On the Tate conjectures for divisors on a fibred variety and on its generic scheme fibre in the case of finite characteristic

We investigate interrelations between the Tate conjecture for divisors on a fibred variety over a finite field and the Tate conjecture for divisors on the generic scheme fibre under the condition that the generic scheme fibre has zero irregularity. Let $\pi:X\to C$ be a surjective morphism of smooth projective varieties over a finite field $\mathbb{F}_q$ of characteristic $p$, $C$ is a curve and the generic scheme fibre of $\pi$ is a smooth variety $V$ over the field $k=\kappa(C)$ of rational functions of the curve $C$, $\overline k$ is an algebraic closure of the field $k$, $k^s$ is its separable closure, $\operatorname{NS}(V)$ is the Néron–Severi group of classes of divisors on the variety $V$ modulo algebraic equivalence, and assume that the following conditions hold: $H^1(V\otimes\overline k,\mathcal O_{V\otimes\,\overline k})=0$, $\operatorname{NS}(V)=\operatorname{NS}(V\otimes\overline k)$. If, for a prime number $l$ not dividing ${\operatorname{Card}}([\operatorname{NS}(V)]_{\operatorname{tors}})$ and different from the characteristic of the field $\mathbb{F}_q$, the following relation holds $\operatorname{NS}(V)\otimes\mathbb{Q}_l\,\,\widetilde{\rightarrow}\,\,[H^2(V\otimes k^{\operatorname{s}},\mathbb{Q}_l(1))]^{\operatorname{Gal}( k^{\operatorname{s}}/k)} $ (in other words, if the Tate conjecture for divisors on $V$ holds), then for any prime number $l\neq\operatorname{char}(\mathbb{F}_q)$ the Tate conjecture holds for divisors on $X$: $\operatorname{NS}(X)\otimes\mathbb{Q}_l\,\,\widetilde{\rightarrow} \,\,[H^2(X\otimes\overline{\mathbb{F}}_q,\mathbb{Q}_l(1))]^{\operatorname{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)}$. In particular, it follows from this result that the Tate conjecture for divisors on an arithmetic model of a $\operatorname{K}3$ surface over a sufficiently large global field of finite characteristic different from $2$ holds as well.