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Toward the $p$-adic Hodge parameters in the potentially crystalline representations of $\mathrm{GL}_n$

Yiqin He

Abstract

Let $p$ be a prime number, $n$ an integer $\geq 2$, and $L$ a finite extension of $\mathrm{Q}_p$. Let $ρ_L$ be an $n$-dimensional (non-critical but not necessary generic) potentially crystalline $p$-adic Galois representation of the absolute Galois groups of $L$ of regular Hodge-Tate weights. By generalizing the previous results and strategy for the crystabelline case of Ding and the recent work of Breuil-Ding, we construct an explicit locally analytic representation $π_{1}(ρ_L)$, and describe explicitly the information of Hodge filtration of $ρ_L$ it determines. When $ρ_L$ comes from a patched $p$-adic automorphic representation, we show that $π_{1}(ρ_L)$ is a subrepresentation of the $\mathrm{GL}_n(L)$-representation globally associated to $ρ_L$, under some mild hypothesis.

Toward the $p$-adic Hodge parameters in the potentially crystalline representations of $\mathrm{GL}_n$

Abstract

Let be a prime number, an integer , and a finite extension of . Let be an -dimensional (non-critical but not necessary generic) potentially crystalline -adic Galois representation of the absolute Galois groups of of regular Hodge-Tate weights. By generalizing the previous results and strategy for the crystabelline case of Ding and the recent work of Breuil-Ding, we construct an explicit locally analytic representation , and describe explicitly the information of Hodge filtration of it determines. When comes from a patched -adic automorphic representation, we show that is a subrepresentation of the -representation globally associated to , under some mild hypothesis.
Paper Structure (26 sections, 57 theorems, 291 equations)

This paper contains 26 sections, 57 theorems, 291 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation
  4. Lg
  5. Lg
  6. Hodge filtration and higher intertwining
  7. Hodge parameters
  8. Lg
  9. Reinterpretation and Supplements for defermations
  10. Hodge parameters and higher intertwining for potentially crystalline case
  11. Locally analytic representations for potentially crystalline case
  12. Lg
  13. Preliminary on extension groups and principal series
  14. Constructions of locally analytic representations
  15. Main theorems
  16. ...and 11 more sections

Key Result

Lemma 1.1

(See Lemmas imageofkappa and studyforkerkappau) For $?\in\{0,\circ\}$, we have $\ker(\kappa^{?}_u)=\ker \kappa^{?}_1\cap {\rm Ext}^{1,?}_{u}({\mathbf{D}},{\mathbf{D}})$ and

Theorems & Definitions (126)

  • Lemma 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 3.1
  • proof
  • ...and 116 more