Isogenies - Elliptic Curves: Number Theory and Cryptography - page 387

K satisfy y 1 = x 1 + A 1 x 1 + B 1 . Raising this equation to the Suppose x 1 ,y 1 ∈

p -th power yields

( y 1 ) 2 =( x 1 ) 3 + A 1 ( x 1 )+ B 1 .
Since x 2 = x 1 and y 2 = y 1 , this means that φ maps E 1 ( K )to E 2 ( K ). It is
easy to see that φ is a homomorphism (as in Lemma 2.20). We have

r 1 ( x )= x p r 2 ( x )=( y 2 ) ( p− 1) / 2 =( x 3 + A 1 x + B 1 ) ( p− 1) / 2 .

and

Therefore, deg( φ )=deg r 1 = p .If Q ,

so Ker( φ ) is trivial. The fact that the degree is larger than th e cardinality of
the kernel corresponds to the fact that φ is not separable.

= ∞ is a point of E 1 ,then φ ( Q ) = ∞

Example 12.3
Let E 1 : y 1 = x 1 + ax 1 + bx 1 be an elliptic curve over some field of characteristic
not 2. We require b =0and a 2 =0inordertohave E 1 nonsingular.

Then (0 , 0) is a point of order 2. Let E 2 be the elliptic curve y 2 = x 2 − 2 ax 2 +
( a 2

− 4 b

− 4 b ) x 2 . Define α by
( x 2 ,y 2 )= α ( x 1 ,y 1 )= y 1

.

y 1 ( x 1 − b )
x 1 x 1 ,

It is straightforward to check that α maps points of E 1 ( K )topointsof E 2 ( K ).
It is more dicult to show that α is a homomorphism. However, this fact
follows from Theorem 12.10 below. (We need to verify that α ( ∞ )= ∞ .For
this, see Exercise 12.4.)
We have

r 1 ( x )= x 3 + ax 2 + bx = x 2 + ax + b

x

,

x 2

so deg α =2and α is separable. This means that there are two points in the
kernel. Writing r 1 ( x )= x + a +( b/x ), we see that these two points must be
∞ and (0 , 0), since all other points have finite images (for another proof that
α (0 , 0) = ∞ , see Exercise 12.5).

THEOREM 12.10
Let E 1 and E 2 be elliptic curves over a field K .Let α : E 1 ( K ) → E 2 ( K )
be a n on con stant m ap given by rationalfunctions. If α ( ∞ )= ∞ ,then α isa
hom om orphism , an d therefore an isogeny.

PROOF Recall that, by Corollary 11.4, there are group isomorphisms

Div 0 ( E i ) / (principal divisors) ψ i : E i ( K ) −→

]. Define α ∗ :Div 0 ( E 1 ) Div 0 ( E 2 )by given by P → [ P ] − [ ∞ →

α ∗ : b j [ P j ]

b j [ α ( P j )] .

−→