Quartic equations and 2-division on elliptic curves

摘要

Let K be a field of characteristic different from 2 and C an elliptic curveover K given by a Weierstrass equation. To divide an element of the group C by2, one must solve a certain quartic equation. We characterise the quarticsarising from this procedure and find how far the quartic determines the curveand the point. We find the quartics coming from 2-division of 2- and 3-torsionpoints, and generalise this correspondence to singular plane cubics. We usethese results to study the question of which degree 4 maps of curves can berealised as duplication of a multisection on an elliptic surface.