Kummer exact sequence of an elliptic curve

The exact sequence of Galois modulesinduces the Kummer map of an elliptic curve from $E(K)/nE(K)$ into $H^1(K,E[n])$.

At a prime of good reduction of an elliptic curve with residue characteristic not dividing $m$, multiplication by $m$ extends to a finite étale map of the smooth proper elliptic group scheme over the valuation ring. A local rational point extends to a section by properness. Its division fibre is therefore a finite étale torsor, so the Kummer map of an elliptic curve gives an unramified cohomology class. This bounds ramification in the Weak Mordell-Weil theorem.

A direct proof describes the coordinates, not only the cohomology class. Reduce $P$ by properness and choose a point $\overline{Q}$ with $[m]\overline{Q}=\tilde{P}$ over a finite extension of the residue field. Over the corresponding unramified extension of the complete local field, lift $\overline{Q}$ by the Hensel lemma. The error $[m]Q_0-P$ is in the kernel of reduction of an elliptic curve; prime-to-residue-characteristic multiplication on a formal group corrects it uniquely. There is no $m$-torsion in that kernel, even after a ramified field extension, so every division point with that reduction equals this unramified lift. Consequently all division points of an elliptic curve above $P$ have unramified coordinate fields. The torsion case is discussed in J. S. Milne's Elliptic Curves, Chapter V, Proposition 8.1.

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 $E(K)/nE(K)$.