points have their denominator divisible by at least n distinct primes. This generalizes Siegel's theorem and more recent results of Everest et al. For multiplication by a prime l, it is shown that if n>1 then either the point is l times a rational point or the elliptic curve admits a rational lisogeny.


Acknowledgements
The author is supported by a Marie Curie Intra European Fellowship (PIEFGA2009235210)


Author information
Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, Nederland
J.M.Reynolds@uu.nl

