Quartic equations and 2-division on elliptic curves

TL;DR

This paper characterizes quartic equations arising from halving points on elliptic curves over fields with characteristic not 2, exploring their properties, relation to torsion points, and applications to maps of elliptic surfaces.

Contribution

It provides a detailed characterization of quartics from 2-division on elliptic curves and extends the analysis to singular cubics and elliptic surface mappings.

Findings

Quartics from 2-division are characterized and classified.

Explicit forms of quartics from 2- and 3-torsion points are derived.

Connections between quartics and elliptic surface maps are established.

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.