= Local-to-global gap in isogeny descent
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 https://arxiv.org/abs/1509.03234[Tom Fisher's paper on higher descents with rational two-torsion].
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