PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (192)
Orphanage
Unclass'd (1)
Unproven (490)
Corrections (5)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
[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:
$\displaystyle \mathbb{C}/\Lambda \to E, \quad z \mapsto (\wp(z;\Lambda),\wp'(z;\Lambda)).$



"examples of elliptic functions" is owned by alozano.
(view preamble)

View style:

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

Also defines:  Eisenstein series

This object's parent.
Log in to rate this entry.
(view current ratings)

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 3938 times total.

Classification:
AMS MSC: 33E05 (Special functions :: Other special functions :: Elliptic functions and integrals)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)