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

TL;DR

This paper generalizes Siegel's theorem by analyzing the prime divisors of denominators of pre-images of rational points under elliptic curve isogenies, revealing new conditions related to Galois orbits and rational isogenies.

Contribution

It extends Siegel's theorem to pre-images under isogenies, linking Galois orbit structure to prime divisors and rational isogenies, providing new insights into elliptic curve rational points.

Findings

Pre-images have denominators divisible by at least n distinct primes.

If n>1 under multiplication by a prime l, the point is either an l-multiple or the curve has a rational l-isogeny.

Generalizes previous results of Everest et al. on elliptic curves.

Abstract

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 such points above a certain height 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 emits a rational l-isogeny.