Tate's conjecture on polarized endomorphisms and étale cohomology

Let $X$ be a smooth projective variety of dimension $d$ over the algebraically closed field $\mathbb{k}$, and let $f$ be a ($q$-)polarized endomorphism of $X$: there is an ample divisor $H_X$ on $X$ such that $f^*H_X\sim qH_X$ for some integer $q>1$. For a prime $\ell\ne\operatorname{char}(\mathbb{k})$, write $H^i_{\operatorname{\acute{e}t}}(X,\mathbf{Q}_\ell)$ for the $i$th $\ell$-adic étale cohomology group. An algebraic integer $\lambda$ is a $q$-Weil integer of weight $w$ if every complex embedding $\iota$ of $\mathbf{Q}(\lambda)$ satisfies $|\iota(\lambda)|^2=q^w$. Tate's conjecture. For every $0\le i\le 2d$, all eigenvalues of

$$f^*|_{H^i_{\operatorname{\acute{e}t}}(X,\mathbf{Q}_\ell)}$$

are $q$-Weil integers of weight $i$, and the linear transformation

$$f^*|_{H^i_{\operatorname{\acute{e}t}}(X,\mathbf{Q}_\ell)}$$

is semisimple. When $\mathbb{k}=\overline{\mathbf{F}}_q$ and $f$ is the geometric Frobenius endomorphism $\mathsf{Frob}_q$ of $X$, the first assertion is Weil's Riemann hypothesis, proved by Deligne. The conjecture is motivated by the Kähler analogue of Weil's Riemann hypothesis and predicts parity and symmetry properties for polarized endomorphisms on cohomology; the semisimplicity assertion remains part of the general conjectural picture.