On the pre-image of a point under an isogeny
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.
Cite
@article{arxiv.0810.0092,
title = {On the pre-image of a point under an isogeny},
author = {Jonathan Reynolds},
journal= {arXiv preprint arXiv:0810.0092},
year = {2008}
}
Comments
4 pages, submitted