The Tate conjecture for abelian varieties

Question

Let $k$ be a number field. Recall that Faltings proved the famous Tate conjecture, which states that for any abelian variety $X$ over $k$ and any prime $\ell$, the natural map $$\mathrm{End}(X) \otimes \mathbb{Q}_\ell \to \mathrm{End}(V_\ell(X))^{\mathrm{Gal}(\bar k/k)},\qquad (*)$$ is an isomorphism, where $V_\ell(X)$ denotes the usual Tate module of $X$ tensored with $\mathbb{Q}_\ell$.

If I understand it correctly however, this is but one part of the Tate conjecture, which in general states that the cycle class map $$A^r(X) \otimes \mathbb{Q}_\ell \to H^{2r}(\bar X, \mathbb{Q}_\ell)^{\mathrm{Gal}(\bar k/k)},$$ is an isomorphism for all $r$. The "Tate conjecture" as I state it in $(*)$ is but the case $r=1$ of this conjecture.

Is the full Tate conjecture still open for abelian varieties? Are there any interesting special cases where it is known?

Answer 1

Let me put it this way, Tate's conjecture for abelian varieties is known to imply the Hodge conjecture for abelian varieties, and the last is very much open for this class. For the implication, see the article by Deligne and Milne on "Hodge cycles on abelian varieties" (you can get a copy off of Milne's website).

There are lot's of interesting cases known for Tate. People like Zarhin, who seems to be around and has done a lot of this, can give more precise statements.