The p-adic weight-monodromy conjecture

Let $K$ be a finite extension of ${\mathbb Q}_p$ with residue field ${\mathbb F}_q$, let $K_0$ be the fraction field of $W({\mathbb F}_q)$, and let $X$ be a proper smooth variety over $K$ with a proper strictly semistable model. For $V=H^w_{\text{\rm \'et}}(X_{\overline K},{\mathbb Q}_p)$, let $D_{\text{st}}(V)$ be its semistable period module, equipped with monodromy and weight filtrations $M_\bullet$ and $W_\bullet$.

The $p$-adic weight-monodromy conjecture. The filtrations satisfy

$$M_iD_{\text{st}}(V)=W_{i+w}D_{\text{st}}(V)\qquad\text{for all }i.$$

The comparison isomorphism identifies this with the corresponding assertion for log-crystalline cohomology. The paper proves the conjecture in its p-adically uniformized setting, while the general semistable case is not settled here.