Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 250

on E ,where

1 = − 8 p 2 y

x 1 =2 p 2 p − x

2 p + x , ( x +2 p ) 2 .

Another calculation shows that

s ) .
Let g ∈ G be such tha t g ( √ 2 p ) = − √ 2 p .Then φ g

φ ( x 1 ,y 1 )=( − t, −

is the transformation

obtained by changing √ 2 p to − √ 2 p in the formulas for φ . Therefore,

φ g ( x, y )=( −t, −s )= φ ( x 1 ,y 1 ) .

We obtain

φ − 1 φ g ( x, y )=( x, y )+( − 2 p, 0) .

G satisfies g √ 2 p =+ √ 2 p .Then φ g = φ ,so Now suppose g ∈

φ − 1 φ g ( x, y )=( x, y ) .

Putting everything together, we see that the pair ( C, φ ) is of the type con-
sidered above. We obtai n an element of H 1 ( G, E [2]) that can be regarded as
an element of H 1 ( G, E ( Q )). The cocycle τ φ is given by
τ φ ( g )= T g = ∞

if g √ 2 p = + √ 2 p

if g √ 2 p = − √ 2 p ( − 2 p, 0)

The cohomology class of τ φ is nontrivial in H 1 ( G, E ( Q )), and hence also in
H 1 ( G, E [2]), because C has no rational points. Note that τ φ is a homomor-
phism from G to E [2]. This corresponds to the fact that G acts trivially on
E [2] in the present case, so H 1 ( G, E [2]) = Hom ( G, E [2]). The kernel of τ is

the subgroup of G of index 2 that fixes Q ( √ 2 p ).

In general, if E is given by y 2 =( x Q ,

then a 2-descent yields curves C a,b,c , as in Section 8.2. These curves yield
elements o f H 1 ( G, E [2]). The curves that have rational points give cocycles
in Z ( G, E ( Q )) that are coboundaries. We also saw in the descent procedure
that a rational point on a curve C a,b,c comes from a rational point on E .This
discussion is summarized by the exact sequence

− e 1 )( x − e 2 )( x − e 3 )with e 1 ,e 2 ,e 3 ∈

0 → E ( Q ) / 2 E ( Q ) → H 1 ( G, E [2]) → H 1 ( G, E ( Q ))[2] → 0 .

All of the preceding applies when Q is replaced by a p -adic field Q p with

p ≤∞ . We have an exact sequence

0 → E ( Q p ) / 2 E ( Q p ) → H 1 ( G p ,E [2]) → H 1 ( G p ,E ( Q p ))[2] → 0 ,

where

G p =Gal( Q p / Q p ) .