TITLE: Can you recommend some books on elliptic function?
QUESTION [18 upvotes]: I plan to study elliptic function.  Can you recommend  some  books? What is the relationship between elliptic function and elliptic curve？Many thanks in advance!

REPLY [8 votes]: First of all 

ever-modern  Course of modern analysis by Whittaker-Watson.

For a more introductory style, I highly recommend

V. Prasolov, Y. Solovyev Elliptic Functions and Elliptic Integrals. 

The relation between elliptic curves and elliptic functions can be sketched as follows. Elliptic curve is topologically a torus which can be realized by cutting a parallelogram in $\mathbb{C}$ and identifying its opposite edges. On the other hand, it can be realized  in $\mathbb{CP}^2$ by an algebraic equation of the form
$$y^2=x^3+ax+b.$$
Elliptic functions provide a map between the two pictures, which is also called uniformization. Essentially, $x,y$ are given by some elementary elliptic functions of $z$ (complex coordinate on the parallelogram).
Compare this with a more familiar example: trigonometric functions $\sin$, $\cos$ provide a uniformization of the circle, which can be defined either via an algebraic equation or in a parametric form:
$$x^2+y^2=1\quad \begin{array}{c}\sin,\cos\\ \Longleftrightarrow \\ \;\end{array}\quad \begin{cases}x=\cos t,\\ y=\sin t,\\ t\in[0,2\pi].\end{cases}$$