Tate–Shafarevich group

Remarks

The group $H^1(K, E)$ classifies principal homogeneous spaces of $E$: curves $C$ over $K$ with a simply transitive action of $E$ defined over $K$. A class is trivial exactly when $C$ has a point over $K$, so $\text{Ш}(E/K)$ consists of the classes of such curves $C$ that have a point over every $K_v$.

Example: Selmer's curve $C \colon 3x^3 + 4y^3 + 5z^3 = 0$ in $\mathbb{P}^2_{\Q}$ has points over $\R$ and over every $\Q_p$, but no point over $\Q$. It is a principal homogeneous space for its Jacobian $E \colon x^3 + y^3 + 60z^3 = 0$, so it gives a nonzero element of $\text{Ш}(E/\Q)$.

Non-example: let $C$ be a principal homogeneous space of $E$ over $\Q$ with $C(\R) = \emptyset$. Its class in $H^1(\Q, E)$ has nonzero restriction at the real place, so it is not in $\text{Ш}(E/\Q)$.

The group $\text{Ш}(E/K)$ is a torsion abelian group, and each of its $n$-torsion subgroups is finite, by the finiteness of the Selmer group. It is conjectured to be finite; when it is finite, the Cassels–Tate pairing shows that its order is a perfect square. The letter Ш is the Cyrillic letter sha.