Selmer group
# Selmer group

In arithmetic geometry, the Selmer group, named in honor of the work of Ernst Sejersted Selmer (1951) by John William Scott Cassels (1962), 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 $A_v[f]$ denotes the f-torsion of $A_v$ 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 $K_v-rational$ 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

    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 (1994) 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 $H^1(G_K,M)$ that have images inside certain given subgroups of $H^1(G_{Kv},M).$

