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On the potential automorphy and the local-global compatibility for the monodromy operators at $p \neq l$ over CM fields

Kojiro Matsumoto

TL;DR

The paper advances the study of local-global compatibility for GL_n over CM fields at p ≠ l by extending potential automorphy methods to higher dimensions and introducing a parity-based, monodromy-aware framework using parahoric levels. It develops a new approach to relate Weil–Deligne monodromy with automorphic realizations, enabling automatic compatibility at non-l places and a density-one set of primes, with unconditional Ramanujan results in dimension two. Through crystalline and ordinary automorphy lifting theorems and patching arguments, it establishes potential automorphy in self-dual and ordinary settings, and obtains applications to the purity of compatible $l$-adic Galois representations and Bloch–Kato Selmer groups. Overall, the work broadens the scope of automorphy-lifting techniques, connecting monodromy data to global automorphy in higher rank and more general weight regimes. It lays groundwork for further unifications of local and global Langlands correspondences in CM-field contexts.

Abstract

Let $F$ be a CM field. In this paper, we prove the local-global compatibility for cohomological cuspidal automorphic representations of $\mathrm{GL}_n(\mathbb{A}_F)$ at $p \neq l$ by using certain potential automorphy theorems in some cases including higher dimensional cases. Moreover, we also prove the Ramanujan conjecture for cohomological cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A}_F )$.

On the potential automorphy and the local-global compatibility for the monodromy operators at $p \neq l$ over CM fields

TL;DR

The paper advances the study of local-global compatibility for GL_n over CM fields at p ≠ l by extending potential automorphy methods to higher dimensions and introducing a parity-based, monodromy-aware framework using parahoric levels. It develops a new approach to relate Weil–Deligne monodromy with automorphic realizations, enabling automatic compatibility at non-l places and a density-one set of primes, with unconditional Ramanujan results in dimension two. Through crystalline and ordinary automorphy lifting theorems and patching arguments, it establishes potential automorphy in self-dual and ordinary settings, and obtains applications to the purity of compatible -adic Galois representations and Bloch–Kato Selmer groups. Overall, the work broadens the scope of automorphy-lifting techniques, connecting monodromy data to global automorphy in higher rank and more general weight regimes. It lays groundwork for further unifications of local and global Langlands correspondences in CM-field contexts.

Abstract

Let be a CM field. In this paper, we prove the local-global compatibility for cohomological cuspidal automorphic representations of at by using certain potential automorphy theorems in some cases including higher dimensional cases. Moreover, we also prove the Ramanujan conjecture for cohomological cuspidal automorphic representations of .
Paper Structure (28 sections, 133 theorems, 72 equations)

This paper contains 28 sections, 133 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Main theorems
  3. New ideas in this paper
  4. Local-global compatibility
  5. Potential automorphy
  6. Organization of this paper
  7. Notation
  8. Some observations on the local-global compatibility for the monodromy operators at $p \neq l$
  9. Irreducible components of local lifting rings
  10. Some commutative algebras
  11. $p \neq l$
  12. $p = l$
  13. Automorphy lifting and local-global compatibility
  14. Preliminaries on automorphy lifting theorems
  15. Global deformation theory
  16. ...and 13 more sections

Key Result

Theorem 1.1

(Theorem ordinary local-global) Let $l$ be a prime such that $l > n^2$, $\iota: \overline{\mathbb{Q}}_l \stackrel{\sim}{\longrightarrow} \mathbb{C}$ be an isomorphism of fields and $\pi$ be an $\iota$-ordinary cohomological cuspidal automorphic representation of $\mathrm{GL}_{n}(\mathbb{A}_F)$. (See

Theorems & Definitions (305)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Definition 2.1
  • Lemma 2.2
  • ...and 295 more