In this lecture we will focus on projective hypersurfaces, with quartic K3s in mind, but the methods are more generic, as at some point we will need to do explicit computations.
Take $f \in \mathbf{Z}[x,y,z,w]$ and $X := Z(f) \subset \mathbf{P}^3_{\mathbf{Z}}$.
We may consider the surface $X_{\mathbf{F}_p} := Z(f \bmod p) \subset \mathbf{P}^3(\mathbf{F}_p)$.
Yesterday, we mostly counted classes, but we also saw the cokernel theorem.
If $X$ and $X_{\mathbf{F}_p}$ are smooth then the specialization map is injective
$$\operatorname{Pic}(X_{\mathbf{Q}^{\mathrm{al}}}) \hookrightarrow \operatorname{Pic}(X_{\mathbf{F}_p^{\mathrm{al}}})$$
and $\rho(X_{\mathbf{Q}^{\mathrm{al}}}) = \rho(X_{\mathbf{Q}_p^{\mathrm{al}}}) \leq \rho(X_{\mathbf{F}_p^{\mathrm{al}}})$.
The specialization map has torsion-free cokernel for $p \neq 2$.
Can we use it without computing $\operatorname{Pic}(X_{\mathbf{F}_p^{\mathrm{al}}})$?