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I am currently doing a self study on Algebraic geometry but my ultimate goal is to study more on elliptic curves. Which are the most recommended textbooks I can use to study? I need something not so technical for a junior graduate student but at the same time I would wish to get a book with authority on elliptic curves. Thanks

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If you want to get into the number theoretic investigations, for a gentle introduction start with Cassels, "Lectures on elliptic curves". You can supplement that later with Knapp's "Elliptic Curves". After you have had a look at both, you can start reading Silverman's book. –  Anweshi Jul 24 '10 at 13:44

4 Answers 4

up vote 21 down vote accepted

Silverman and Tate to start, then Silverman, and finally Silverman again. These are basically canonical references for the subject.

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BTW, if you end up looking for a nice easy introduction to the relationship between modular forms and elliptic curves, then Knapp is pretty good: books.google.com/books?id=-e_qVoKF8H8C –  Steve Huntsman Mar 15 '10 at 22:01
You may also find McKean and Moll ( books.google.com/books?id=ovGVquPwo7oC ) has a nice flavor. But this should be considered as a complementary reference. McKean's books typically have somewhat unusual but extremely tasteful takes on subjects, and this is such a case. –  Steve Huntsman Mar 15 '10 at 22:06
And you might like the book of Cassels too. –  Marty Mar 15 '10 at 23:10
I sometimes feel that the elementary approach of Silverman obscures the arguments. If he had assumed that readers know more algebraic geometry and group cohomology, the result could have been made more readable. Of course this is a compromise to enlarge the user base. But it would be nice to have a kind of more highbrow Silverman. –  Andrea Ferretti Jul 21 '10 at 15:53
A saying in Bonn is "Silverman hat durchsichtige Ohren" - "Silverman has transparent ears.", a weird pejorative phrase which references to the fact that he does not use scheme theory, which not only obscures proofs, but at places makes them even uncomplete (eg the additivity of the dual isogeny if I remember correctly). But all in all it is, of course, a fine book, which has concrete formulas, but is taken a mild algebraic geometric view. –  Lennart Meier Jul 21 '10 at 17:35

The other book suggestions are all so far excellent; the only caveat with them is that they all get into the number theoretic aspects very soon. I am taking the guess that you are more geometrically minded since you are starting with algebraic geometry rather than with number theory.

Also I am taking the guess that you are reading algebraic geometry from the standard book of Hartshrone. I assume you are reading the first chapter.

My advise to you would be to first understand affine and projective varieties as given in Chap I. of Hartshorne, and then move straight ahead to chapter IV on algebraic curves. You would have to take a few things like the Riemann-Roch theorem(rather, Serre duality theorem) for granted and you would have to replace any occurrence of "scheme" with variety, and there may be a few gaps. I suggest that you ignore these and read it. This will give you a very solid and rather modern introduction into the subject algebraic curves, and to elliptic curves in particular.

Afterwards you can go back to chaps. II and III and read the theory of schemes and the machinery of sheaf cohomology, if you wish to further pursue algebraic geometry.

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I highly recommend Elliptic Curves by Alain Robert. It is very clearly written, has few prerequisites, yet brings the reader straight into the connection between the complex analytic side of algebraic curves and the algebro-geometric side. It eventually discusses $p$-adic curves and their relation to $p$-adic analytic functions, as well as using these to prove the main theorem of complex multiplication (that $j(\tau)$ is an algebraic integer).

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The books by Silverman can't be beat and I won't simply repeat what's been said below.

Adding to the books already mentioned, Lawrence Washington's recent text is supposed to be excellent, but I haven't seen it yet.

Miles Ried also wrote a beautiful set of lecture notes that was used at Cambridge for many years-they may or may not still be available online for download.

And of course no introduction to algebraic geometry through elliptic curves would be complete without mentioning the classic introduction to algebraic curves by William Fulton, which is available online free for download by googling it.

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