Number Theory: An Introduction to Mathematics

3 Cubic Curves 553

If K has characteristic= 2 , 3 , then it is projectively equivalent to the projective
completionC=Ca,bof an affine curve of the form

Y^2 −(X^3 +aX+b).

It is easily seen that, conversely, for any choice ofa 1 ,...,a 6 ∈Kthe curveW,and
in particularCa,b, is irreducible overKand that 0 , the unique point at infinity, is a flex.
For anyu,r,s,t∈Kwithu=0, the invertible linear change of variables

X=u^2 X′+r,

Y=u^3 Y′+su^2 X′+t

replaces the curveW=W(a 1 ,...,a 6 )by a curveW′=W′(a′ 1 ,...,a 6 ′)of the same
form. The numbering of the coefficients reflects the fact that ifr=s=t=0, then

a 1 =ua′ 1 , a 2 =u^2 a′ 2 , a 3 =u^3 a 3 ′,

a 4 =u^4 a′ 4 , a 6 =u^6 a 6 ′.

In particular, for any nonzerou∈K, the invertible linear change of variables

X=u^2 X′,

Y=u^3 Y′

replacesCa,bbyCa′,b′,where

a=u^4 a′,

b=u^6 b′.

By replacingXbyx+XandYbyy+Y, we see that if aK-point (x,y)ofCa,b
is singular, then

3 x^2 +a=y= 0 ,

which implies 4a^3 + 27 b^2 = 0. Thus the curveCa,bhas no singular points if
4 a^3 + 27 b^2 =0.
We will call

d:= 4 a^3 + 27 b^2

thediscriminantof the curveCa,b. It is not difficult to verify that if the cubic polyno-
mialX^3 +aX+bhas rootse 1 ,e 2 ,e 3 ,then

d=−[(e 1 −e 2 )(e 1 −e 3 )(e 2 −e 3 )]^2.

Ifd = 0,a = 0, then the polynomialX^3 +aX+bhas the repeated root
x 0 =− 3 b/ 2 aandP =(x 0 , 0 )is the unique singular point. Ifd=a =0, then
b=0andP=( 0 , 0 )is the unique singular point.