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Eigenvarieties over CM fields and trianguline representations

Vaughan McDonald

TL;DR

The work proves that Galois representations attached to points on derived GL$_n$-eigenvarieties over CM fields are trianguline at $p$ with Sen weights dictated by eigenvariety data, aligning with Hansen’s conjectural framework. The authors achieve this by lifting to a unitary group $ ilde{G}$ of signature $(n,n)$, establishing an analytic continuation for its eigenvariety localized at an Eisenstein ideal, and then transferring trianguline-structure information back to GL$_n$ via a degree-shifting strategy that leverages boundary and middle-degree cohomology. Central to the argument are (i) a Satake-transfer–based embedding of GL$_n$ eigenvarieties into unitary-middle-degree eigenvarieties, (ii) a small-slope/torsion-vanishing analysis of boundary cohomology, and (iii) a deformation-theoretic trianguline framework that yields local-global compatibility and explicit triangulation parameters. The results extend Hansen’s conjecture to a CM-field setting without requiring the field to avoid imaginary quadratic components, and they illuminate how middle-degree unitary–GL transfers control trianguline deformations. This work advances p-adic Langlands program goals by linking eigenvariety geometry, trianguline varieties, and Galois-deformation spaces in a CM-field context, with potential applications to wider classes of groups lacking discrete series.

Abstract

We show that the Galois representations associated to points on certain (derived) eigenvarieties for $\operatorname{GL}_n$ over a CM field are trianguline with the expected Sen weights, verifying an analogue of a conjecture of Hansen in many cases. The proof follows the strategy of passing to a larger unitary group $\widetilde{G}$ of signature $(n,n)$, where the key new input is an analytic continuation result for an eigenvariety for $\widetilde{G}$ localised at an Eisenstein maximal ideal. We also discuss the (subtle) relation of eigenvarieties for $\operatorname{GL}_n$ with the trianguline variety.

Eigenvarieties over CM fields and trianguline representations

TL;DR

The work proves that Galois representations attached to points on derived GL-eigenvarieties over CM fields are trianguline at with Sen weights dictated by eigenvariety data, aligning with Hansen’s conjectural framework. The authors achieve this by lifting to a unitary group of signature , establishing an analytic continuation for its eigenvariety localized at an Eisenstein ideal, and then transferring trianguline-structure information back to GL via a degree-shifting strategy that leverages boundary and middle-degree cohomology. Central to the argument are (i) a Satake-transfer–based embedding of GL eigenvarieties into unitary-middle-degree eigenvarieties, (ii) a small-slope/torsion-vanishing analysis of boundary cohomology, and (iii) a deformation-theoretic trianguline framework that yields local-global compatibility and explicit triangulation parameters. The results extend Hansen’s conjecture to a CM-field setting without requiring the field to avoid imaginary quadratic components, and they illuminate how middle-degree unitary–GL transfers control trianguline deformations. This work advances p-adic Langlands program goals by linking eigenvariety geometry, trianguline varieties, and Galois-deformation spaces in a CM-field context, with potential applications to wider classes of groups lacking discrete series.

Abstract

We show that the Galois representations associated to points on certain (derived) eigenvarieties for over a CM field are trianguline with the expected Sen weights, verifying an analogue of a conjecture of Hansen in many cases. The proof follows the strategy of passing to a larger unitary group of signature , where the key new input is an analytic continuation result for an eigenvariety for localised at an Eisenstein maximal ideal. We also discuss the (subtle) relation of eigenvarieties for with the trianguline variety.

Paper Structure

This paper contains 35 sections, 60 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. History
  3. Proof method
  4. Related work for CM fields
  5. What our work does
  6. Contents
  7. Notation and conventions
  8. Preliminaries on (co)homology
  9. Locally symmetric spaces and completed (co)homology
  10. Completed (co)homology
  11. Complexes for completed cohomology
  12. Review of $G$-extensions and derived Jacquet functors
  13. Jacquet functors in terms of homology
  14. Setting of CM fields
  15. Hecke algebras
  16. ...and 20 more sections

Key Result

Theorem 1.1

Suppose $\mathfrak{m} \subset {\mathbf T}(K^p)$ is decomposed generic (in the sense of Caraiani--Scholze) and non-Eisenstein, $F$ contains an imaginary quadratic field $F_0$, and $p$ is a prime that splits in $F_0$. Then for any point $(x,\delta = (\delta_1,\dots, \delta_n)) \in \mathcal{E}^i(K^p)_{

Theorems & Definitions (133)

  • Theorem 1.1
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7: Completed versus overconvergent cohomology
  • Remark 1.8
  • Remark 2.2
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • ...and 123 more