We study a variant of the Hasse principle for finite Galois modules, allowing exceptional sets of positive density. For a Galois module whose underlying abelian group is isomorphic to Fp⊕r (r ≤ 2), we show that the product of the restriction maps for places in a set of places S is injective if the Dirichlet density of S is strictly larger than 1-p-r. We give applications to the local-global divisibility problem for elliptic curves and the Hasse principle for flexes on plane cubic curves.
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