Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 299

Therefore, by Lemmas 9.32 and 9.31,

ψ 2 n
ψ n

ω n

1

2 ℘ ( nz )= 1

=

ψ n . 2

This completes the proof of the theorem when the characteristic of the field
is 0.
Suppose now that E is defined over a field K of arbitrary characteristic
(not 2) by y 2 = x 3 + Ax + B .Let( x, y ) E ( K ). Let α , β ,and X be three

independent transcendental elements of C and let Y satisfy Y 2 = X 3 + αX + β .
There is a ring homomorphism

∈

ρ : Z [ α, β, X, Y ] −→ K ( x, y )

such that

g ( α, β, X, Y ) → g ( A, B, x, y )

for all polynomials g .Let R = Z [ α, β, X, Y ]andlet E be the elliptic curve
over R defined by y 2 = x 3 + αx + β . We want to say that by Corollary 2.33,
ρ induces a homomorphism

E ( R )

ρ : −→ E ( K ( x, y )) .

But we need to have R satisfy Conditions (1) and (2) of Section 2.11. The
easiest way to accomplish this is to let M be the kernel of the map R →
K ( x, y ). Since K ( x, y )isafield, M is a maximal ideal of R .Let R M be the
localization of R at M (this means, we invert all elements of R not in M ).
Then R ⊆ R M and the map ρ extends to a map

ρ : R M −→ K ( x, y ) .

Since R M is a local ring, and projective modules over local rings are free, it
can be shown that R M satisfies Condition (2). Since we are assuming that
K ( x, y ) has characteristic not equal to 2, it follows that 2 is not in , hence

is invertible in R M . Therefore, R M satisfies Condition (1). Now we can apply
Corollary 2.33.
The point n ( X, Y )in E ( R M ) is described by the polynomials ψ j , φ j ,and ω j ,
which are polynomials in X, Y with coecients in Z [ α, β ]. Applying ρ shows
that these polynomials, regarded as polynomials in x, y with coecients in K ,
describe n ( x, y )on E . Therefore, the theorem holds for E .

M

As an application of the division polynomials, we prove the following result,
which will be used in Chapter 11.

PROPOSITION 9.34
L et E be an elliptic curve over a field K .Let f ( x, y ) be a function fro m E to
K ∪{∞} and let n ≥ 1 be an integer not divisiblebythe characteristicof K .