Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 246

First, we give a way to construct elements of H 1 ( G, E ( Q )). Let C be a
curve defined over Q such that C is isomorphic to E over Q . This means th at
thereisamap φ : E C given by rational functions with coe cients in Q

and an invers e f unction φ − 1

→

: C → E also given by rational functions with

G ,andlet φ g denote the map obtained by applying
g to the coecients of the rational functions defining φ .Since C is defined
over Q ,themap φ g

coe cients in Q .Let g ∈

maps E to gC = C . Note that

g ( φ ( P )) = ( φ g )( gP )

(8.19)

for all P ∈ E ( Q ), since the expression g ( φ ( P )) means we apply g to ev-
erything, while φ g means applying g to the coecients of φ and gP means

applying g to P .
We have to be a little careful when applying g 1 g 2 . The rule is

φ g 1 g 2 =( φ g 2 ) g 1 ,

since applying g 1 g 2 to the coe cients of φ means first applying g 2 , then ap-
plying g 1 to the result.
We say that a map φ is defined over Q if φ g ( P )= φ ( P ) for all P

∈ E ( Q )

and all g G (this is equivalent to saying that the coe cients of the rational

functions defining φ can be taken to be in Q , though proving this requires
results such as Hilbert's Theorem 90).
The map φ − 1 φ g gives a map from E to E .Wea ss umethefollowing:
A ssu m ption: Assume that there is a point T g ∈

∈

E ( Q ) such that

φ − 1 ( φ g ( P )) = P + T g

(8.20)

for all P ∈ E ( Q ). Equation (8.20) can be rewritten as

φ g ( P )= φ ( P + T g ) (8.21)

for all P ∈ E ( Q ). If we let P =( φ g ) − 1 ( Q ) for a point Q ∈ C ( Q ), then the
assumption becomes

φ − 1 ( Q )=( φ g ) − 1 ( Q )+ T g ,

(8.22)

which says that φ − 1 and ( φ g ) − 1 differ by a translation. We'll give an example
of such a map φ below.

LEMMA 8.32
D efine τ φ : G → E ( Q ) by τ φ ( g )= T g .Then τ φ ∈ Z ( G, E ( Q )) .