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§5. An Explicit 2-Isogeny 99

ϕ E

E → E →
{0,(0, 0)}
is a homomorphism of groups.

(5.4) Proposition. On the curve E[a, b] we have
b by

(0, 0) + (x, y) = , − .
x x 2
Proof. For (x 1 , y 1 )+(0, 0) = (x 2 , y 2 ) consider the line y = (x 1 /x 1 )x through (0, 0)
and (x 1 , y 1 ). For the third point of intersection we compute

2
y
1 x = x + ax + bx
3
2
2
x 2
1
2
y
3 1 2
0 = x − − a x + bx.
x 2
1
This means that
2
2
y 2 1 y − ax − x 3 1 bx 1 b
1
1
x 2 = 2 − a − x 1 = 2 = 2 = .
x x x x 1
1 1 1
2
Moreover, y 2 =−(y 1 /x 1 )x 2 =−by 1 /x which proves the proposition.
1
(5.5) Remark. We can show that ϕ((x, y) + (0, 0)) = ϕ(x, y) by the direct calcu-
lation
2 2 2 2 2 2
−by/x b y x y
= =
(b/x) 2 x 4 b 2 x 2
and
2 2 2 2 2 2
−by/x b /x − b x by bx − b y 2
= · = x − b .
(b/x) 2 b 2 x 2 x 2 x 2
In order to study the image of ϕ, we introduce the following homomorphism
from the curve to the multiplicative group of the field modulo squares.
∗ 2
∗
(5.6) Proposition. The function α : E[a, b] → k /(k ) defined by α(0) = 1,
∗ 2
∗ 2
α((0, 0)) = b mod(k ) , and α((x, y)) = x mod(k ) is a grouphomomorphism.
Proof. For three points (x 1 , y 1 ), (x 2 , y 2 ),and (x 3 , y 3 ) on a line y = λx + ν and on
the curve E[a, b] the roots of the cubic equation
2 3 2
(λx + ν) = x + ax + bx