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Towards the $p$-adic Hodge parameters in semistable representations of $\mathrm{GL}_n(\mathrm{Q}_p)$

Yiqin He

Abstract

Let $ρ_p$ be an $n$-dimensional non-critical semistable $p$-adic Galois representation of the absolute Galois group of $\mathrm{Q}_p$ with regular Hodge--Tate weights. Let $\mathrm{D}$ be the associated $(\varphi,Γ)$-module over the Robba ring. By combining Ding's and Breuil--Ding's methods for the crystalline case with Qian's computation of higher extension groups of locally analytic generalized Steinberg representations, we capture the full information of the $p$-adic Hodge parameters of $ρ_p$ on the automorphic side by considering several Steinberg subquotients of $\mathrm{D}$ and the "crystalline" Hodge parameters between them. These results also admit geometric and Lie-algebraic reformulations on flag varieties related to the moduli space of Hodge parameters. We then construct an explicit locally analytic representation $π_{1}(ρ_p)$ and explicitly describe which Hodge-parameters information of $ρ_p$ it determines. In particular, if the monodromy rank of $ρ_p$ is at most $1$, $π_{1}(ρ_p)$ determines $ρ_p$. When $ρ_p$ comes from a $p$-adic automorphic representation, we show that $π_{1}(ρ_p)$ is a subrepresentation of the $\mathrm{GL}_n(\mathrm{Q}_p)$-representation globally associated to $ρ_p$, under mild hypotheses. Although it is still difficult to construct an explicit representation $π_{1}(ρ_p)$ that determines $ρ_p$, our results provide new evidence for the $p$-adic Langlands program in general semistable cases and demonstrate the broad applicability of Ding's, Breuil--Ding's, and Qian's methods.

Towards the $p$-adic Hodge parameters in semistable representations of $\mathrm{GL}_n(\mathrm{Q}_p)$

Abstract

Let be an -dimensional non-critical semistable -adic Galois representation of the absolute Galois group of with regular Hodge--Tate weights. Let be the associated -module over the Robba ring. By combining Ding's and Breuil--Ding's methods for the crystalline case with Qian's computation of higher extension groups of locally analytic generalized Steinberg representations, we capture the full information of the -adic Hodge parameters of on the automorphic side by considering several Steinberg subquotients of and the "crystalline" Hodge parameters between them. These results also admit geometric and Lie-algebraic reformulations on flag varieties related to the moduli space of Hodge parameters. We then construct an explicit locally analytic representation and explicitly describe which Hodge-parameters information of it determines. In particular, if the monodromy rank of is at most , determines . When comes from a -adic automorphic representation, we show that is a subrepresentation of the -representation globally associated to , under mild hypotheses. Although it is still difficult to construct an explicit representation that determines , our results provide new evidence for the -adic Langlands program in general semistable cases and demonstrate the broad applicability of Ding's, Breuil--Ding's, and Qian's methods.

Paper Structure

This paper contains 25 sections, 51 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation
  4. Lg
  5. Reinterpretation of $p$-adic Hodge parameters
  6. Notations and precise statements for general philosophy
  7. "Crystalline" Hodge parameters inside $\mathbf{D}$
  8. Capture all the Hodge parameters via different refinements
  9. Deformations of semistable $(\varphi,\Gamma)$-modules
  10. Deformations of $\mathcal{F}_{u}$ and $\mathcal{F}_{S_0(u)}$ for $u\in \mathscr W_n^{S_0}$
  11. Reinterpretation and supplements for deformations
  12. Higher extension groups and Breuil-Schraen $\mathscr L$-invariants
  13. On ${\rm Ext}^{\bullet}$ between parabolic inductions of locally analytic generalized Steinberg representations
  14. Capture $p$-adic Hodge parameters through Breuil-Schraen $\mathscr L$-invariants
  15. Locally analytic representations for semistable case
  16. ...and 10 more sections

Key Result

Theorem 1.1

(Theorem twoblocksPare1 and a special case of Proposition twoblocksPare2) If $s=2$, $\mathcal{C}_{\mathrm{ST}}$ determines $\underline{\mathscr L}(\mathbf{D})$. If $s=3$, $\mathcal{C}_{\mathrm{ST}}$ determines $\underline{\mathscr L}(\mathbf{D})$ up to one Hodge parameter. $\blacktriangleleft$$\bla

Theorems & Definitions (85)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Proposition 1.7
  • Theorem 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 75 more