examples of elliptic functions

Examples of Elliptic Functions

Let $\Lambda \subset \Complex$ be a lattice generated by $w_1,w_2$ . Let $\Lambda^{\ast}$ denote $\Lambda-\{ 0 \}$ .

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. The derivative of the Weierstrass $\wp$ -function is also an elliptic function $$\wp'(z;\Lambda)=-2\sum_{w\in\Lambda^{\ast}}\frac{1}{(z-w)^3}$$
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 $\wp$ , $\wp'$ and $\mathcal{G}_{2k}$ are related by the following important equation:$$\left( \wp'(z;\Lambda) \right)^2 = 4 \wp(z;\Lambda)^3 - g_2(\Lambda) \wp(z;\Lambda) - g_3(\Lambda$$ In particular, the previous equation provides an isomorphism between $\Complex/\Lambda$ and the elliptic curve $E : y^2=4x^3-g_2x-g_3$ given by: $$\Complex/\Lambda \to E, \quad z \mapsto (\wp(z;\Lambda),\wp'(z;\Lambda)).$$