elliptic function

Let Λ be a lattice in the sense of number theoryMathworldPlanetmath, i.e. a 2-dimensional free groupMathworldPlanetmath over which generates over .

An elliptic functionMathworldPlanetmath ϕ, with respect to the lattice Λ, is a meromorphicPlanetmathPlanetmath funtion ϕ: which is Λ-periodicPlanetmathPlanetmath:


Remark: An elliptic function which is holomorphic is constant. Indeed such a functionMathworldPlanetmath would induce a holomorphic function on /Λ, which is compact (and it is a standard result from Complex Analysis that any holomorphic function with compact domain is constant, this follows from Liouville’s Theorem).

Example: The Weierstrass -function (see elliptic curveMathworldPlanetmath) is an elliptic function, probably the most important. In fact:

Theorem 1.

The field of elliptic functions with respect to a lattice Λ is generated by and (the derivative of ).


See [2], chapter 1, theorem 4. ∎


  • 1 James Milne, Modular FunctionsDlmfMathworld and Modular FormsMathworldPlanetmath, online course notes. http://www.jmilne.org/math/CourseNotes/math678.htmlhttp://www.jmilne.org/math/CourseNotes/math678.html
  • 2 Serge Lang, Elliptic Functions. Springer-Verlag, New York.
  • 3 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
Title elliptic function
Canonical name EllipticFunction
Date of creation 2013-03-22 13:47:03
Last modified on 2013-03-22 13:47:03
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 7
Author alozano (2414)
Entry type Definition
Classification msc 33E05
Related topic ExamplesOfEllipticFunctions
Related topic WeierstrassSigmaFunction
Related topic ModularDiscriminant
Related topic WeierstrassWpFunction
Related topic TableOfMittagLefflerPartialFractionExpansions
Related topic PeriodicFunctions