# Reference: Elliptic curves as complex tori

I'm looking for books which contain a more or less self-contained description of how elliptic curves over $\mathbb{C}$ - that is, nonsingular plane cubic curves - can be realized as a quotient of the complex plane by some lattice.

I'm currently trying to get into Husemollers book on elliptic curves, but it is not detailed enough for my skill set and I like to have several sources at hand. Lecture notes are of course also welcome (I do have some background in complex analysis, but only up to the very basics of elliptic functions). Thanks!

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For elliptic curves over $\mathbb{C}$, you can look at Chapter 6 in Silverman's Arithmetic of Elliptic Curves. Also, in Diamond and Shurman's A first course in Modular forms, Chapter 2 (or is it Chapter 1?) covers the construction of elliptic curves as quotient of $\mathbb{C}$ by a lattice. Finally, Andrew Granville of Université de Montréal taught a course on elliptic curves and modular forms last semester (Fall 2011), and you can look at Chapter 14 in his online lecture notes.

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Very nice, thank you. I'll keep this open for a short while in case someone wants to add more recommendations, but I think these should be enough. –  Paul Apr 18 '12 at 12:29
@Paul By the way, Husemöller contains a lot of very important topics, but it is quite difficult to digest when first learning about elliptic curves. I would suggest putting it aside for a moment and reading Silverman instead. –  M Turgeon Apr 18 '12 at 12:50
If you're first learning about the topic, I'd go with a book by a Math.StackExchange user, Elliptic Curvers, Modular Forms, and Their L-Functions by Alvaro Lozano-Robledo. This is targeted at advanced undergrads and will give you an overview of the whole topic. –  Graphth Apr 18 '12 at 13:35
@Graphth I guess it depends on the meaning of "first learning", but this book looks great for the big-picture-without-too-many-proofs kind of first learning! –  M Turgeon Apr 18 '12 at 13:49

This is how Koblitz deals with elliptic curves in Introduction to Elliptic Curves and Modular Forms. But, Husemoller does NOT deal with elliptic curves in this fashion, at least not at first. And, since you say you're "trying to get into" his book, I assume you're at the beginning.

By the way, if you happen to be on a campus that has Springer link, you can search for elliptic curves, narrow the search to books, and you will find Silverman's book (i.e., you can legally download it for free). Search for modular forms and you'll find Diamond and Shimura's book as well as the 1-2-3 of Modular Forms, which doesn't deal with things in the way you are talking about but is a good book.

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+1 for the Springer link reference: My university has access to this, something I just found out. This is awesome. And yes, I just started with Husemoller's book and skimmed over it until I got to the part I needed, but it seemed way too terse for me. –  Paul Apr 18 '12 at 17:32

You may also want to have a look at Silverman's "Advanded Topics in the Arithmetic of Elliptic Curves". Chapter 1 is about elliptic and modular functions, and it works out many details about elliptic curves over $\mathbb{C}$, lattices, etc.

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