Elliptic Curves
Computing with Elliptic Curves over C
Practical computation with elliptic curves over the complex numbers: periods, the torus map, and heights.
Engineering / MathematicsElliptic Curves2 min readKV-MATH-0643
Analytic methods over the complex numbers supply practical tools for curves over the rationals: periods, elliptic logarithms, and the real-place contribution to heights.
Computing periods
The period lattice is computed by the arithmetic-geometric mean, which converges quadratically. The real period and the complex one follow from the roots of the cubic.
Computing the period lattice
- Find the rootsOf the cubic defining the curve, numerically — see root finding.
- Apply the AGMTo appropriate combinations of the roots.
- Assemble the latticeThe periods follow from the limits.
The elliptic logarithm
Mapping a point on the curve back to the torus is the elliptic logarithm. It is the inverse of the Weierstrass parametrisation and is computed by numerical integration or by the AGM.
Heights
- Naive height
- Measures the size of the coordinates. Easy to compute, not quite additive under the group law.
- Canonical height
- A quadratic form on the Mordell-Weil group, obtained by a limiting process from the naive height. Genuinely quadratic.
- Local decomposition
- The canonical height decomposes into contributions from each place, with the real place computed analytically.
Finding points
Searching for rational points is guided by height bounds. Points of bounded canonical height lie in a bounded region of the torus, which converts the search into a lattice point enumeration — see short vector search.
Precision
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 7.4.1. Structural reference unverified: the source file was not available during authoring; chapter and section numbers are taken from the published edition and have not been checked against a physical copy.