 Isogeny Selmer group

For an isogeny of elliptic curves $\varphi :E\to E^{\prime }$ over a number field $K$, this finite subgroup of $H^1(K,E[\varphi ])$ consists of Galois cohomology classes whose restriction at every place comes from $E^{\prime }(K_v)/\varphi E(K_v)$. It fits intowhere the last group is the kernel of the induced map on Tate–Shafarevich groups. Thus local computations bound the rational-point quotient without necessarily determining it. For multiplication by $n$, this recovers the Selmer group of an elliptic curve.