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[parent] examples of elliptic functions (Example)

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. The Weierstrass $ \wp$-function is defined by the series
    $\displaystyle \wp(z;\Lambda)=\frac{1}{z^2}+\sum_{w\in\Lambda^{\ast}}\frac{1}{(z-w)^2}-\frac{1}{w^2}$
  2. The derivative of the Weierstrass $ \wp$-function is also an elliptic function
    $\displaystyle \wp'(z;\Lambda)=-2\sum_{w\in\Lambda^{\ast}}\frac{1}{(z-w)^3}$
  3. The Eisenstein series of weight $ 2k$ for $ \Lambda$ is the series
    $\displaystyle \mathcal{G}_{2k}(\Lambda)=\sum_{w\in\Lambda^{\ast}}w^{-2k}$
    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:
    $\displaystyle g_2=60\cdot\mathcal{G}_4(\Lambda),\quad g_3=140\cdot\mathcal{G}_6(\Lambda)$

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

$\displaystyle \left( \wp'(z;\Lambda) \right)^2 = 4 \wp(z;\Lambda)^3 - g_2(\Lambda) \wp(z;\Lambda) - g_3(\Lambda)$
In particular, the previous equation provides an isomorphism between $ \mathbb{C}/\Lambda$ and the elliptic curve $ E : y^2=4x^3-g_2x-g_3$ given by:
$\displaystyle \mathbb{C}/\Lambda \to E, \quad z \mapsto (\wp(z;\Lambda),\wp'(z;\Lambda)).$



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See Also: elliptic function, $\wp$-function

Also defines:  Eisenstein series

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Cross-references: isomorphism, equation, elliptic curves, theory, weight, elliptic function, derivative, series, generated by, lattice
There are 5 references to this entry.

This is version 4 of examples of elliptic functions, born on 2003-08-25, modified 2006-11-04.
Object id is 4649, canonical name is ExamplesOfEllipticFunctions.
Accessed 3908 times total.

Classification:
AMS MSC: 33E05 (Special functions :: Other special functions :: Elliptic functions and integrals)
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