Semi-stable representations as limits of crystalline representations

TL;DR

This paper constructs explicit limits of crystalline Galois representations converging to semi-stable representations, providing tools for understanding reductions and invariants in p-adic Hodge theory.

Contribution

It explicitly constructs sequences of crystalline representations converging to semi-stable representations within the trianguline framework, and relates this to invariants and reduction computations.

Findings

Constructed explicit sequences of crystalline representations converging to semi-stable ones.

Reproduced and extended formulas for the ${\ m L}$-invariant.

Provided new insights into reductions of crystalline representations for small weights.

Abstract

We construct an explicit sequence $V_{k_n,a_n}$ of crystalline representations of exceptional weights converging to a given irreducible two-dimensional semi-stable representation $V_{k,{\mathcal{L}}}$ 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. Also, we recover a formula of Stevens expressing the ${\mathcal{L}}$-invariant as a logarithmic derivative. Our result can be used to compute the reduction of $V_{k,{\mathcal{L}}}$ in terms of the reductions of the $V_{k_n,a_n}$. For instance, using the zig-zag conjecture we recover (resp. extend) the work of Breuil-M\'ezard and Guerberoff-Park computing the reductions of the $V_{k,{\mathcal{L}}}$ for weights at most $p-1$ (resp. $p+1$), at least on the inertia subgroup. In the cases where zig-zag is known, we are further able to obtain some new information about the reductions for small odd weights. Finally, we explain some apparent violations to local constancy in the weight of the reductions of crystalline representations of small weight.