nLab
Weierstrass elliptic function

Contents

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 11 and τ\tau satisfying Im(τ)>0Im(\tau) \gt 0, so that (z)=(z+1)\wp(z) = \wp(z + 1) and (z+τ)=(z)\wp(z + \tau) = \wp(z)) that exhibits an explicit parametrization of the form

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

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

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

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

See at elliptic curve and at Möbius transformation for more.

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

Last revised on July 28, 2020 at 12:36:47. See the history of this page for a list of all contributions to it.