Tate–Shafarevich group ($\mathrm{Sha}(E/K)$)

The Tate–Shafarevich group consists of the principal homogeneous spaces for $E$ over $K$ that have a point over every completion of $K$. Its $n$-torsion is the quotient of the $n$-Selmer group by $E(K)/nE(K)$.