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?
Is this clear to you ?
For an affine elliptic curve defined by $C:y^2=x^3+Ax+B$ then the line passing through $P,Q$ is defined by some equation $ax+by+c=0$ (assuming $P,Q$ are distinct otherwise we are considering the tangent line at $P$)
$ax+by+c$ is a rational function on the curve and its divisor (zeros & poles) on the projective closure (ie. $E=C\cup O$ where $O$ is the point at infinity) is $P+Q+R-3O$,
where $R$ is the 3rd point of the line.
So $P+ Q+R=3O$ in the Picard group $Pic(E)$, choosing $O$ as the neutral element gives $P+Q=-R=R'$ in $Pic(E)/\langle O\rangle$.
Where $R'=(x_R,-y_R)$ because the divisor of the rational function $x-x_R$ is $R+R'-2O$.