The obstruction map [C-Sertöz]

Compute a $p$-adic approximation of the obstruction map

$\pi : \operatorname{Pic}(X_{\mathbf{F}_p}) \subset H^2_{\mathrm{crys}}(X/\mathbf{Z}_p) \longrightarrow H^2_{\mathrm{crys}}(X/\mathbf{Z}_p) / F^1 H^2_{\mathrm{crys}}(X/\mathbf{Z}_p)$

If $\pi(C) \neq 0$, then $C \notin \operatorname{Pic}(X)$.   (analogous to $\operatorname{Pic}(X_{\mathbf{C}}) = H^{1,1}(X_{\mathbf{C}}) \cap H^2(X, \mathbf{Z})$)

1. compute a $p$-adic approximation of $\operatorname{Frob}_p$
2. compute an approximation of $\operatorname{Pic}(X_{\mathbf{F}_p})_{\mathbf{Q}_p} = \ker( \operatorname{Frob}_p - p \cdot \operatorname{id} \mid H^2_{\mathrm{dR}}(X/\mathbf{Q}_p))$
3. compute an approximation of $\pi_{\mathbf{Q}_p} : \operatorname{Pic}(X_{\mathbf{F}_p})_{\mathbf{Q}_p} \rightarrow H^2_{\mathrm{dR}}(X/\mathbf{Q}_p) / F^1 H^2_{\mathrm{dR}}(X/\mathbf{Q}_p)$
4. $\dim \operatorname{Pic}(X) \leq \dim_{\mathbf{Q}_p} \ker \pi_{\mathbf{Q}_p}$

By picking a basis that respects the Hodge filtration, the map $H^2_{\mathrm{dR}}(X/\mathbf{Q}_p) \rightarrow H^2_{\mathrm{dR}}(X/\mathbf{Q}_p)/F^1_{\mathbf{Q}_p}$ is a coordinate projection.