Elliptic Integrals and Elliptic Functions

Elliptic curves are named for elliptic integrals, which arose from computing arc length on an ellipse. Inverting those integrals produces doubly periodic functions, and the curve is the image of that inversion.

The integrals

Arc length on an ellipse, and on the lemniscate, leads to integrals of a rational function divided by the square root of a cubic or quartic. These cannot be evaluated in elementary terms.

Inversion

The decisive step, due to Abel and Jacobi, was to invert the integral rather than evaluate it. The inverse function turns out to be doubly periodic in the complex plane.

Elliptic functions

Doubly periodic

Invariant under translation by two independent complex periods.

Meromorphic

Analytic except for poles. A non-constant elliptic function must have poles, by Liouville's theorem.

The field of elliptic functions

Generated by the Weierstrass function and its derivative, which satisfy a cubic relation — the curve equation.

The connection to the curve

Why this matters computationally

The analytic picture supplies practical algorithms: periods are computed by the arithmetic-geometric mean, points are located via the torus, and complex multiplication theory rests entirely on this correspondence — see computing over the complexes.

Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 7.1.2. 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.