Number Theory: An Introduction to Mathematics

3 Cubic Curves 551

to be the projective line defined byaX+bY+cZ. It follows from Euler’s theorem on
homogeneous functions that (x,y,z) is itself a point of the tangent.
It is easily seen that ifC ̄is the projective completion of an affine plane curve
C,andifz=0, then (x,y,z) is a non-singular point ofC ̄if and only if (x/z,y/z)
is a non-singular point ofC. Moreover, if the tangent toC ̄at (x,y,z)isthe
projective line

( ̄X,Y,Z)=aX+bY+cZ,

then the tangent toCat (x/z,y/z) is the affine line defined by

(X,Y)=aX+bY+c.

LetCbe an affine plane curve over the fieldK, defined by the polynomialf(X,Y),
and let (x,y) be a non-singularK-point ofC. Then we can write

f(x+X,y+Y)=aX+bY+f 2 (X,Y)+···,

wherea,bare not both zero,f 2 (X,Y)is a homogeneous polynomial of degree 2, and
all unwritten terms have degree>2. The non-singular point (x,y)issaidtobean
inflection pointor, more simply, aflexofCiff 2 (X,Y)is divisible byaX+bY.
Similarly we can define a flex for a projective plane curve. Let (x,y,z) be a non-
singular point of the projective plane curve over the fieldK, defined by the homoge-
neous polynomialF(X,Y,Z). Then we can write

F(x+X,y+Y,z+Z)=aX+bY+cZ+F 2 (X,Y,Z)+···,

wherea,b,care not all zero,F 2 (X,Y,Z)is a homogeneous polynomial of degree 2,
and all unwritten terms have degree>2. The non-singular point (x,y,z)issaidtobe
aflexifF 2 (X,Y,Z)is divisible byaX+bY+cZ.
Two more definitions are required before we embark on our study of cubic
curves. A projective curve over the fieldK, defined by the homogeneous polynomial
F(X,Y,Z)of degreed>0, is said to be reducible overKif

F(X,Y,Z)=F 1 (X,Y,Z)F 2 (X,Y,Z),

whereF 1 andF 2 are homogeneous polynomials of degree less thandwith coefficients
fromK.TheK-points of the curve defined byFare then just theK-points of the curve
defined byF 1 , together with theK-points of the curve defined byF 2. A curve is said
to beirreducible over Kif it is not reducible overK.
Two projective curves over the fieldK, defined by the homogeneous polynomials
F(X,Y,Z)andG(X′,Y′,Z′), are said to beprojectively equivalentif there exists an
invertible linear transformation

X=a 11 X′+a 12 Y′+a 13 Z′

Y=a 21 X′+a 22 Y′+a 23 Z′

Z=a 31 X′+a 32 Y′+a 33 Z′