Trisections and cubes in elliptic curves
We show how to compute the pre-images of multiplication by 3 in the group of points of an elliptic curve E over a field k of characteristic different from 2 and 3 such that E(k) = E. We show Q = (xQ, yQ) ∈ E(k) if and only if yQ − d − m(xQ − c) is a cube in k for every 3-torsion point (c, d) ∈ E, where m is the slope of the tangent to E at (c, d). We reduce our problem to the computation of at most 2 independent cubic roots plus a few polynomial operations.
Journal or Serie
Journal of Pure and Applied Algebra, 2023, vol. 227, núm. 4, 107259