Logarithmic geometry and Frobenius, II

Based on the strong analogy between the category of log mixed Hodge structures and the category ${\mathcal{A}}_X$ of $\mathrm{\ell }$-adic nature, which we have introduced in the previous part and is closely related to the weight-monodromy conjecture, we prove the $\mathrm{\ell }$-adic analogues of some theorems in Hodge theory related to the SL(2)-orbit theorem.