 Three-isogeny descent

Let $\varphi :E\to E^{\prime }$ be a degree-three isogeny over a field $K$ of characteristic different from three. Its connecting map injectsTo identify the target with $K^{\times }/(K^{\times }{)}^3$, one needs an identification of the Galois module $E[\varphi ]$ with ${\mu }_3$. This occurs, for example, when the kernel of an isogeny of the dual isogeny has a rational generator: the perfect isogeny pairing induced by the Weil pairing then identifies $E[\varphi ]$ with the homomorphisms from that constant cyclic group to ${\mu }_3$. Kummer theory gives $H^1(K,{\mu }_3)=K^{\times }/(K^{\times }{)}^3$; this cohomological statement does not require ${\mu }_3\subset K$.

Over a number field, the resulting classes in the cube-class group of a field are unramified outside finitely many places, including primes over three and primes of bad reduction. Their valuations away from that finite set are divisible by three, placing the image in an S-unramified power class group. Local image conditions then define the corresponding isogeny Selmer group. Without the kernel identification, one must retain the possibly twisted Galois module in the cohomology target.