##  [Tate–Shafarevich Group](/tate-shafarevich-group) 

  ##  [Tate–Shafarevich Group](https://puremath.quantumdictionary.io/tate-shafarevich-group-0) 

  

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**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

For an abelian variety A over a global field, the Tate–Shafarevich group (commonly denoted Sha) is the group of principal homogeneous spaces (torsors) for A that are locally trivial at every completion of the base field but may fail to have a global rational point; it measures the precise obstruction to the Hasse principle for A.