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Change of weights operations for triangulated $({\varphi},Γ)$-modules

Zichuan Wang

TL;DR

The work develops a general theory of change of weights for triangulated $(\varphi,\Gamma_K)$-modules in families, via pullback operations that adjust Sen weights and act compatibly with triangulations. It situates these weight shifts inside the analytic Emerton–Gee stack framework, defining weight-uniform trianguline substacks and showing their rigidity under the pullbacks $p_{i,\sigma}$ when regularity is preserved. For non-critical crystabelline cases with $K=\mathbb{Q}_p$, it proves a precise intertwining with translation functors on locally analytic $\mathrm{GL}_n(K)$-representations, establishing a 1-1 correspondence between weight shifts and automorphic translations using Ding’s construction. The arguments blend factorization of Sen polynomials, triangulation theory, Beilinson–Beilinson–Beilinson style universal extensions, and stack-theoretic pullbacks to connect $(\varphi,\Gamma_K)$-module operations with automorphic translation, yielding a robust, geometric framework for weight-shifting phenomena.

Abstract

Zhixiang Wu has shown the existence of "change of weights" operation on $(\varphi,Γ)$-modules in families (arxiv:2405.16637). We interpret it in the trianguline case as pullbacks with a discussion on related stacks. Finally, we prove that it intertwines well with translation functors via a 1-1 correspondence defined by Yiwen Ding (arXiv:2407.21237) in the non-critical crystabelline case.

Change of weights operations for triangulated $({\varphi},Γ)$-modules

TL;DR

The work develops a general theory of change of weights for triangulated -modules in families, via pullback operations that adjust Sen weights and act compatibly with triangulations. It situates these weight shifts inside the analytic Emerton–Gee stack framework, defining weight-uniform trianguline substacks and showing their rigidity under the pullbacks when regularity is preserved. For non-critical crystabelline cases with , it proves a precise intertwining with translation functors on locally analytic -representations, establishing a 1-1 correspondence between weight shifts and automorphic translations using Ding’s construction. The arguments blend factorization of Sen polynomials, triangulation theory, Beilinson–Beilinson–Beilinson style universal extensions, and stack-theoretic pullbacks to connect -module operations with automorphic translation, yielding a robust, geometric framework for weight-shifting phenomena.

Abstract

Zhixiang Wu has shown the existence of "change of weights" operation on -modules in families (arxiv:2405.16637). We interpret it in the trianguline case as pullbacks with a discussion on related stacks. Finally, we prove that it intertwines well with translation functors via a 1-1 correspondence defined by Yiwen Ding (arXiv:2407.21237) in the non-critical crystabelline case.

Paper Structure

This paper contains 25 sections, 32 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Background and main results
  3. Change of weights for triangulated $(\varphi,\Gamma_K)$-modules
  4. Using Sen polynomials
  5. The point of view of the analytic Emerton-Gee stacks
  6. Relation with translation functor under Ding's crystabelline correspondence
  7. Structure of the paper
  8. Acknowledgements
  9. Preliminaries
  10. Notations and conventions
  11. Extensions of $(\varphi,\Gamma_K)$-modules
  12. Triangulations on trianguline $(\varphi,\Gamma)$-modules
  13. Very generic case
  14. Crystabelline case
  15. More general cases
  16. ...and 10 more sections

Key Result

Lemma 2.2

Let ${\bf k}=(k_\sigma)_\sigma \in {\mathbb N}^{[K:{\mathbb Q}_p]}$ and let be an exact sequence of $(\varphi,\Gamma_K)$-modules over ${\mathcal{R}}_{K,A}$. Then, the pushout $\iota_{\bf k}(D)$ of $D$ along $D_1\hookrightarrow t^{\bf -k}D_1$ and the pullback $p_{\bf k}(D)$ of $D$ along $t^{\bf k} D_2\hookrightarrow D_2$ are related by a commutative diagram \begin{tikzcd} 0

Theorems & Definitions (87)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Example 2.4
  • Theorem 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • ...and 77 more