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Regarding counterexamples, there is Sarnak-Wang, and then the results of Bjorn Poonen. Regarding proofs that the Hasse principle holds, I don't know the best result over ALL global fields. Of course global function fields are $C_2$, thus the Hasse principle trivially holds over global function fields for hypersurfaces of degree $d$ in $\mathbb{P}^n$ satisfying $d^2 \leq n$, because these hypersurfaces have rational points.