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Ihara's Lemma for $\mathrm{GL}_d$: the limit case

Pascal Boyer

TL;DR

The paper advances the generalization of Ihara's lemma to $\mathrm{GL}_d$ in the limit case, a key step for non-minimal $R=\mathbb{T}$ theorems in higher rank. It develops a geometric framework using KHT-Shimura varieties, nearby cycles, and Newton filtrations, and leverages level-raising results (BLGGT and Gee–Annalen) to control non-generic contributions. The main result confirms Clozel–Harris–Taylor conjectures in the limit setting, establishing genericity for all irreducible $\mathrm{GL}_d(F_v)$-submodules in the localized middle cohomology and building a robust lattice-theoretic viewpoint. This work deepens the connection between automorphic forms, Galois representations, and geometric methods, with potential implications for $R=\mathbb{T}$-type results and arithmetic applications such as special values of adjoint L-functions and local-global compatibility phenomena.

Abstract

Clozel, Harris and Taylor proposed conjectural generalizations of the classical Ihara's lemma for $\mathrm{GL}_2$, to higher dimensional similitude groups. We prove these conjectures in the so called limit case, which after base change is the essential one, under any hypothesis allowing level raising.

Ihara's Lemma for $\mathrm{GL}_d$: the limit case

TL;DR

The paper advances the generalization of Ihara's lemma to in the limit case, a key step for non-minimal theorems in higher rank. It develops a geometric framework using KHT-Shimura varieties, nearby cycles, and Newton filtrations, and leverages level-raising results (BLGGT and Gee–Annalen) to control non-generic contributions. The main result confirms Clozel–Harris–Taylor conjectures in the limit setting, establishing genericity for all irreducible -submodules in the localized middle cohomology and building a robust lattice-theoretic viewpoint. This work deepens the connection between automorphic forms, Galois representations, and geometric methods, with potential implications for -type results and arithmetic applications such as special values of adjoint L-functions and local-global compatibility phenomena.

Abstract

Clozel, Harris and Taylor proposed conjectural generalizations of the classical Ihara's lemma for , to higher dimensional similitude groups. We prove these conjectures in the so called limit case, which after base change is the essential one, under any hypothesis allowing level raising.
Paper Structure (25 sections, 182 equations, 4 figures)

This paper contains 25 sections, 182 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Ihara's lemma: origin and proofs
  3. Generalisations of Ihara's Lemma
  4. Main results
  5. Preliminaries
  6. Representations of $\GL_d(M)$
  7. Weil--Deligne inertial types
  8. Kottwiz--Harris--Taylor Shimura varieties
  9. Nearby cycles and filtrations
  10. Filtrations of stratification of $\Psi_\varrho$
  11. The canonical filtration of $H^0(\sh_{K,\bar{s}_v},\Psi_{K,\overline \Zm_l})_{\mathfrak m}$ is strict
  12. Local and global monodromy
  13. Uniformity of automorphic sub-lattices
  14. Infinite level at $v$
  15. Typicity
  16. ...and 10 more sections

Figures (4)

  • Figure 1: Exchange process
  • Figure 2: Filtrations of stratification of $\Psi_{K,\varrho}$
  • Figure 3: Three filtrations of $H^0(\sh_{K^v(\oo),\bar{s}_v},\Psi_\varrho)_{\mathfrak m}$ when $s_\varrho=3$
  • Figure 4: First step

Theorems & Definitions (9)

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