Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 300

Suppose f ( P + T )= f ( P ) for all P

∈ E ( K ) and all T ∈ E [ n ] .Thenthere is

afunction h on E su ch that f ( P )= h ( nP ) for all P .

PROOF The case n = 1 is trivial, so we assume n> 1. Let T ∈ E [ n ].
There are rational functions R ( x, y ) ,S ( x, y ) depending on T such that

( x, y )+ T =( R ( x, y ) ,S ( x, y )) .

Let y 2 = x 3 + A x + B be the equation of E and regard K ( x, y ) as the quadratic

extension of K ( x ) given by adjoining √ x 3 + Ax + B .Since( R, S ) lies on E ,
we have S 2 = R 3 + AR + B .Themap

σ T : K ( x, y ) → K ( x, y )
f ( x, y ) → f ( R, S )

is a homomorphism from K ( x, y ) to itself. Since σ −T is the inverse of σ T ,the
map σ T is an automorphism. Because ( x, y )+ T =( x, y )+ T when T = T ,
we have σ T ( x, y ) = σ T ( x, y )when T = T . Therefore, we have a gro up of n 2
distinct automorphisms σ T ,where T runs through E [ n ], acting on K ( x, y ). A
basic result in Galois theory says that if G is a group of distinct automorphisms
of a field L , then the fixed field F of G satisfies [ L : F ]= . Therefore, the

field F of functions f satisfying the conditions of the proposition satisfies

| G |

[ K ( x, y ): F ]= n 2 .

(9.26)

Let n ( x, y )=( g n ( x ) ,yh n ( x )) for rational functions g n ,h n .Then

K ( g n ( x ) ,yh n ( x )) ⊆ F. (9.27)

Moreover,

[ K ( g n ( x ) ,yh n ( x )) : K ( g n ( x ))] ≥ 2 (9.28)

since clearly yh n ( x ) ∈ K ( g n ( x )). Therefore, by (9.26), (9.27), and (9.28),

[ K ( x, y ): K ( g n ( x ))] ≥ 2 n 2 .

From Theorem 3.6,

g n ( x )= φ n

ψ n ,

and φ n and ψ n are polynomials in x . Therefore, X = x is a root of the

polynomial

g n ( x ) ψ n ( X ) P ( X )= φ n ( X ) − ∈ K [ g n ( x )][ X ] .

By Lemma 3.5,

φ n ( X )= X n 2 + ···