The Basic Theory - Elliptic Curves: Number Theory and Cryptography - page 23

F is identically 0, then we let ord L,P ( F )=

If . It is not hard to show that

ord L,P ( F ) is independent of the choice of parameterization of the line L . Note
that v = v 0 = 1 corresponds to the nonhomogeneous situation above, and the
definitions coincide (at least when z

∞

= 0). The advantage of the homogeneous
formulation is that it allows us to treat the points at infinity along with the
finite points in a uniform manner.

LEMMA 2.3
Let L 1 and L 2 be lines intersecting inapo nt P ,and,for i =1 , 2 , et
L i ( x, y, z ) be a linear polynom ial defining L i .Then ord L 1 ,P ( L 2 )=1 unless
L 1 ( x, y, z )= αL 2 ( x, y, z ) for som e constant α ,inwh ch case ord L 1 ,P ( L 2 )=
∞ .

PROOF When we substitute the parameterization for L 1 into L 2 ( x, y, z ),
we obtain L 2 , which is a linear expression in u, v .Let P correspond to ( u 0 :
v 0 ). Since L 2 ( u 0 ,v 0 ) = 0, it follows that L 2 ( u, v )= β ( v 0 u − u 0 v )forsome
constant β .If β =0,thenord L 1 ,P ( L 2 )=1. If β = 0, then all points on
L 1 lie on L 2 .Sincetwopointsin P 2 K determine a line, and L 1 has at least
three points ( P 1 K always contains the points (1 : 0) , (0 : 1) , (1 : 1)), it follows
that L 1 and L 2 are the same line. Therefore L 1 ( x, y, z ) is proportional to
L 2 ( x, y, z ).

Usually, a line that intersects a curve to order at least 2 is tangent to the
curve. However, consider the curve C defined by

F ( x, y, z )= y 2 z x 3 =0 .

−

Let

x = au, y = bu, z = v

be a line through the point P = (0 : 0 : 1). Note that P corresponds to
( u : v ) = (0 : 1). We have

F ( u, v )= u 2 ( b 2 v

a 3 u ), so every line through P
intersects C to order at least 2. The line with b = 0, which is the best choice
for the tangent at P , intersects C to order 3. The a ne part of C is the curve
y 2 = x 3 , which is pictured in Figure 2.7. The point (0 , 0) is a singularity of
the curve, which is why the intersections at P have higher orders than might
be expected. This is a situation we usually want to avoid.

−

DEFINITION 2.4 Acurve C in P 2 K defined by F ( x, y, z )=0 issaidtobe
nonsingular at a point P ifatleast on e of the partialderivatives F x ,F y ,F z
is nonzero at P .

For example, consider an elliptic curve defined by F ( x, y, z )= y 2 z − x 3 −

Axz 2 − Bz 3 = 0, and assume the characteristic of our field K is not 2 or 3.