Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 21 / 1 / b

The obstruction is the local-to-global gap in isogeny descent. A two-isogeny descent quartic can have points over $\mathbb{R}$ and every p-adic field without a rational point. Local tests compute an isogeny Selmer group, which can strictly contain the rational-point quotient. For an isogeny of elliptic curves $\varphi :E\to E^{\prime }$, the comparison isThe last group is the kernel of $\varphi :\mathrm{Sha}(E/\mathbb{Q})\to \mathrm{Sha}(E^{\prime }/\mathbb{Q})$, so it records elements of the Tate–Shafarevich group invisible to all local tests. Searching a locally soluble covering which has no rational points never succeeds. Thus a Selmer group of an elliptic curve gives an upper bound and found independent points give a lower bound, and these need not coincide. Higher descent can improve the bounds, but this particular procedure has no unconditional guarantee of closing every gap.