examples of elliptic functions
Let Λ⊂ℂ be a lattice generated by
w1,w2. Let Λ∗ denote Λ-{0}.
-
1.
The Weierstrass ℘-function
is defined by the series
℘(z;Λ)=1z2+∑w∈Λ∗1(z-w)2-1w2 -
2.
The derivative of the Weierstrass ℘-function is also an elliptic function
℘′(z;Λ)=-2∑w∈Λ∗1(z-w)3 -
3.
The Eisenstein series of weight 2k for Λ is the series
𝒢2k(Λ)=∑w∈Λ∗w-2k The Eisenstein series of weight 4 and 6 are of special relevance in the theory of elliptic curves. In particular, the quantities g2 and g3 are usually defined as follows:
g2=60⋅𝒢4(Λ),g3=140⋅𝒢6(Λ)
Remark: The elliptic functions ℘, ℘′ and 𝒢2k are related by the following important equation:
(℘′(z;Λ))2=4℘(z;Λ)3-g2(Λ)℘(z;Λ)-g3(Λ) |
In particular, the previous equation provides an isomorphism between ℂ/Λ and the elliptic curve E:y2=4x3-g2x-g3 given by:
ℂ/Λ→E,z↦(℘(z;Λ),℘′(z;Λ)). |
Title | examples of elliptic functions |
---|---|
Canonical name | ExamplesOfEllipticFunctions |
Date of creation | 2013-03-22 13:54:04 |
Last modified on | 2013-03-22 13:54:04 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 7 |
Author | alozano (2414) |
Entry type | Example |
Classification | msc 33E05 |
Related topic | EllipticFunction |
Related topic | WeierstrassWpFunction |
Defines | Eisenstein series |