Selmer group

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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

\mathrm{Sel}^{(f)}(A/K)=\bigcap_v\mathrm{ker}(H^1(G_K,\mathrm{ker}(f))\rightarrow H^1(G_{K_v},A_v[f])/\mathrm{im}(\kappa_v))

where Av[f] denotes the f-torsion of Av and \kappa_v is the local Kummer map B_v(K_v)/f(A_v(K_v))\rightarrow H^1(G_{K_v},A_v[f]). Note that H^1(G_{K_v},A_v[f])/\mathrm{im}(\kappa_v) is isomorphic to H^1(G_{K_v},A_v)[f]. Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have Kv-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

0 → B(K)/f(A(K)) → Sel(f)(A/K) → Ш(A/K)[f] → 0.

The Selmer group in the middle of this exact sequence is finite and effectively computable. This implies the weak Mordell–Weil theorem that its subgroup B(K)/f(A(K)) is finite. There is a notorious problem about whether this subgroup can be effectively computed: there is a procedure for computing it that will terminate with the correct answer if there is some prime p such that the p-component of the Tate–Shafarevich group is finite. It is conjectured that the Tate–Shafarevich group is in fact finite, in which case any prime p would work. However, if (as seems unlikely) the Tate–Shafarevich group has an infinite p-component for every prime p, then the procedure may never terminate.

Ralph Greenberg has generalized the notion of Selmer group to more general p-adic Galois representations and to p-adic variations of motives in the context of Iwasawa theory.

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