2.11. Selmer and Sha 67
Finally, the Shafarevich-Tate group is the quotient of the Selmer group
by its subgroup E(Q)/2E(Q). Thus, each element of the Shafarevich-
Tate group is represented by C(1, 1) or by a homogeneous space that
is solvable everywhere locally but does not have a rational point:
X2(E/Q) = {C(1, 1)}
{C(δ1,δ2) Sel2(E/Q) without points over Q}.
These three groups, Selmer, X (or “Sha”) and E/2E, fit in a short
exact sequence
0 −→ E(Q)/2E(Q) −→ Sel2(E/Q) −→ X2(E/Q) −→ 0.
In other words, the map ψ : E(Q)/2E(Q) Sel2(E/Q) is injective,
the map φ : Sel2(E/Q) X2(E/Q) is surjective, and the kernel of
φ is the image of ψ.
Remark 2.11.1. Here for simplicity we have defined what number
theorists would usually refer to as the 2-part of the Selmer group
(Sel2(E/Q) above) and the 2-torsion of X (the group X2 as de-
fined above). For the definition of the full Selmer and X groups, see
[Sil86], Ch. X, §4.
Example 2.11.2. Let E : y2 = x3 1156x, as in Example 2.10.4.
The full group of homogeneous spaces H has 64 elements:
H = {C(δ1,δ2) : δi = ±1, ±2, ±17, ±34}.
The spaces in H with δ2 0 do not have points over R, so they
do not belong to Sel2(E/Q). Moreover, we showed that the spaces
(δ1,δ2) = (2,2), (17,2), (34,2), (−1,1), (−2,1), (−17,1), (−34,1),
(−1,17), (−2,17), (−17,17), (−34,17), (1,34), (2,34), (17,34), and
(34,34) do not have points over Q2. Therefore, they do not belong
to Sel2(E/Q) either. All other spaces have rational points; therefore,
they are everywhere locally solvable, so they all belong to Sel2(E/Q).
Sel2(E/Q) = {C(δ1,δ2) : (δ1,δ2) =
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