New York Journal of Mathematics
Volume 17 (2011) 163-172


Jonathan Reynolds

On the pre-image of a point under an isogeny and Siegel's theorem

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

Consider a rational point on an elliptic curve under an isogeny. Suppose that the action of Galois partitions the set of its pre-images 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 l-isogeny.


The author is supported by a Marie Curie Intra European Fellowship (PIEF-GA-2009-235210)

Author information

Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, Nederland