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. == Selmer group of an isogeny == The Selmer group of an abelian variety A {\displaystyle A} with respect to an isogeny f : A → B {\displaystyle f:A\to B} of abelian varieties can be defined in terms of Galois cohomology as Sel ( f ) ( A / K ) = ⋂ v ker ( H 1 ( G K , ker ( f ) ) → H 1 ( G K v , A v [ f ] ) / im ( κ v ) ) {\displaystyle \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 ] {\displaystyle A_{v}[f]} denotes the f {\displaystyle f} -torsion of A v {\displaystyle A_{v}} and κ v {\displaystyle \kappa _{v}} is the local Kummer map B v ( K v ) / f ( A v ( K v ) ) → H 1 ( G K v , A v [ f ] ) .