Selmer group

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

The Selmer group of an isogeny
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


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

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 $$H^1(G_{K_v},A_v[f])/\operatorname{im}(\kappa_v)$$ 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.

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.

The Selmer group of a finite Galois module
More generally one can define the Selmer group of a finite Galois module M (such as the kernel of an isogeny) as the elements of H1(GK,M) that have images inside certain given subgroups of H1(GK v ,M).