The Basic Theory - Elliptic Curves: Number Theory and Cryptography - page 39

The points at infinity have w = 0. To find them, we set w =0andget0= u 4 ,
which means u = 0. We thus find only the point ( u : v : w )=(0:1:0). But
we have two points, namely (2 , 0) and ( 2 , 0) in the corresponding Weierstrass

model. The problem is that ( u : v : w ) = (0 : 1 : 0) is a singular point in the
quarticmodel. Atthispointwehave

−

F u = F v = F w =0 .

What is happening is that the curve intersects itself at the point ( u : v :

w ) = (0 : 1 : 0). One branch of the curve is v =+ u 2 1+(1 /u ) 4 and the

other is v = −u 2 1+(1 /u ) 4 . For simplicity, let's work with real or complex
numbers. If we substitute the second of these expressions into x =2( v +1) /u 2
and take the limit as u →∞ ,weobtain

= 2(1 − u 2 1+(1 /u ) 4 )
u 2

x = 2( v +1)
u 2

→− 2 .

If we use the other branch, we find x → +2. So the transformation that
changes the quartic equation into the Weierstrass equation has pulled apart
the two branches (the technical term is “resolved the singularities”) at the
singular point.

2.5.4 Intersection of Two Quadratic Surfaces

The intersection of two quadratic surfaces in three-dimensional space, along
with a point on this intersection, is usually an elliptic curve. Rather than work
in full generality, we'll consider pairs of equations of the form

au 2 + bv 2 = e, cu 2 + dw 2 = f,

where a, b, c, d, e, f are nonzero elements of a field K of characteristic not 2.
Each separate equation may be regarded as a surface in uvw -space, and they
intersect in a curve. We'll show that if we have a point P in the intersection,
then we can transform this curve into an elliptic curve in Weierstrass form.
Before analyzing the intersection of these two surfaces, let's consider the
first equation by itself. It can be regarded as giving a curve C in the uv -
plane. Let P =( u 0 ,v 0 )beapointon C .Let L be the line through P with
slope m :

u = u 0 + t, v = v 0 + mt.

We want to find the other point where L intersects C . See Figure 2.6.
Substitute into the equation for C and use the fact that au 0 + bv 0 = e to

obtain

a (2 u 0 t + t 2 )+ b (2 v 0 mt + m 2 t 2 )=0 .