Let $X$ be a K3 over $\overline{\mathbb{F}_p}$. The (crystalline version's) Tate conjecture predicts:

$c_1: Pic(X)\otimes\mathbb{Q}_p\rightarrow H^2_{crys}(X/W)^{\Phi=p}\otimes\mathbb{Q}_p$

is an isomorphism.

If $\lambda=1-\frac{1}{h}$ is the smallest slope of $H^2_{crys}(X/W)$, then $H^2_{crys}(X/W)$ must be isogenous to $H_{\lambda}\oplus H_1^{22-2h} \oplus H_{2-\lambda}$, where $H_\alpha$ is the F-crystal that $\Phi^t=p^s$ for $\alpha=s/t$. The $\Phi=p$ parts of the latter F-crystal apparently has rank $22-2h$, so the $\Phi=p$ part of $H^2_{crys}(X/W)$ must also has rank $22-2h$? Which means the above conjecture is equivalent to $\rho=22-2h$ always hold true? But the later one is not true in general, as a generic K3 in char $p$ should have Picard number $2<22-2h$ in general.

Where was I wrong?

Let $X$ be a K3 over $\overline{\mathbb{F}_p}$. The (crystalline version's) Tate conjecture predicts:

$c_1: Pic(X)\otimes\mathbb{Q}_p\rightarrow H^2_{crys}(X/W)^{\Phi=p}\otimes\mathbb{Q}_p$

is an isomorphism.

If $\lambda=1-\frac{1}{h}$ is the smallest slope of $H^2_{crys}(X/W)$, then $H^2_{crys}(X/W)$ must be isogenous to $H_{\lambda}\oplus H_1^{22-2h} \oplus H_{2-\lambda}$, where $H_\alpha$ is the F-crystal that $\Phi^t=p^s$ for $\alpha=s/t$. The $\Phi=p$ parts of the latter F-crystal apparently has rank $22-2h$, so the $\Phi=p$ part of $H^2_{crys}(X/W)$ must also has rank $22-2h$? Which means the above conjecture is equivalent to $\rho=22-2h$ always hold true? But the later one is not true in general, as a general K3 in char $p$ should have Picard number $2$, which is usually less than $22-2h$.

Where was I wrong?

Let $X$ be a K3 over $\overline{\mathbb{F}_q}$. The (crystalline version's) Tate conjecture predicts:

$c_1: Pic(X)\otimes\mathbb{Q}_p\rightarrow H^2_{crys}(X/W)^{\Phi=p}\otimes\mathb{Q}_p$

is an isomorphism.

Let $X$ be a K3 over $\overline{\mathbb{F}_p}$. The (crystalline version's) Tate conjecture predicts:

$c_1: Pic(X)\otimes\mathbb{Q}_p\rightarrow H^2_{crys}(X/W)^{\Phi=p}\otimes\mathbb{Q}_p$

is an isomorphism.

If $\lambda=1-\frac{1}{h}$ is the smallest slope of $H^2_{crys}(X/W)$, then $H^2_{crys}(X/W)$ must be isogenous to $H_{\lambda}\oplus H_1^{22-2h} \oplus H_{2-\lambda}$, where $H_\alpha$ is the F-crystal that $\Phi^t=p^s$ for $\alpha=s/t$. The $\Phi=p$ parts of the latter F-crystal apparently has rank $22-2h$, so the $\Phi=p$ part of $H^2_{crys}(X/W)$ must also has rank $22-2h$? Which means the above conjecture is equivalent to $\rho=22-2h$ always hold true? But the later one is not true in general, as a generic K3 in char $p$ should have Picard number $2<22-2h$ in general.

Where was I wrong?