Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From $p$-modular to $p$-adic Langlands correspondences for $\operatorname{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$: deformations in the non-supercuspidal case

Ramla Abdellatif, Agnès David, Beth Romano, Hanneke Wiersema

Abstract

This paper surveys what is known about (conjectural) $p$-adic and $p$-modular semisimple Langlands correspondences in the non-supercuspidal setting for the unramified quasi-split unitary group $\operatorname{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$. It focuses in particular on the potential of deformation theory to relate these correspondences.

From $p$-modular to $p$-adic Langlands correspondences for $\operatorname{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$: deformations in the non-supercuspidal case

Abstract

This paper surveys what is known about (conjectural) -adic and -modular semisimple Langlands correspondences in the non-supercuspidal setting for the unramified quasi-split unitary group . It focuses in particular on the potential of deformation theory to relate these correspondences.

Paper Structure

This paper contains 30 sections, 18 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Non-supercuspidal representations of $G$ over $\overline{\mathbb F}_p$
  3. Principal series representations and characters of $G$
  4. Special series representations
  5. Classification of non-supercuspidal representations of $G$
  6. A non-supercuspidal semisimple Langlands correspondence
  7. Galois representations and dual groups associated with $G$
  8. From Langlands parameters to $C$-parameters
  9. A Langlands correspondence for non-supercuspidal representations
  10. Deforming non-supercuspidal representations of $G$
  11. Deforming parabolically induced representations
  12. Deformations of special series representations
  13. Deforming Galois parameters
  14. Definition of $C$-parameters in characteristic zero
  15. Inertial types and generic $C$-parameters
  16. ...and 15 more sections

Key Result

Lemma 2.3

Let $(r, \lambda) \in \mathbb Z \times \overline{\mathbb{F}}_{p}^{\times}$ with $0 \leq r \leq p^{2} - 2$. The smooth character $\mu_{\lambda}\omega^{r} : T \to \overline{\mathbb{F}}_{p}^{\times}$ extends to a smooth character $\chi : G \to \overline{\mathbb{F}}_{p}^{\times}$ if, and only if, $\lamb

Theorems & Definitions (74)

  • Remark 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Theorem 2.5
  • proof
  • Remark 2.6
  • Definition 3.1
  • ...and 64 more