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?
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.