I'm trying t find good references to understand "why" the elliptic curves have a group structure. The canonical answer seems to be that:
- Every curve has a group associated to it, the picard group
- the elliptic curves' set of solutions in projective $\mathbb Q$ space is in set theoretic bijection with the Picard group.
- Hence, we pullback the group structure of the "correct object" (the Picard group) onto the elliptic curve itself, giving the "weird geometric group law" of:
draw a line between $P$ and $Q$, find third point of intersection $R$, reflect $R$ to $R'$. Define $P \cdot Q \equiv R'$.
Can I find an elementary exposition of this somewhere? Silverman expects one to know sophisticated algebraic geometry as far as I can tell. I found picard groups in Hartshorne as well, but it once again seems like quite a lot of effort to get to the idea of a picard group in the book.
Is there an elementary (read: undergrad who knows a first course in varieties/diffgeo/algebra/number theory/topology) source to understand the structure of the Picard group of the elliptic curve, and how it relates to the elementary group law one is taught?