examples of elliptic functions

Examples of Elliptic Functions

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

1. The Weierstrass $\mathrm{\wp }$-function is defined by theseries

   $$\mathrm{\wp }(z;\mathrm{\Lambda })=\frac{1}{z^2}+\sum _{w\in {\mathrm{\Lambda }}^{\ast }}\frac{1}{{(z-w)}^2}-\frac{1}{w^2}$$

2. The derivative of the Weierstrass $\mathrm{\wp }$-function is also anelliptic function

   $${\mathrm{\wp }}^{\prime }(z;\mathrm{\Lambda })=-2\sum _{w\in {\mathrm{\Lambda }}^{\ast }}\frac{1}{{(z-w)}^3}$$

3. The Eisenstein series of weight $2k$ for $\mathrm{\Lambda }$ isthe series

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

   The Eisenstein series of weight $4$ and $6$ are of specialrelevance in the theory of elliptic curves. In particular, the quantities $g_2$ and $g_3$ are usually defined as follows:

   $$g_2=60\cdot {𝒢}_4(\mathrm{\Lambda }),g_3=140\cdot {𝒢}_6(\mathrm{\Lambda })$$

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

$${\left({\mathrm{\wp }}^{\prime }(z;\mathrm{\Lambda })\right)}^2=4\mathrm{\wp }{(z;\mathrm{\Lambda })}^3-g_2(\mathrm{\Lambda })\mathrm{\wp }(z;\mathrm{\Lambda })-g_3(\mathrm{\Lambda })$$

In particular, the previous equation provides an isomorphism between $ℂ/\mathrm{\Lambda }$ and the elliptic curve $E:y^2=4x^3-g_{2}x-g_3$ given by:

$$ℂ/\mathrm{\Lambda }\to E,z\mapsto (\mathrm{\wp }(z;\mathrm{\Lambda }),{\mathrm{\wp }}^{\prime }(z;\mathrm{\Lambda })).$$