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# 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 elliptic function $\phi$, 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, Modular Functions and Modular Forms, online course notes. http://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.

## Mathematics Subject Classification

33E05*no label found*

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