3 problems

Principal homogeneous space Hasse principle. If $Y(F_\nu)\neq\varnothing$ for every $\nu\in\Omega_F$, then $Y(F)\neq\varnothing$.

Projective homogeneous space Hasse principle. If $Y(F_\nu)\neq\varnothing$ for every $\nu\in\Omega_F$, then $Y(F)\neq\varnothing$.

Let $\widetilde{\varphi}:\widetilde{Y}\to\widetilde{X}$ be the morphism of smooth projective curves over the residue field, and let $\varphi_1$ and $\varphi_2$ be two lifts of…