(I asked this on math-stackexchange, but it seems more appropriate to this forum, so I took it off from there and am posting it here)

After the great answer I got for my previous question about the Tate conjectures What is the intuition behind the concept of Tate twists?, I'm ready for my next one:

Let $X$ be an abelian variety defined over a number field $k$. I am given to believe that there is some relationship between the Tate conjectures and the finiteness of the Tate-Shafarevich group. I imagine that this is because the Tate-Shafarevich group is equal to the Manin obstruction $X(\mathbb{A}_k)^{Br(X)}$ (where $\mathbb{A}_k$ denotes the adeles), and that the Brauer-Grothendieck group of $X$ has something to do with the Tate conjectures.

The relationship between the Tate conjectures and the Brauer-Grothendieck group is not one I understand well. If I understand "Conjectures on Algebraic Cycles in $l$-adic Cohomology" (written by Tate) correctly, the conjecture he calls $T^1(Y)$ (the first Tate conjecture on the variety $Y$) is equivalent to $Br(Y)$ being finite IF $Y$ is a variety over a finite field.

I don't know how to understand this relationship in any way that would be coherent. Is it true that the finiteness of the Tate-Shafarevich group of an abelian variety over a number field is implied by the Tate conjecutres on that abelian variety. Is the reverse true? Why is there even a relationship between these seemingly very different statements?

(I asked this on math-stackexchange, but it seems more appropriate to this forum, so I took it off from there and am posting it here)

After the great answer I got for my previous question about the Tate conjectures What is the intuition behind the concept of Tate twists?, I'm ready for my next one:

Let $X$ be an abelian variety defined over a number field $k$. I am given to believe that there is some relationship between the Tate conjectures and the finiteness of the Tate-Shafarevich group. I imagine that this is because the Tate-Shafarevich group is equal to the Manin obstruction $X(\mathbb{A}_k)^{Br(X)}$ (where $\mathbb{A}_k$ denotes the adeles), and that the Brauer-Grothendieck group of $X$ has something to do with the Tate conjectures.

The relationship between the Tate conjectures and the Brauer-Grothendieck group is not one I understand well. If I understand "Conjectures on Algebraic Cycles in $l$-adic Cohomology" (written by Tate) correctly, the conjecture he calls $T^1(Y)$ (the first Tate conjecture on the variety $Y$) is equivalent to $Br(Y)$ being finite IF $Y$ is a variety over a finite field.

I don't know how to understand this relationship in any way that would be coherent. Is it true that the finiteness of the Tate-Shafarevich group of an abelian variety over a number field is implied by the Tate conjecutres on that abelian variety. Is the reverse true? Why is there even a relationship between these seemingly very different statements?