Quartic equations and 2-division on elliptic curves
Algebraic Geometry
2007-07-02 v1
Abstract
Let K be a field of characteristic different from 2 and C an elliptic curve over K given by a Weierstrass equation. To divide an element of the group C by 2, one must solve a certain quartic equation. We characterise the quartics arising from this procedure and find how far the quartic determines the curve and the point. We find the quartics coming from 2-division of 2- and 3-torsion points, and generalise this correspondence to singular plane cubics. We use these results to study the question of which degree 4 maps of curves can be realised as duplication of a multisection on an elliptic surface.
Keywords
Cite
@article{arxiv.0706.4379,
title = {Quartic equations and 2-division on elliptic curves},
author = {George H. Hitching},
journal= {arXiv preprint arXiv:0706.4379},
year = {2007}
}