n-Selmer group (${\mathrm{Sel}}^{(n)}(E/K)$)

The $n$-Selmer group consists of the classes in $H^1(K,E[n])$ whose restriction at every place lies in the image of the local Kummer map of an elliptic curve. It is finite and contains $E(K)/nE(K)$.

For an elliptic curve over a number field $K$, global and local Kummer maps of an elliptic curve give $0\to E(K)/nE(K)\to {\mathrm{Sel}}^{(n)}(E/K)\to \mathrm{Sha}(E/K)[n]\to 0$. Indeed a Selmer class maps to a locally trivial torsor; conversely any locally trivial torsor killed by $n$ has a global Kummer-sequence lift whose local restrictions lie in the local Kummer images. For a prime $\ell$, the Mordell-Weil theorem therefore gives ${\dim }_{{\mathbb{F}}_{\ell }}{\mathrm{Sel}}^{(\ell )}=\mathrm{rank}E(K)+{\dim }_{{\mathbb{F}}_{\ell }}E(K)[\ell ]+{\dim }_{{\mathbb{F}}_{\ell }}\mathrm{Sha}(E/K)[\ell ]$. Thus a Selmer computation supplies a rigorous upper rank bound, while independent rational points supply a lower bound; the gap need not represent undiscovered rational points.

For a split CM prime $\mathfrak{p}$, use $E[{\mathfrak{p}}^{\infty }]$ in the Galois cohomology definition of the Selmer group of an elliptic curve, with local conditions given by its Kummer maps. The decomposition of the CM $p$-power torsion splits the full $p$-primary Selmer group into the two prime-primary pieces. Class-field descriptions must retain the CM-character twist and distinguish relaxed local conditions from the actual Selmer conditions.

The Tate–Shafarevich group consists of the principal homogeneous spaces for $E$ over $K$ that have a point over every completion of $K$. Its $n$-torsion is the quotient of the $n$-Selmer group by $E(K)/nE(K)$.