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?
Best Answer
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 $R+R'-2O$.