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

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 l-isogeny.