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66 3. Elliptic Curves and Their Isomorphisms

(c) L(k ) ∩ C(k ) ={P}, one point where i(P; L, C) = 3 and in this case

PP = P and P is either a point of inflection or a singular point.
In particular, the chord-tangent composition is a function C(k) × C(k) → C(k).
(1.2) Theorem. Let C be a nonsingular cubic curve defined over a field k, and let
O be a point on C(k). Then the law of composition defined by P + Q = O(PQ) on
C(k) makes C(k) into an abelian groupwith O as zero element and −P = P(OO).
Moreover, O is a point of inflection if and only if P + Q + R = 0 whenever P, Q,
and R are the three intersection points of C with a line. In this case we also have
−P = PO and OO = O.

Proof. The above group law was introduced in §5 of the Introduction. All of the
group axioms were considered except for the associative law, and, in fact, they are
immediate from the definition. The statement about O being an inflection point was
also considered in the Introduction.
The associativity relation would follow if we knew that P(Q + R) = (P + Q)R
since composing this with O yields our composition law.
We begin with the case where P, Q,and R are distinct. To form P(Q + R),we
find first QR, join that to O, and take the third intersection point Q + R. Now join
Q + R to P, which gives the point P(Q + R), and we need to show that it is the
same as (P + Q)R. In the figure each of the points O, P, Q, R, PQ, P + Q, QR,
and Q + R lies on one dotted line and one solid line.

We have nine points, O, P, Q, R, PQ, P + Q, QR, Q + R, and the intersection
T of the line joining P to Q + R and the line joining P + Q to R. We must show
that T is on the cubic.
For this, observe that there are two degenerate cubics which go through the nine
points, namely the union C 1 of the three dotted lines and the union C 2 of the three