Isogeny Selmer group (${\mathrm{Sel}}^{\varphi }(E/K)$)

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.

Coverings surviving all local solubility tests need not have rational points. The quotient of an isogeny Selmer group by the rational-point image is a kernel in the Tate–Shafarevich group. A nonzero class therefore passes every local test but contributes no rational point to a search. This explains why two-isogeny descent can leave a gap between a certified upper bound and found lower bound for the rank of an elliptic curve. Higher descents address finer obstructions; see Tom Fisher's paper on higher descents with rational two-torsion.