examples of elliptic functions

Let Λ be a latticeMathworldPlanetmath generated by w1,w2. Let Λ denote Λ-{0}.

  1. 1.

    The Weierstrass -functionMathworldPlanetmath is defined by the series

  2. 2.

    The derivative of the Weierstrass -function is also an elliptic functionMathworldPlanetmath

  3. 3.

    The Eisenstein series of weight 2k for Λ is the series


    The Eisenstein series of weight 4 and 6 are of special relevance in the theory of elliptic curves. In particular, the quantities g2 and g3 are usually defined as follows:


Remark: The elliptic functions , and 𝒢2k are related by the following important equation:


In particular, the previous equation provides an isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath between /Λ and the elliptic curve E:y2=4x3-g2x-g3 given by:

Title examples of elliptic functions
Canonical name ExamplesOfEllipticFunctions
Date of creation 2013-03-22 13:54:04
Last modified on 2013-03-22 13:54:04
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 7
Author alozano (2414)
Entry type Example
Classification msc 33E05
Related topic EllipticFunction
Related topic WeierstrassWpFunction
Defines Eisenstein series