= Selmer exact sequence and rank bound
{c}
{title2=$\operatorname{Sel}^{(n)}(E/K)$}
For an <elliptic curve> over a number field $K$, global and local <Kummer maps of an elliptic curve> give $0\to E(K)/nE(K)\to\operatorname{Sel}^{(n)}(E/K)\to\operatorname{Sha}(E/K)[n]\to0$. 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 $\dim_{\mathbb F_\ell}\operatorname{Sel}^{(\ell)}=\operatorname{rank}E(K)+\dim_{\mathbb F_\ell}E(K)[\ell]+\dim_{\mathbb F_\ell}\operatorname{Sha}(E/K)[\ell]$. 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.
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