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Semi-stable representations as limits of crystalline representations

Anand Chitrao, Eknath Ghate, Seidai Yasuda

TL;DR

This work develops a geometric bridge between crystalline and semi-stable two-dimensional Galois representations by embedding crystalline families $V_{k_n,a_n}$ in the Colmez–Chenevier trianguline moduli space and proving their convergence to a prescribed semi-stable limit $V_{k,\mathcal{L}}$. Central to the approach is a detailed rigid-analytic blow-up $\widetilde{U}_r$ and a precise description of the exceptional fiber, which yields an explicit formula for the $\mathcal{L}$-invariant as a logarithmic derivative along a tangent direction: $\mathcal{L} = -\log(\chi(\gamma))\cdot (\lambda/\mu) = 2a_p'(k)/a_p(k)$ in suitable normalizations. The paper then leverages this geometric framework to compute reductions of semi-stable representations from crystalline approximants, relying on the zig-zag conjecture to describe inertia-restricted reductions and connecting with Breuil–Mézard and Guerberoff–Park results for small weights. Additionally, the authors establish a general local-constancy perspective, provide a robust method for computing $\mathcal{L}$-invariants via Benois’ basis, and prove a bounded-Hodge-Tate-weights stability result, contributing a new geometric mechanism for understanding reductions in families of $p$-adic Galois representations.

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-Mézard 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.

Semi-stable representations as limits of crystalline representations

TL;DR

This work develops a geometric bridge between crystalline and semi-stable two-dimensional Galois representations by embedding crystalline families in the Colmez–Chenevier trianguline moduli space and proving their convergence to a prescribed semi-stable limit . Central to the approach is a detailed rigid-analytic blow-up and a precise description of the exceptional fiber, which yields an explicit formula for the -invariant as a logarithmic derivative along a tangent direction: in suitable normalizations. The paper then leverages this geometric framework to compute reductions of semi-stable representations from crystalline approximants, relying on the zig-zag conjecture to describe inertia-restricted reductions and connecting with Breuil–Mézard and Guerberoff–Park results for small weights. Additionally, the authors establish a general local-constancy perspective, provide a robust method for computing -invariants via Benois’ basis, and prove a bounded-Hodge-Tate-weights stability result, contributing a new geometric mechanism for understanding reductions in families of -adic Galois representations.

Abstract

We construct an explicit sequence of crystalline representations of exceptional weights converging to a given irreducible two-dimensional semi-stable representation of . 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 -invariant as a logarithmic derivative. Our result can be used to compute the reduction of in terms of the reductions of the . For instance, using the zig-zag conjecture we recover (resp. extend) the work of Breuil-Mézard and Guerberoff-Park computing the reductions of the for weights at most (resp. ), 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.

Paper Structure

This paper contains 24 sections, 22 theorems, 142 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Limits of crystalline representations
  4. Computing the $\mathcal{L}$-invariant
  5. Reductions of semi-stable representations
  6. Relation with local constancy
  7. The blow-up of $U_r$
  8. The blow-up as a rigid analytic variety
  9. The blow-up map $\pi : \widetilde{U}_r \to U_r$
  10. Proof that $\pi : \widetilde{U}_r \to U_r$ is the blow-up
  11. Points in the exceptional fiber as tangent directions
  12. Explicit bases of $H^1(\mathcal{R}_{\mathbb{Q}_p}(x^r\chi))$ and $\mathcal{L}$-invariants
  13. An explicit description of $G(\vert x \vert, r + 1)$
  14. A partial description of $G'(\vert x \vert, r + 1)$
  15. Another basis of $H^1(\mathcal{R}_{\mathbb{Q}_p}(x^r\chi))$
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $k \geq 3$, $r = k-2$ and $\mathcal{L} \in \mathbb{P}^1(E)$. For $n \geq 1$, let Then i.e., the sequence of points in $\widetilde{\mathcal{T}}_2$ associated to the crystalline representations $V^*_{k_n,a_n}$ converges to the point in $\widetilde{\mathcal{T}}_2$ associated to the semi-stable representation $V^*_{k, \mathcal{L}}$.

Figures (1)

  • Figure 1: Exceptional fiber of the blow-up in the rigid setting.

Theorems & Definitions (47)

  • Theorem 1.1
  • Conjecture 1.2: Zig-Zag Conjecture
  • Theorem 1.3
  • Theorem 1.6: Ber
  • Corollary 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 37 more