The 
-Selmer group of an elliptic
curve 
over a number field 
is the finite abelian group 
of Galois
cohomology classes associated with division by a positive
integer 
that satisfy a local solubility condition at every completion
of 
. It fits into the exact
sequence
![0->E(K)/nE(K)->Sel^((n))(E/K)->Sha(E/K)[n]->0,](/images/equations/SelmerGroup/NumberedEquation1.svg) |
where 
is the Tate-Shafarevich group, which measures
the failure of the local-to-global principle for
certain curves associated with 
(Silverman 2009).
More generally, a Selmer group can be defined for an isogeny of elliptic curves. Selmer groups are effectively
computable in many cases and give upper bounds for the group
rank of the group 
of rational points. They played an important role
in Andrew Wiles' proof of Fermat's last theorem.
