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}.
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