# Contents

## Idea

The Weierstrass elliptic function $\wp$ is a doubly periodic meromorphic function on the complex numbers $\mathbb{C}$ (with the periods typically normalized to $1$ and $\tau$ satisfying $Im(\tau) \gt 0$, so that $\wp(z) = \wp(z + 1)$ and $\wp(z + \tau) = \wp(z)$) that exhibits an explicit parametrization of the form

$(\wp, \wp'): \mathbb{C}/L \to C$

where $C \subset \mathbb{P}^2(\mathbb{C})$ is the set of solutions to the cubic Weierstrass equation, and $L \subset \mathbb{C}$ is the lattice $\langle 1, \tau \rangle$. In other words, we have a cubic relation of type

$(\wp')^2 = 4\wp^3 + a\wp + b$

for some constants $a, b$, providing an explicit parametrization of an elliptic curve (a nonsingular projective cubic curve $C$ considered over $\mathbb{C}$) by a complex torus $\mathbb{C}/L$.

See at elliptic curve and at Möbius transformation for more.

## References

Named after Karl Weierstrass.

Last revised on April 17, 2017 at 06:42:38. See the history of this page for a list of all contributions to it.