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!
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1Take a look at [this paper](http://www.math.vt.edu/people/brown/doc/ellipses_not_ecs.pdf). It gives a brief history of elliptic curves including the answer to your second question. – John Douma May 29 '13 at 05:28
3 Answers
McKean and Moll have written the nice book Elliptic Curves: Function Theory, Geometry, Arithmetic that cleanly illustrates the connection between elliptic curves and elliptic/modular functions. If you haven't seen the book already, you should.
As for elliptic functions proper, my suggested books tend to be a bit on the old side, so pardon me if I don't know the newer treatments. Anyway, I quite liked Lawden's Elliptic Functions and Applications and Akhiezer's Elements of the Theory of Elliptic Functions. An oldie but goodie is Greenhill's classic, The Applications of Elliptic Functions; the notation is a bit antiquated, but I have yet to see another book that has a wide discussion of the applications of elliptic functions to physical problems.
At one time... every young mathematician was familiar with $\mathrm{sn}\,u$, $\mathrm{cn}\,u$, and $\mathrm{dn}\,u$, and algebraic identities between these functions figured in every examination.
– E.H. Neville
Finally, I would be remiss if I did not mention the venerable Abramowitz and Stegun, and the successor work, the DLMF. The chapters on the Jacobi and Weierstrass elliptic functions give a nice overview of the most useful identities, and also point to other fine work on the subject.
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I bought a copy of McKean and Moll based on this recommendation, and although it seems like a fine book, I would caution others that it is a graduate textbook and not at all easy to read. My background is that I have an undergraduate degree in math and a graduate degree in physics, and I reviewed my complex analysis before starting in on McKean and Moll. The book is written in a style that clearly assumes the reader is taking a graduate course in the subject, with the willingness to commit the implied amount of time. It's hard to find the main thread of the story they want to tell. – Sep 02 '18 at 16:35
First of all
 evermodern Course of modern analysis by WhittakerWatson.
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: $$x^2+y^2=1\quad \begin{array}{c}\sin,\cos\\ \Longleftrightarrow \\ \;\end{array}\quad \begin{cases}x=\cos t,\\ y=\sin t,\\ t\in[0,2\pi].\end{cases}$$
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I've forgotten to mention Prasolov/Solovyev; indeed, very nice! – J. M. ain't a mathematician May 29 '13 at 08:35

The link for Prasolov is broken. A link that works is https://www.amazon.com/FunctionsIntegralsTranslationsMathematicalMonographs/dp/0821805878/ref=cm_cr_arp_d_product_top?ie=UTF8 – Sep 02 '18 at 16:44
There is a classical 3 volume series by C.L. Siegel. It is wellwritten, though the perspective is a little bit outdated. I guess (no book at hand) you can find treatments by Serge Lang in the GTM series as well. I am not sure if Stein's book on complex analysis studied this topic.
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