On the pre-image of a point under an isogeny

TL;DR

This paper investigates the field of definition of pre-images of rational points under isogenies on curves, proving that for m=2 the existence of a rational pre-image always coincides with a quadratic factor, and exploring the case m=3.

Contribution

It confirms that for m=2, the existence of a rational pre-image always corresponds to a quadratic factor, and analyzes the m=3 case, identifying conditions for counterexamples.

Findings

For m=2, rational pre-images always exist when quadratic factors are present.

For m=3, counterexamples only occur when a rational point of order three exists.

Results hold over fields with characteristic not two or three.

Abstract

Given a rational point on a curve in a rational isogeny class, a natural question concerns the field of definition of its pre-images. The multiplication by m endomorphism is a powerful and much-used tool. The pre-images for this map are found by factorizing a monic polynomial of degree m^2. For m = 2, Everest and King gave examples where the existence of a quadratic factor coincided with the existence of a rational pre-image via a 2-isogeny. Nelson Stephens asked if this always happens and the question is answered in the affirmative. It is also shown that the analogue for m = 3 can only be false when there exists a rational point of order three and a small number of counterexamples are found. The results are proven over any field with characteristic not two or three.