# What are the recommended books for an introductory study of elliptic curves?

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

• 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

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

• 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
• As someone who's never really done more than glance at a few early sections of Hartshorne, I am personally glad for that approach. I think that elliptic curves and toric varieties are two of the best topics for motiviating AG and that an approach that takes this into account is good for AG. Indeed, this is the only thing that has caused me (and probably others like me with an applied bent) to even contemplate learning about schemes. – Steve Huntsman Jul 21 '10 at 18:01

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 Hartshorne. I assume you are reading the first chapter.

My advice 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.

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).

Elliptic curves lie at the intersection of several areas of mathematics which have been approached in different ways. So the problem in learning the subject is deciding which approach to adopt.

Lawrence Washington's book is very approachable and doable, written by a serious mathematician with the novice in mind, and it is realistic to expect that it can be worked out.

Knapp's notes are very readable, and after the outlines of the subject are familiar furnish enrichment and orientation-they are compulsively readable-and can mislead one into thinking that his notes can be easily mastered!

Silverman's two volumes are the definitive treatment of the arithmetic aspects of the subject for the average researcher. As to the quibbles about scheme theory being omitted in his treatment, one has to ask oneself what one is learning the subject for. As Silverman remarks, it is easy to write down a particular elliptic curve and pursue interesting arithmetic research questions without using scheme theory.

One could view Mordell's finite generation theorem as either being "elementary" or "interesting". Both points of view have a measure of truth in them. This is a foundational theorem in the subject, and its proof requires no more than high school mathematics! This is of course a testament to the fact that elliptic curves are amazing objects. Not sure Andrew Wiles used scheme theory in his proof of FLT. He may well have relied on deep results obtained using scheme theory, but that does not make his proof less valid.

Another example is Mazur's theorem on the torsion subgroup for elliptic curves over Q. The result is very useful, even if we do not understand its proof, it is easy to apply. Very few of us have the time to learn algebraic geometry in its rigorous modern formulation. Silverman remarks on this matter in his introduction. One has to view the study of this subject as a prerequisite to engaging in research which really is the point of any serious study.

I have been lucky enough to write and publish papers on the subject with very limited knowledge. Elliptic curves are deep mathematical objects especially when viewed from an arithmetic perspective, but interesting problems can be pursued with modest equipment. So unless one wants to be an algebraic geometer, it is possible to pursue arithmetic questions as topics for research without knowing about schemes.

I have found myself referring to Silverman's books and to Knapp's notes when the "chips were down" and I was tackling research problems. My problems may have been superficial, but the results were published - so there must be a community that finds them interesting. Maybe I am slow, but I am leary of advice that gives a sequence of books to be read for learning something. I suspect most of us will read very few books thoroughly - and skimming a mathematics book doesn't seem like a useful pursuit to me.

As Goethe (?) said, "Art is long. Life is short. Decisions are difficult, and opportunities are fleeting." My two cents for whatever it is worth.

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 Reid 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.

If you are also interested in exercises on elliptic curves with solutions I recommend the book "Computational Commutative Algebra and Algebraic Geometry: Course and exercises with detailed solutions" by Yengui, see https://www.amazon.com/dp/1096374447?ref_=pe_3052080_397514860