3 Cubic Curves 555X=T^2 ,Y=T^3. In the node case, if we putT=Y/(X+ 3 b/ 2 a),thenithasthe
parametrization
X=T^2 + 3 b/a,Y=T^3 + 9 bT/ 2 a.Thus in both cases the cubic curve is in fact elementary.
We now restrict attention to non-singular cubic curves, i.e. curves which do not
have a singular point.
TwoK-points of a projective cubic curve determine a projective line, which inter-
sects the curve in a thirdK-point. This procedure for generating additionalK-points
was used implicitly by Diophantus and explicitly by Newton. There is also another
procedure, which may be regarded as a limiting case: the tangent to a projective cubic
curve at aK-point intersects the curve in anotherK-point. The combination of the
two procedures is known as the ‘chord and tangent’ process. It will now be described
analytically for the cubic curveCa,b.
IfOis the unique point at infinity of the cubic curveCa,band ifP=(x,y)is any
finiteK-point, then the affine line determined byOandPisX−xand its other point
of intersection withCa,bisP∗=(x,−y).
Now letP 1 =(x 1 ,y 1 )andP 2 =(x 2 ,y 2 )be any two finiteK-points. Ifx 1 =x 2 ,
then the affine line determined byP 1 andP 2 is
Y−mX−c,where
m=(y 2 −y 1 )/(x 2 −x 1 ), c=(y 1 x 2 −y 2 x 1 )/(x 2 −x 1 ),and its third point of intersection withCa,bisP 3 =(x 3 ,y 3 ),where
x 3 =m^2 −x 1 −x 2 , y 3 =mx 3 +c.Ifx 1 =x 2 ,buty 1 =y 2 , then the affine line determined byP 1 andP 2 isX−x 1 and its
other point of intersection withCa,bisO. Finally, ifP 1 =P 2 , it may be verified that
the tangent toCa,batP 1 is the affine line
Y−mX−c,where
m=( 3 x^21 +a)/ 2 y 1 , c=(−x 13 +ax 1 + 2 b)/ 2 y 1 ,and its other point of intersection withCa,bis the pointP 3 =(x 3 ,y 3 ),wherex 3 and
y 3 are given by the same formulas as before, but with the new values ofmandc(and
withx 2 =x 1 ).
It is rather remarkable that theK-points of a non-singular projective cubic curve
can be given the structure of an abelian group. That this is possible is suggested by the
addition theorem for elliptic functions.
Suppose thatK=Cis the field of complex numbers and that the cubic curve is
the projective completionCλof the affine curve
Y^2 −gλ(X),