556 XIII Connections with Number Theory
where
gλ(X)= 4 λX^3 − 4 ( 1 +λ)X^2 + 4 Xis Riemann’s normal form andλ = 0 ,1. IfS(u)is the elliptic function defined in
§3 of Chapter XII, thenP(u)=(S(u),S′(u))is a point ofCλfor anyu∈C.Ifwe
define the sum ofP(u)andP(v)to be the pointP(u+v), then the set of allC-points
ofCλbecomes an abelian group, withP( 0 )=( 0 , 0 )as identity element and with
P(−u)=(S(u),−S′(u))as the inverse ofP(u). In order to carry this construction
over to the cubic curveCa,bandtootherfieldsthanC, we interpret it geometrically.
It was shown in (10) of Chapter XII that
S(u+v)= 4 S(u)S(v)[S(v)−S(u)]^2 /[S′(u)S(v)−S′(v)S(u)]^2.The points(x 1 ,y 1 )=(S(u),S′(u))and(x 2 ,y 2 )=(S(v),S′(v))determine the affine
line
Y−mX−c,where
m=[S′(v)−S′(u)]/[S(v)−S(u)],
c=[S′(u)S(v)−S′(v)S(u)]/[S(v)−S(u)].The third point of intersection of this line with the cubicCλis the point (x 3 ,y 3 ), where
x 3 =c^2 / 4 λx 1 x 2
=[S′(u)S(v)−S′(v)S(u)]^2 / 4 λS(u)S(v)[S(v)−S(u)]^2
= 1 /λS(u+v).On the other hand, the points( 0 , 0 ) =(S( 0 ),S′( 0 ))and(x∗ 3 ,y∗ 3 )= (S(u+v),
S′(u+v))determine the affine lineY−(y 3 ∗/x∗ 3 )Xand its third point of intersec-
tion withCλis the point(x 4 ,y 4 ),wherex 4 = 1 /λx∗ 3 =x 3. Evidentlyy 42 =y 32 ,andit
may be verified that actuallyy 4 =y 3. Thus(x∗ 3 ,y 3 ∗)is the third point of intersection
withCλof the line determined by the points( 0 , 0 )and(x 3 ,y 3 ).
The origin( 0 , 0 )may not be a point of the cubic curveCa,bbutO, the point at
infinity, certainly is. Consequently, as illustrated in Figure 2, we now define thesum
P 1 +P 2 of twoK-pointsP 1 ,P 2 ofCa,bto be theK-pointP 3 ∗,whereP 3 is the third
point ofCa,bon the line determined byP 1 ,P 2 andP 3 ∗is the third point ofCa,bon the
line determined byO,P 3 .IfP 1 =P 2 , the line determined byP 1 ,P 2 is understood to
mean the tangent toCa,batP 1.
It is simply a matter of elementary algebra to deduce from the formulas previously
given that, if addition is defined in this way, the set of allK-points ofCa,bbecomes
an abelian group, withOas identity element and with−P=(x,−y)as the inverse of
P=(x,y).Since−P=Pif and only ify=0, the elements of order 2 in this group
are the points(x 0 , 0 ),wherex 0 is a root of the polynomialX^3 +aX+b(if it has any
roots inK).
Throughout the preceding discussion of cubic curves we restricted attention to
those with a flex. It will now be shown that in a sense this is no restriction.