I'm interested in learning some stuff about elliptic curves. I've been learning scheme theory, and I'm interested in seeing these tools "in action". It seems that the standard introduction to elliptic curves is Silverman's book, which doesn't make use of schemes at all. So I'm curious, is there an introduction to elliptic curves from the point of view of schemes?
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6$\begingroup$ Chapter 2 of the book of Katz and Mazur on the Arithmetic of Elliptic Curves is a concise review of the most important facts. It does not cover as much as Silverman's book, though... $\endgroup$ – Matthieu Romagny Apr 5 '17 at 16:31

4$\begingroup$ Have you looked at "Arithmetic moduli of elliptic curves" by Katz and Mazur? There is a pdf at math.bu.edu/people/jsweinst/AWS/Files/…. You might or might not find it digestible. $\endgroup$ – Neil Strickland Apr 5 '17 at 16:32

$\begingroup$ @NeilStrickland I wasn't aware of this book, but it looks like a nice reference. Thanks $\endgroup$ – Alex Mathers Apr 5 '17 at 17:55

2$\begingroup$ Another good choice is Hida's "Geometric Modular Forms and Elliptic Curves" as it has a nice long chapter 1 explaining all the machinery. $\endgroup$ – Stiofáin Fordham Apr 5 '17 at 18:49

2$\begingroup$ Mumford's book on Abelian Varieties is from the scheme point of view. Elliptic Curves are just 1dimensional abelian varieties. (I think any good treatise on abelian varieties will be from the scheme point of view.) $\endgroup$ – Artur Jackson Apr 6 '17 at 1:03
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Not exactly a book, but there are course notes on abelian varieties from a course that Brian Conrad taught a few years ago. It is definitely from the perspective of scheme theory/functor of points.

$\begingroup$ Is this a redacted version of Mumford's book? It seems most topics and proofs are very similar. $\endgroup$ – Bombyx mori Apr 6 '17 at 4:40

5$\begingroup$ @Bombyxmori: The basic order of development is similar (it was the course text) but Picard schemes are used to give alternative proofs in some key places and descent theory is used more thoroughly to carry everything out over a general field from the outset (whereas Mumford works over an algebraically closed field; e.g., the proof of Poincare reducibility over a general field involves some issues that don't arise over an algebraically closed field). $\endgroup$ – nfdc23 Apr 6 '17 at 5:04
