Isogenies - Elliptic Curves: Number Theory and Cryptography - page 394

a p th power, where p is the characteristic of K , then for all but finitely many
a , these n zeros are distinct (if F is a p th power, then F a 1 /p ) p ,

so the roots cannot be distinct; that is why this case is excluded). We say
that n isthedegreeof F .If F

a =( F 1 /p − −

a has n distinct zeros for each a and F has
n distinct poles, then we say that F is unramified.

−

PROPOSITION 12.18 (Riemann-Hurwitz)
Let C 1 , C 2 be curves of genus g 1 ,g 2 defined over an algebraicallyc osed field
K ,and let F : C 1 →C 2 be an u n ram ified rationalm ap of degree n .Then

2 g 1 − 2= n (2 g 2 − 2) .

PROOF See [49]. More generally, the Riemann-Hurwitz formula can be
extended to cover the case where F is ramified.

In our case, F is a function from the elliptic curve E , which has genus 1,
to the projective line P 1 , which has genus 0. By the above discussion, F is
unramified of degree n . Therefore, 0 = 2 n , which is a contradiction.

We conclude that e 1 ,e 2 ,e 3 must be distinct and therefore that E 2 is non-
singular. This completes the proof of Lemma 12.17.

−

We have shown that α :( x, y ) → ( X, Y ) gives a map from E to E 2 .Equa-
tions (12.2), (12.3) show that the points in the subgroup C are exactly the
points mapping to ∞ . In particular, since ∞ maps to ∞ , Theorem 12.10
shows that α is an isogeny. Its kernel is C . By Exercise 12.8, α is separable.
This completes the proof of Theorem 12.16.

Example 12.4
Let E be given by y 2 = x 3 + ax 2 + bx ,with b =0and a 2 − 4 b = 0 (these

conditions make the curve nonsingular). The point (0 , 0) is a point of order 2,
so this point, along with ∞ , gives a subgroup of order 2. The set S is { (0 , 0) } .
For Q =(0 , 0), we have v Q = g Q = a 4 = b and g Q =0,so u Q = 0. Therefore,

b
x ,

by
x 2 .

X = x + Y = y −

The curve E 2 is given by the equation

Y 2 = X 3 + aX 2 − 4 bX − 4 ab.

Let

y 2
x 2 ,

= y x 2

X 3 = X + a = x + ax + b
x

by
x 2

− b
y

= 3 = Y = y − .