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 ] ) . {\displaystyle 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 ] ) / im ( κ v ) {\displaystyle H^{1}(G_{K_{v}},A_{v}[f])/\operatorname {im} (\kappa _{v})} is isomorphic to H 1 ( G K v , A v ) [ f ] {\displaystyle H^{1}(G_{K_{v}},A_{v})[f]} . Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have K v {\displaystyle K_{v}} -rational points for all places v {\displaystyle v} of K {\displaystyle 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 ( 2 ) ( A / K ) → {\displaystyle 0\to B(K)/f(A(K))\to \operatorname {Sel} ^{(2)}(A/K)\to \;} Ш(A/K) → 0 {\displaystyle \,\to 0}
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