Semistable representations as limits of crystalline representations

Chitrao, Anand; Ghate, Eknath; Yasuda, Seidai

doi:10.2140/ant.2025.19.1049

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Semistable representations as limits of crystalline representations

Anand Chitrao, Eknath Ghate and Seidai Yasuda

Vol. 19 (2025), No. 6, 1049–1097

DOI: 10.2140/ant.2025.19.1049

Abstract

We construct an explicit sequence $V_{k_{n}, a_{n}}$ of crystalline representations of exceptional weights converging to a given irreducible two-dimensional semistable representation $V_{k, ℒ}$ of $Gal ({\bar{ℚ}}_{p}$ / $ℚ_{p})$ . 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 $ℒ$ -invariant as a logarithmic derivative.

Our result can be used to compute the reduction of $V_{k, ℒ}$ 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 and Mézard and Guerberoff and Park computing the reductions of the $V_{k, ℒ}$ for weights $k$ 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.

Keywords

Galois representations, ($\varphi$, $\Gamma$)-modules, $\mathcal{L}$-invariants, rigid geometry, blow-ups

Mathematical Subject Classification

Primary: 11F80, 14G22

Milestones

Received: 19 July 2022

Revised: 29 February 2024

Accepted: 29 April 2024

Published: 14 May 2025

Authors

Anand Chitrao
School of Mathematics Tata Institute of Fundamental Research Mumbai India

Eknath Ghate
School of Mathematics Tata Institute of Fundamental Research Mumbai India

Seidai Yasuda
Department of Mathematics Hokkaido University Hokkaido Japan

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Vol. 19, No. 6, 2025