Semi-stable representations as limits of crystalline representations

We construct an explicit sequence of crystalline representations converging to a given irreducible two-dimensional semi-stable representation of $\mathrm{Gal}({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$. The convergence takes place in the blow-up space of two-dimensional trianguline representations studied by Colmez and Chenevier. The process of blow-up is described in detail in the rigid analytic setting and may be of independent interest. Our convergence result can be used to compute the reductions of any irreducible two-dimensional semi-stable representation in terms of the reductions of certain crystalline representations of exceptional weight. In particular, this provides an alternative approach to computing the reductions of irreducible two-dimensional semi-stable representations that circumvents the somewhat technical machinery of integral $p$-adic Hodge theory. For instance, using the zig-zag conjecture made in [Gha21] on the reductions of crystalline representations of exceptional weights, we recover completely the work of Breuil-M\'ezard and Guerberoff-Park on the reductions of irreducible semi-stable representations of weights at most $p+1$, at least on the inertia subgroup. In the cases where the zig-zag conjecture is known, we are further able to obtain some new information about the reductions for small odd weights. Finally, we use the above ideas to explain some apparent violations to local constancy in the weight of the reductions of crystalline representations of small weight which were noticed in [Gha21].