Selmer group

In arithmetic geometry, the Selmer group, named in honor of the work of Selmer (1951) by Cassels (1962), is a group constructed from an isogeny of abelian varieties.

The Selmer group of an abelian variety A with respect to an isogeny f : A → B of abelian varieties can be defined in terms of Galois cohomology as

where A_v[f] denotes the f-torsion of A_v and ${\displaystyle \kappa _{v}}$ is the local Kummer map ${\displaystyle B_{v}(K_{v})/f(A_{v}(K_{v}))\rightarrow H^{1}(G_{K_{v}},A_{v}[f])}$. Note that ${\displaystyle H^{1}(G_{K_{v}},A_{v}[f])/\mathrm {im} (\kappa _{v})}$ is isomorphic to ${\displaystyle H^{1}(G_{K_{v}},A_{v})[f]}$. Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have K_v-rational points for all places v of K. The Selmer group is finite. This implies that the part of the Tate–Shafarevich group killed by f is finite due to the following exact sequence

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