elliptic function

Let $\Lambda\in\mathbb{C}$ be a lattice in the sense of number theory, i.e. a 2-dimensional free group over ${\mathbb{Z}}$ which generates $\mathbb{C}$ over $\mathbb{R}$.

An $\phi$, with respect to the lattice $\Lambda$, is a meromorphic funtion $\phi:\mathbb{C}\to\mathbb{C}$ which is $\Lambda$-periodic:

 $\phi(z+\lambda)=\phi(z),\quad\forall z\in\mathbb{C},\quad\forall\lambda\in\Lambda$

Remark: An elliptic function which is holomorphic is constant. Indeed such a function would induce a holomorphic function on ${\mathbb{C}/\Lambda}$, 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 $\wp$-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 $\Lambda$ is generated by $\wp$ and $\wp^{\prime}$ (the derivative of $\wp$).

Proof.

See [2], chapter 1, theorem 4. ∎

References

• 1 James Milne, , 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, . 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