Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 235

It can be shown that ψ ◦ ψ is multiplication by 2 on E . Fermat's descent

procedure can be analyzed in terms of the maps ψ and ψ .
More generally, if E is an elliptic curve given by y 2 = x 3 + Cx 2 + Ax and E
is given by y 2 = x 3 2 Cx 2 +( C 2 4 A ) x , then there are maps ψ : E E − − →

given by
( x ,y )= ψ ( x, y )= y 2

,

y ( x 2 − A )

x 2 ,

ψ (0 , 0) = ψ ( ∞ )= ∞,

x 2

and ψ : E → E given by

( x, y )= ψ ( x ,y )= y 2

,

y ( x 2

− C 2 +4 A )
8 x 2

ψ (0 , 0) = ψ (

4 x 2 ,

∞ )= ∞ .

The composition ψ ◦ ψ is multiplication by 2 on E . Itispossibletodo
descent and prove the Mordell-Weil theorem using the maps ψ and ψ .This
is a more powerful method than the one we have used since it requires only
one two-torsion to be rational, rather than all three. For details, see [114],
[109].
Th e maps ψ and ψ can be shown to be homomorphisms between E ( Q )and
E ( Q ) and are described by rational functions. In gene ral , for elli pt ic curves
E 1 and E 2 over a field K , a homomorphism from E 1 ( K )to E 2 ( K )thatis
given by rational functions is called an isogeny .

8.7 2-Selmer Groups; Shafarevich-Tate Groups

Let's return to the basic descent procedure of Section 8.2. We start with
an elliptic curve E defined over Q by

y 2 =( x − e 1 )( x − e 2 )( x − e 3 )

with all e i ∈ Z . This leads to equations

x − e 1 = au 2
x − e 2 = bv 2
x − e 3 = cw 2 .

These lead to the equations

au 2 − bv 2 = e 2 − e 1 ,

2

− cw 2 = e 3 − e 1 .

This defines a curve C a,b,c in u, v, w . In fact, it is the intersection of two
quadratic surfaces. If it has a rational point, then it can be changed to an