66 2. Elliptic curves

everywhere locally (Qp, R) but not globally (Q) is the main obstacle

for the descent method to fully work.

2.11. Selmer and Sha

In Example 2.10.5, we found a type of homogeneous space that made

our approach to finding generators of E(Q)/2E(Q) break down. In

this section, we study everywhere locally solvable spaces in detail.

Let E : y2 = (x − e1)(x − e2)(x − e3) be an elliptic curve with

ei ∈ Z and e1 + e2 + e3 = 0. Let Γ be defined as in Corollary 2.9.7,

i.e.:

Γ = {n ∈ Z : 0 = n is square-free and if p | n, then p |

Δ}/(Z×)2

where Δ = (e1 − e2)(e2 − e3)(e1 − e3). We define H as the following

set of homogeneous spaces:

H := {C(δ1,δ2) : δ1,δ2 ∈ Γ }.

Some homogeneous spaces in H have rational points that correspond

to rational points on E(Q); see Prop. 2.10.3. Other homogeneous

spaces do not have points (e.g. C(1, 2) in Example 2.10.4, or C(−1, 2)

in Example 2.10.5). For each elliptic curve, we define two different

sets of homogeneous spaces, the Selmer group and the Shafarevich-

Tate group, as follows. The Selmer group is

Sel2(E/Q) := {C(δ1,δ2) with points over R and Qp for all primes p}.

In other words, the Selmer group is the set of all homogeneous spaces

that are solvable everywhere locally, i.e., over R and over all fields of

p-adic numbers. The group operation on

Sel2(E/Q) is defined by

C(δ1,δ2) · C(γ1,γ2) = C(δ1γ1,δ2γ2).

Notice that, due to Prop. 2.10.3, E(Q)/2E(Q) injects into H

via δ and the homogeneous spaces in the image of δ, i.e. δ(E) ⊆

H, have rational points. Since Q ⊆ Qp for all primes p ≥ 2, the

spaces in the image of δ belong to Sel2(E/Q). Hence, Sel2(E/Q)

has a subgroup formed by those homogeneous spaces in Sel2(E/Q)

that have rational points as well (i.e., over Q), and this subgroup is

isomorphic to E(Q)/2E(Q):

E(Q)/2E(Q) = {C(δ1,δ2) with points defined over Q}.