= Solution
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> $\phi:E\to E'$, the comparison is
$$
0\longrightarrow E'(\mathbb Q)/\phi E(\mathbb Q)
\longrightarrow\operatorname{Sel}^{\phi}(E/\mathbb Q)
\longrightarrow\operatorname{Sha}(E/\mathbb Q)[\phi]
\longrightarrow0.
$$
The last group is the kernel of $\phi:\operatorname{Sha}(E/\mathbb Q)\to\operatorname{Sha}(E'/\mathbb Q)$, 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.
Back to article page
