elliptic function
Let $\mathrm{\Lambda}\in \u2102$ be a lattice in the sense of number
theory^{}, i.e. a 2-dimensional free group^{} over $\mathbb{Z}$ which
generates $\u2102$ over $\mathbb{R}$.
An elliptic function^{} $\varphi $, with respect to the lattice $\mathrm{\Lambda}$, is a meromorphic^{} funtion $\varphi :\u2102\to \u2102$ which is $\mathrm{\Lambda}$-periodic^{}:
$$\varphi (z+\lambda )=\varphi (z),\forall z\in \u2102,\forall \lambda \in \mathrm{\Lambda}$$ |
Remark: An elliptic function which is holomorphic is constant. Indeed such a function^{} would induce a holomorphic function on $\u2102/\mathrm{\Lambda}$, 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 $\mathrm{\wp}$-function (see elliptic curve^{}) is an elliptic function, probably the most important. In fact:
Theorem 1.
The field of elliptic functions with respect to a lattice $\mathrm{\Lambda}$ is generated by $\mathrm{\wp}$ and ${\mathrm{\wp}}^{\mathrm{\prime}}$ (the derivative of $\mathrm{\wp}$).
Proof.
See [2], chapter 1, theorem 4. ∎
References
- 1 James Milne, Modular Functions^{} and Modular Forms^{}, 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 |