The Hasse principle for finite Galois modules allowing exceptional sets of positive density

We study a variant of the Hasse principle for finite Galois modules, allowing exceptional sets positive density. For module whose underlying abelian group is isomorphic to [Formula: see text] ([Formula: text]), we show that product restriction maps places in set injective if Dirichlet density strictly larger than text]. give applications local–global divisibility problem elliptic curves and flexes on plane cubic curves.