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).

Hence,

Sel2(E/Q) = {C(δ1,δ2) : (δ1,δ2) =

(1,1),(−1,34),(−34,2),(34,17),

(1,17),(34,1),(−34,34),(−2,2),

(17,1),(−17,34),(2,1),(−2,34),

(−17,2),(17,17),(−1,2),(2,17)}.