Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Endoscopy on $\mathrm{SL}_2$-eigenvarieties

Christian Johansson, Judith Ludwig

Abstract

In this paper, we study p-adic endoscopy on eigenvarieties for $\mathrm{SL}_2$ over totally real fields, taking a geometric perspective. We show that non-automorphic members of endoscopic L-packets of regular weight contribute eigenvectors to overconvergent cohomology at critically refined endoscopic points on the eigenvariety, and we precisely quantify this contribution. This gives a new perspective on and generalizes previous work of the second author. Our methods are geometric, and are based on showing that the $\mathrm{SL}_2$-eigenvariety is locally a quotient of an eigenvariety for $\mathrm{GL}_2$, which allows us to explicitly describe the local geometry of the $\mathrm{SL}_2$-eigenvariety. In particular, we show that it often fails to be Gorenstein at such points.

Endoscopy on $\mathrm{SL}_2$-eigenvarieties

Abstract

In this paper, we study p-adic endoscopy on eigenvarieties for over totally real fields, taking a geometric perspective. We show that non-automorphic members of endoscopic L-packets of regular weight contribute eigenvectors to overconvergent cohomology at critically refined endoscopic points on the eigenvariety, and we precisely quantify this contribution. This gives a new perspective on and generalizes previous work of the second author. Our methods are geometric, and are based on showing that the -eigenvariety is locally a quotient of an eigenvariety for , which allows us to explicitly describe the local geometry of the -eigenvariety. In particular, we show that it often fails to be Gorenstein at such points.
Paper Structure (41 sections, 76 theorems, 256 equations)

This paper contains 41 sections, 76 theorems, 256 equations.

Table of Contents

  1. Introduction
  2. Heuristics on how to compute eigenspaces of overconvergent automorphic forms
  3. Results and methods
  4. Previous results
  5. Failure of classicality and $p$-adic endoscopy
  6. Eigenvarieties at endoscopic points, and singularities
  7. Outline of the paper
  8. Notation
  9. Automorphic and Galois representations
  10. Locally symmetric spaces and local systems
  11. Automorphic representations
  12. The Galois representation associated to a cuspidal cohomological automorphic representation
  13. L-packets and classical functoriality from $\mathrm{GL}\xspace_2$ to $\mathrm{SL}\xspace_2$
  14. Pseudorepresentations
  15. Deformation Theory
  16. ...and 26 more sections

Key Result

Theorem A

The following holds:

Theorems & Definitions (148)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Remark 1.2.1
  • Lemma 2.1.1
  • proof
  • Proposition 2.1.2
  • proof
  • ...and 138 more