Selmer group

Summary

The Selmer group of an isogeny of elliptic curves over a number field is a group of Galois cohomology classes. These classes satisfy local conditions at every place, and the group is finite and computable. It contains a copy of the quotient $E'(K)/\phi(E(K))$, and the difference is the $\phi$-torsion of the Tate–Shafarevich group. Computing a Selmer group is the standard way to bound the rank of $E(K)$ from above. It is the effective core of the proof of Mordell's theorem.