Artin–Tate conjecture for smooth proper surfaces over finite fields

Let $k=\mathbb F_q$ be a finite field, let $T=\operatorname{Spec} k$, and let $X$ be a smooth proper surface over $T$. Let $P_2(X,t)$ be the characteristic polynomial of Frobenius on $H^2(X\times_T\bar T,\mathbb Q_\ell)$, let $\rho(X)$ be the rank of $\operatorname{NS}(X)$, let $\operatorname{Br}(X)$ be the Brauer group, and let $\alpha(X)=\chi(X,\mathcal O_X)-1+\dim(A)$, where $A=\operatorname{Pic}^{\mathrm{red},0}_{X/k}$. Write $\Delta_{\mathrm{ar}}(\operatorname{NS}(X))$ for the discriminant of the height pairing on $\operatorname{NS}(X)$. Artin–Tate conjecture. The group $\operatorname{Br}(X)$ is finite, $\operatorname{ord}_{s=1}P_2(X,q^{-s})=\rho(X)$, and

$$P^*_2(X,q^{-1})=\lim_{s\to1}\frac{P_2(X,q^{-s})}{(s-1)^{\rho(X)}}$$

satisfies

$$P^*_2(X,q^{-1})=[\operatorname{Br}(X)]\cdot\Delta_{\mathrm{ar}}(\operatorname{NS}(X))\cdot q^{-\alpha(X)}.$$

This conjecture relates the order of the pole of the surface zeta function to algebraic cycles and predicts its leading coefficient in terms of the Brauer group and the discriminant of the arithmetic height pairing. Its status is not resolved in the supplied source.