# examples of elliptic functions

Let $\Lambda\subset\mathbb{C}$ be a lattice  generated by $w_{1},w_{2}$. Let $\Lambda^{\ast}$ denote $\Lambda-\{0\}$.

1. 1.

The Weierstrass $\wp$-function  is defined by the series

 $\wp(z;\Lambda)=\frac{1}{z^{2}}+\sum_{w\in\Lambda^{\ast}}\frac{1}{(z-w)^{2}}-% \frac{1}{w^{2}}$
2. 2.

The derivative of the Weierstrass $\wp$-function is also an elliptic function  $\wp^{\prime}(z;\Lambda)=-2\sum_{w\in\Lambda^{\ast}}\frac{1}{(z-w)^{3}}$
3. 3.

The Eisenstein series of weight $2k$ for $\Lambda$ is the series

 $\mathcal{G}_{2k}(\Lambda)=\sum_{w\in\Lambda^{\ast}}w^{-2k}$

The Eisenstein series of weight $4$ and $6$ are of special relevance in the theory of elliptic curves. In particular, the quantities $g_{2}$ and $g_{3}$ are usually defined as follows:

 $g_{2}=60\cdot\mathcal{G}_{4}(\Lambda),\quad g_{3}=140\cdot\mathcal{G}_{6}(\Lambda)$

Remark: The elliptic functions $\wp$, $\wp^{\prime}$ and $\mathcal{G}_{2k}$ are related by the following important equation:

 $\left(\wp^{\prime}(z;\Lambda)\right)^{2}=4\wp(z;\Lambda)^{3}-g_{2}(\Lambda)\wp% (z;\Lambda)-g_{3}(\Lambda)$

In particular, the previous equation provides an isomorphism      between $\mathbb{C}/\Lambda$ and the elliptic curve $E:y^{2}=4x^{3}-g_{2}x-g_{3}$ given by:

 $\mathbb{C}/\Lambda\to E,\quad z\mapsto(\wp(z;\Lambda),\wp^{\prime}(z;\Lambda)).$
Title examples of elliptic functions ExamplesOfEllipticFunctions 2013-03-22 13:54:04 2013-03-22 13:54:04 alozano (2414) alozano (2414) 7 alozano (2414) Example msc 33E05 EllipticFunction WeierstrassWpFunction Eisenstein series