Isogenies - Elliptic Curves: Number Theory and Cryptography - page 385

an analytic map). Let ω ∈ L 1 . Then the function

f ( z + ω ) − f ( z )

reduces to 0 mod L 2 . Since it is continuous and takes values in the discrete
set L 2 , it is constant. Therefore, its derivative is 0, so f ( z + ω )= f ( z )for
all z . This means that f is a holomorphic doubly periodic function, hence
constant, by Theorem 9.1. Therefore,

f ( z )= αz + β for some α, β , as desired.

In anticipation of the algebraic situation, and recalling that endomorphisms
of elliptic curves are given by rational functions, we prove the following.

PROPOSITION 12.7
Let E 1 = C /L 1 and E 2 = C /L 2 be elliptic curves over C ,let ℘ i ( z ) be the
We erstra ss ℘ -function for E i ,and let [ α ] be an isogeny fro m E 1 to E 2 .Then
there are rationalfunctions R 1 ( x ) ,R 2 ( x ) su ch that

℘ 2 ( αz )= ℘ 1 ( z ) R 2 ( ℘ 1 ( z )) . ℘ 2 ( αz )= R 1 ( ℘ 1 ( z )) ,

PROOF We have αL 1 ⊆ L 2 .Let f ( z )= ℘ 2 ( αz ). Let ω ∈ L 1 .Then

αω ∈ L 2 ,so

f ( z + ω )= ℘ 2 ( αz + αω )= ℘ 2 ( αz )= f ( z )

℘ 2 ( αz ) is a rational function of ℘ 1 and ℘ 1 by
Theorem 9.3. In fact, the end of the proof of Theorem 9.3 shows that, since
℘ 2 ( αz ) is an even function, it is a rational function of ℘ 1 ( z ). Differentiation
yields the statement about ℘ 2 ( αz ).

for all z . Therefore, z →

Recall that z mod L 1 corresponds to ( ℘ 1 ( z ) ,℘ 1 ( z ) )onthecurve E 1 : y 1 =
4 x 1 − g 2 x 1 − g 3 . The proposition says that [ α ]: E 1 → E 2 corresponds to

( x 1 ,y 1 ) −→ ( x 2 ,y 2 )=( R 1 ( x 1 ) ,y 1 R 2 ( x 1 )) .

12.2 The Algebraic Theory
Let E 1 : y 1 = x 1 + A 1 x 1 + B 1 and E 2 : y 2 = x 2 + A 2 x 2 + B 2 be elliptic
curves over a field K (later we will also work with generalized Weierstrass
equatio ns) . An is og eny from E 1 to E 2 is a nonconstant homomorphism
α : E 1 ( K ) → E 2 ( K ) that is given by ratio nal functions. This means that
α ( P + Q )= α ( P )+ α ( Q ) for all P, Q ∈ E 1 ( K ) and that there are rational