I have a specific question about a geometric proof that the group law on elliptic curves is associative, as carried out in Fulton's Algebraic Curves (Section 5.6) or here.
The main thrust is to begin with a curve $\mathcal{C}$, and then to find two sets of 9 points (with 8 points in common) on 3 lines on this curve $\mathcal{C}$, to posit that each of these sets of points determines some elliptic curve that is not $\mathcal{C}$ (let us call these curves $\mathcal{C}'$ and $\mathcal{C}''$), and then to apply the Cayley-Bacharach Theorem to show that the 9th point on both $\mathcal{C}'$ and $\mathcal{C}''$ must be the same.
What I do not understand – surely Cayley – Bacharach applies only if $\mathcal{C}'$ and $\mathcal{C}''$ are distinct curves, and I do not understand why we are able to say that there MUST be another curve going through 9 points of $\mathcal{C}$ that is not equal to $\mathcal{C}$. Also, how is the proof valid once we recognize that the "different" curves $\mathcal{C}'$ and $\mathcal{C}''$, for which the only things we know is that they have the same nine points, share these same nine points in common?
Best Answer
I will try to isolate the idea of the proof of the associativity, hope this answers the unclear points. First, we are doing the following, when we use the Cayley-Bacharach link in the OP.
The definition of the sum:
We start with $O$, fixed (rational) point on some fixed cubic curve $$ E\subset \Bbb P^2\ , $$ (all spaces defined over a fixed field,) then consider two other points, $P,Q$, and use a specific receipt to define the point $P+Q$. The notations in loc. cit are rather irritating, and i will never use something like $PQ$ (for a point). The point P+Q is defined uniquely as in the "picture":
Here, points in triple joined in the picture through a line correspond to points on a line in the geometry (of the affine space where the elliptic curve also lives in). Now consider a further point $R$, we want to show: $$(P+Q)+R = P+(Q+R)\ .$$ This means the equality of the following points "in the middle"
X
andX'
of the diagrams:and
(Above, starting from
X
, and respectivelyX'
, we have to intersect the linesOX
andOX'
with the curve, the "third point" is then $(P+Q)+R$, and respectively $P+(Q+R)$.)So it is natural to show that both diagrams fit in the same picture:
The question is explicitly, if the points
* = X
and respectively* = X'
, constructed as follows starting with the eight points "on the margin", $A,P,A,R,B,Q+R,O,P+Q$ do coincide:The proof forgets now everything about $X,X'$, introduces a new point, $Y$, defined as the intersection of the lines $L_2$ and $M_2$. (A priori, this point may or may not lie on the elliptic curve. In the end, all three points $X,X',Y$ coincide.) We are now in the position now to apply in the generic case (eight distinct points) the theorem Cayley-Bacharach for the (degenerated) cubic curves $$ \begin{aligned} C_l &:& l_1l_2l_3&=0\ ,\\ C_m &:& m_1m_2m_3&=0\ , \end{aligned} $$ and the given elliptic curve $E$.The $8+1$ points are $A,P,A,R,B,Q+R,O,P+Q$ plus $Y$. It follows, that $Y$ is also on the cubic $E$. We get by construction $X=Y=X'$, the relation we wanted.
In case some of the points coincide, we have to use multiplicities, this leads to the solution from [Fulton, Algebraic Curves], a sort of intersection number (as part of an intersection theory) is needed.