Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mod $\ell$ multiplicities in certain $U(4)$ Shimura varieties

Jeffrey Manning

Abstract

We use the Taylor-Wiles-Kisin patching method to investigate the multiplicities with which Hecke eigensystems appear in the mod-$\ell$ cohomology of unitary Shimura sets, associated to central simple algebras of the form $B=M_2(D)$, for $D$ a nonsplit quaternion algebra over a $CM$ field. We follow a similar strategy to the one used in our prior work for Shimura curves, exploiting the natural self-duality in this setting. Our method requires a careful analysis of certain irreducible components of local deformation rings. We introduce and analyze a new local model for local deformation rings specific to the case of a banal prime, which is significantly better behaved than the standard local models for local deformation rings. Our main result is a "multiplicity $2^a$ result", in the case where quaternion algebra $D$ ramifies only at banal primes, where $a$ is the number of places in the discriminant of $D$ which satisfy a certain Galois theoretic condition. We also prove an additional statement about the endomorphism ring of the cohomology of the Shimura set, which has an application to the study of congruence modules.

Mod $\ell$ multiplicities in certain $U(4)$ Shimura varieties

Abstract

We use the Taylor-Wiles-Kisin patching method to investigate the multiplicities with which Hecke eigensystems appear in the mod- cohomology of unitary Shimura sets, associated to central simple algebras of the form , for a nonsplit quaternion algebra over a field. We follow a similar strategy to the one used in our prior work for Shimura curves, exploiting the natural self-duality in this setting. Our method requires a careful analysis of certain irreducible components of local deformation rings. We introduce and analyze a new local model for local deformation rings specific to the case of a banal prime, which is significantly better behaved than the standard local models for local deformation rings. Our main result is a "multiplicity result", in the case where quaternion algebra ramifies only at banal primes, where is the number of places in the discriminant of which satisfy a certain Galois theoretic condition. We also prove an additional statement about the endomorphism ring of the cohomology of the Shimura set, which has an application to the study of congruence modules.

Paper Structure

This paper contains 32 sections, 79 theorems, 267 equations.

Table of Contents

  1. Introduction
  2. Local Theory
  3. Types and the local Langlands Correspondence
  4. Inertial Types
  5. The local Langlands correspondence
  6. The local Jacquet--Langlands correspondence
  7. Local Galois Deformation Rings
  8. Local Deformation Rings for $v\nmid \ell$
  9. Fixed type deformation rings
  10. Unipotent deformation rings
  11. The banal case
  12. Models for local deformation rings in the banal case
  13. Modeling $R^\square(\overline{r})$ via $\mathcal{N}^{\underline{n}}$
  14. The irreducible components of $\mathcal{N}^{\underline{n}}$
  15. Irreducible components with unipotent types
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $F^+$ be a totally real number field with $F/F^+$ a quadratic CM extension. Let $\ell \ge 5$ be a prime which does not ramify in $F$ and such that all primes of $F^+$ lying above $\ell$ split in $F$. Let $E/\mathbb{Q}_\ell$ be a finite extension with ring of integers $\mathcal{O}$ and residue fi Let $\Delta$ be the set of finite places of $F^+$ lying below a prime of $F$ at which $D$ does not

Theorems & Definitions (151)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • ...and 141 more