Elliptic Curves
Lattices, Complex Tori and the Weierstrass p-Function
The identification of elliptic curves over the complex numbers with complex tori, and the Weierstrass function that realises it.
Engineering / MathematicsElliptic Curves2 min readKV-MATH-0637
Over the complex numbers, every elliptic curve is the quotient of the plane by a lattice. This identification is the source of the group law and of complex multiplication.
The correspondence
The Weierstrass function
The isomorphism is realised by the Weierstrass function, constructed as a sum over the lattice designed to converge while remaining doubly periodic.
Together with its derivative it satisfies a cubic relation, which is the Weierstrass equation of the corresponding curve. The lattice invariants determine the curve coefficients.
Lattices and curves
| Lattice notion | Curve notion |
|---|---|
| Lattice | Elliptic curve over C |
| Homothetic lattices | Isomorphic curves |
| Lattice invariants | Curve coefficients |
| Ratio of periods | Point in the upper half plane |
| Sublattice of finite index | Isogeny |
Computing periods
Given a curve over the rationals or reals, its periods are computed by the arithmetic-geometric mean, which converges quadratically and is the standard method.
What the correspondence gives
- A conceptual proof of the group law and its associativity.
- The classification of endomorphism rings — see isogenies.
- Complex multiplication theory, which links curves to imaginary quadratic fields — see complex multiplication.
- Practical algorithms for locating points and computing heights.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 7.2.1. 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.