For an elliptic curve 
over a number field 
, the Tate-Shafarevich group (also called the Shafarevich-Tate
group) is the group kernel
![Sha(E/K)=ker[H^1(K,E)->product_(v)H^1(K_v,E)],](/images/equations/Tate-ShafarevichGroup/NumberedEquation1.svg) |
where 
ranges over all places of 
, 
is the completion of 
at 
, and 
denotes the first Galois
cohomology group. Its elements correspond to principal
homogeneous spaces for 
that have a point over every 
. Nonzero elements correspond to such spaces having no 
-rational point, so 
measures failure of the Hasse
principle (Silverman 2009, Milne 2021).
The Tate-Shafarevich group is a torsion abelian group, meaning that it consists entirely of torsion
elements, and is conjectured to be a finite group.
For each positive integer 
, its 
-torsion subgroup occurs in
the exact sequence
![0->E(K)/nE(K)->Sel^((n))(E/K)->Sha(E/K)[n]->0,](/images/equations/Tate-ShafarevichGroup/NumberedEquation2.svg) |
where 
is the 
-Selmer group. The conjectured finiteness of 
is part of the Swinnerton-Dyer
conjecture.
