Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 249

If C has a rational point Q , choose P such that φ ( P )= Q .Then gQ = Q for
all g implies that

T g = g ( −P ) − ( −P )

for all g ∈ G . Conversely, if T g = g ( −P ) − ( −P ) for all g then gφ ( P )= φ ( P )
for all g ∈ G ,so φ ( P ) is a rational point.

Propositions 8.33 and 8.34 give us a reinterpretation in terms of cohomol-
ogy groups of the fundamental question of when certain curves have rational
points.

Example 8.12
Consider the curve C 1 ,p,p from Section 8.8. It was given by the equations

x = u 2
x − 2 p = pv 2
x +2 p = pw 2 .

These were rewritten as

w 2 − v 2 =4 ,

2

− pv 2 =2 p.

The method of Section 2.5.4 changes this to

C : s 2 =2 p ( t 4 +6 t 2 +1) .

Finally, the transformation
t = √ 2 p ( x +2 p )
y

− 2 p + 2 t 2 ( x − p ) = 2 p x 2 +4 px − 4 p 2

x ( x − 2 p )

, s = √ 2 p

(use the formulas of Section 2.5.3, plus a minor change of variables) changes
the equation to

E : y 2 = x ( x

− 2 p )( x +2 p ) .

We want to relate the curve C 1 ,p,p from Section 8.8 to a cohomology class in
H 1 ( G, E ( Q )). The map

φ : E → C
( x, y ) → ( t, s )

gives a map from E to C . Since the equations for E and C have coecients
in Q , these curves are defined over Q . However, φ is not defined over Q .
A short computation shows that

( x, y )+( − 2 p, 0) = ( x 1 ,y 1 )