The Tate-Shafarevich group (also called the Shafarevich-Tate group) of an elliptic curve 
over a number field 
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 field 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 rational point over every 
. Nonzero elements
correspond to such principal homogeneous
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 subgroup ![Sha(E/K)[n]](/images/equations/Tate-ShafarevichGroup/Inline14.svg)
of elements annihilated
by 
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.
