Selmer exact sequence and rank bound (${\mathrm{Sel}}^{(n)}(E/K)$)

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.