What is the relationship between the finiteness of the Tate-Shafarevich group and the Tate conjectures?

Question

(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?

Answer 1

Consider an elliptic curve $X$ over a number field $F$. The Tate conjecture for $X$ is trivially true (By this is meant the statement that Galois invariant subspace of $\ell$-adic cohomology is spanned by algebraic cycles on $X$). But proving the finiteness of the Tate-Shafarevich group is still open in general; the situation is better for $F = Q$.

Even the validity of Tate conjecture for all powers of $X$ does not suffice to prove the finiteness of the Tate-Shafarevich group; there are examples of CM elliptic curves for which the Tate conjecture is known for all powers (work of Milne, Zarhin, Kumar Murty) but the finiteness is as yet unknown.

For function fields, there are results known to the effect that the Tate conjecture implies the finiteness of the Tate-Shafarevich group (but the $p$-part - here $p$ is the characteristic of the function field - is notorious to handle, I am informed); see papers of Zarhin, Schneider, Milne, Ulmer, Trihan-Kato, Geisser.

Answer 2

I believe that Tate was led to formulate his conjecture (first over finite fields, and then, by analogy, over finitely generated fields, in particular number fields) by considering the function field case of BSD. A careful examination of BSD for an elliptic curve over the function field of a curve over a finite field shows that the statement that order of vanishing of the $L$-function equals the Mordell--Weil rank is equivalent to the Tate conjecture (on divisors) for the corresponding elliptic surface over the finite field(the one obtained by taking the minimal regular model of the elliptic curve).

Off the top of my head, I don't remember what happens if one goes further, and tries to understand the leading term of the $L$-function in geometric terms.

You should probably look at the original Bourbaki seminar on this, as well as at the papers mentioned by SGP in their answer.