UC Berkeley

Let $C$ be a smooth projective geometrically connected curve over a finite field with function field $K$ or the spectrum of the ring of integers in a number field $K$. Let $X$ be a smooth projective geometrically connected curve over $K$. Let $\pi:\mathcal{X}\longrightarrow C$ be a proper regular model of $X/K$. Artin and Grothendieck proved that there is an isomorphism $Sha(\Pic^{0}_{X/K})\cong \mathrm{Br}(\mathcal{X})$ up to finite groups. As a result, this implies that the BSD conjecture for $\Pic^{0}_{X/K}$ is equivalent to the Tate conjecture for the surface $\mathcal{X}$ when $K$ is in positive characteristic. In this thesis, we generalize this result to fibrations $\pi:\mathcal{X}\longrightarrow C$ of arbitraryrelative dimensions fibered over $C$, where $C$ is a smooth projective curve over arbitrary finitely generated fields or the spectrum of the ring of integers in a number field. As a consequence, we reprove the reduction theorem of the Tate conjecture for divisors due to Andr e and Ambrosi, and give a simpler proof of a theorem of Geisser who proved it using the etale motivic cohomology theory. We also reduce Artin's question on the finiteness of Brauer groups of proper regular schemes to dimension at most $3$.