nLab Weierstrass elliptic function

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.

Related concepts

• elliptic curve

• elliptic function

• pillowcase orbifold

References

Named after Karl Weierstrass.

Lecture notes:

• Motohico Mulase, Section 1.3 of: Lectures on the combinatorial structure of the moduli spaces of Riemann surfaces, 2004 (pdf)

See also

• Wikipedia, Weierstrass’s elliptic functions