Isogenies - Elliptic Curves: Number Theory and Cryptography - page 388

Clearly, α ∗ is a group homomorphism.

LEMMA 12.11
α ∗ m aps principal divisors toprincipal divisors.

PROOF Writing ( x 2 ,y 2 )= α ( x 1 ,y 1 ), where ( x i ,y i ) are coordinates for
E i , allows us to regard K ( x 2 ,y 2 ) as a subfield of K ( x 1 ,y 1 ) (see the proof
of Proposition 12.12). The norm map for this extension maps elements of
K ( x 1 ,y 1 ) × to elements of K ( x 2 ,y 2 ) × , and yields a map from principal divisors
on E 1 to principal divisors on E 2 . The main part of the proof of the lemma is
showingthatthisnormmapisthesameasthemap α ∗ on principal divisors.
For this, see [43, Prop. 1.4].

Therefore, α ∗ gives a well-defined map

α ∗ :Div 0 ( E 1 ) / (principal divisors) Div 0 ( E 2 ) / (principal divisors) . −→

If P ∈ E 1 ( K ), then

α ∗ ( ψ 1 ( P )) = α ∗ ([ P ] − [ ∞ ]) = [ α ( P )] − [ ∞ ]= ψ 2 ( α ( P )) .

Therefore,

α = ψ − 1

◦ α ∗ ◦ ψ 1 .

2

Since all three maps on the right are homomorphisms, so is α .

The following tells us that an elliptic curve isogenous to an elliptic curve E
is essentially uniquely determined by the kernel of the isogeny to it. This may
seem obvious from the viewpoint of group th eory since the group of points
on the isogenous curve is isomorphic to E ( K ) /C ,where C is the kernel of
the isogeny. But we are asking for more: we want the uniqueness of the
curve as an algebraic variety. We say that two elliptic cu rves E 2 ,E 3 are
isomor ph ic if th er e are group homomorphisms β : E 2 ( K ) → E 3 ( K )and
γ : E 3 ( K ) → E 2 ( K ) such that β and γ are given by rational functions and
such that γ ◦ β =idon E 2 and β ◦ γ =idon E 3 .

PROPOSITION 12.12
Let E 1 ,E 2 ,E 3 be elliptic curves over a field K and suppose that the re e xist
separable sogenies α 2 : E 1 → E 2 and α 3 : E 1 → E 3 defined over K .If
Ker α 2 =Ker α 3 ,then E 2 isisom orphicto E 3 over K .Infact,there isan
isom orphism β : E 2 →

E 3 su ch that β ◦ α 2 = α 3 .

PROOF This proof will use some concepts from field theory and Galois
theory. It may be skipped by readers unfamiliar with these subjects.