On the Conjectures of Artin and Shafarevich–Tate

For an arithmetic model $\pi\colon X\to\operatorname{Spec}A$ of a smooth projective variety $V$ over a number field $k$, the interrelations between the conjecture of Artin about the finiteness of $\mathrm{Br}(X)$ and the conjecture of Shafarevich–Tate about the finiteness of $\text{III}(\operatorname {Spec}A,\mathrm{Pic}^0(V))$ are studied.