Number Theory: An Introduction to Mathematics

3 Cubic Curves 557

y

x

P

P

P*=P +P

0

1

2

3

3 12

P

Fig. 2.Addition onCa,b.

LetCbe a projective cubic curve over the fieldK, defined by the homogeneous
polynomialF 1 (X,Y,Z), and suppose thatChas a non-singularK-pointP. Without
loss of generality we assume thatP=( 1 , 0 , 0 )and that the tangent atPis the projec-
tive lineZ.ThenF 1 has no term inX^3 or inX^2 Y:

F 1 (X,Y,Z)=aY^3 +bY^2 Z+cY Z^2 +dZ^3 +eX^2 Z+gXY^2 +hXZ^2.

Heree=0, sincePis non-singular, and we may supposeg=0, since otherwisePis
a flex. If we replacegX+aYbyX, this assumes the form

F 2 (X,Y,Z)=XY^2 +bY^2 Z+cY Z^2 +dZ^3 +eX^2 Z+gXY Z+hXZ^2 ,

with new values for the coefficients. If we now replaceX+bZbyX, this assumes the
form

F 3 (X,Y,Z)=XY^2 +cY Z^2 +dZ^3 +eX^2 Z+gXY Z+hXZ^2 ,

again with new values for the coefficients. The projective cubic curveDover the field
K, defined by the homogeneous polynomial

F 4 (U,V,W)=VW^2 +cV^2 W+dUV^2 +eU^3 +gU V W+hU^2 V,

has a flex at the point( 0 , 0 , 1 ). Moreover,

F 3 (U^2 ,VW,UV)=U^2 VF 4 (U,V,W),

F 4 (XZ,Z^2 ,XY)=XZ^2 F 3 (X,Y,Z).

This shows thatany projective cubic curve over the field K with a non-singular K-point
isbirationally equivalentto one with a flex.