Tate–Shafarevich group

Summary

Let $E$ be an elliptic curve over a number field $K$. Its Tate–Shafarevich group $\text{Ш}(E/K)$ measures the failure of the Hasse principle for twisted forms of $E$. Its elements are classes of genus one curves that become isomorphic to $E$ over an algebraic closure. These curves have points over every completion of $K$, and the trivial class is that of curves with a point over $K$. It is conjectured to be finite, and its order enters the Birch and Swinnerton-Dyer conjecture.