Tate conjecture

Let $k$ be a field that is finitely generated over its prime field, such as a number field or a finite field. Fix a separable closure $\bar k$ and let $G_k=\operatorname{Gal}(\bar k/k)$. Let $\ell$ be a prime different from $\operatorname{char}k$. Let $X$ be a smooth projective geometrically irreducible variety over $k$, let $i\ge0$, and write

$$W^{i}_\ell(X)=H^{2i}_{\mathrm{\acute et}}\bigl(X_{\bar k},\mathbb Q_\ell(i)\bigr),$$

where $\mathbb Q_\ell(i)$ is the $i$-th Tate twist. This is a finite-dimensional $\mathbb Q_\ell$-vector space with a continuous action of $G_k$. Let $Z^i(X)$ be the group of algebraic cycles of codimension $i$ on $X$ defined over $k$. The $\ell$-adic cycle class map sends $Z^i(X)$ into the Galois invariants, so it gives a $\mathbb Q_\ell$-linear map

$$c^i_\ell\colon Z^i(X)\otimes_{\mathbb Z}\mathbb Q_\ell\longrightarrow W^{i}_\ell(X)^{G_k}.$$

Conjecture $T^i(X,\ell)$ (Tate). The map $c^i_\ell$ is surjective. In words, every Galois-invariant class in $H^{2i}_{\mathrm{\acute et}}(X_{\bar k},\mathbb Q_\ell(i))$ is a $\mathbb Q_\ell$-linear combination of classes of algebraic cycles defined over $k$.

A solution must prove $T^i(X,\ell)$ for every such field $k$, every prime $\ell\ne\operatorname{char}k$, every $X$ and every $i$. Replacing $k$ by a finite extension, the conjecture also predicts that every class fixed by an open subgroup of $G_k$ is algebraic over some finite extension. The cases $i=0$ and $i=\dim X$ are trivial. The case $i=1$ (divisors) is already open in general, both over number fields and over finite fields.

Background

Tate stated the conjecture in 1963 at the Woods Hole and Purdue conferences [1]. It is the $\ell$-adic analogue of the Hodge conjecture. Tate also proposed two companion statements over finite fields: the $G_k$-action on $W^i_\ell(X)$ is semisimple, and the order of the pole of the zeta function $Z(X,q^{-s})$ at $s=i$ equals the rank of the group of cycles of codimension $i$ modulo numerical equivalence.

For divisors on a surface $X$ over a finite field, $T^1(X,\ell)$ is equivalent to finiteness of the $\ell$-primary part of the Brauer group of $X$. For abelian varieties $A$ and $B$ over $k$, $T^1(A\times B,\ell)$ is equivalent to Tate's isogeny conjecture: the natural map $\operatorname{Hom}(A,B)\otimes\mathbb Z_\ell\to\operatorname{Hom}_{G_k}(T_\ell A,T_\ell B)$ is an isomorphism. The conjecture would also show that the motivic Galois group is determined by $\ell$-adic Galois representations.