Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Irreducibility and Monodromy of Automorphic Galois Representations of $\mathrm{GL}(4)$

Alireza Shavali

Abstract

We prove that over totally real fields, the $p$-adic Galois representations attached to non-self-dual regular algebraic cuspidal automorphic representations of $\mathrm{GL}(4)$ are irreducible. We then develop the theory of extra-twists in a general setting and use it to compute the monodromy group (over $\mathbb{Q}$) of these Galois representations, in both self-dual and non-self-dual settings, and prove $p$-adic and residual big image results.

Irreducibility and Monodromy of Automorphic Galois Representations of $\mathrm{GL}(4)$

Abstract

We prove that over totally real fields, the -adic Galois representations attached to non-self-dual regular algebraic cuspidal automorphic representations of are irreducible. We then develop the theory of extra-twists in a general setting and use it to compute the monodromy group (over ) of these Galois representations, in both self-dual and non-self-dual settings, and prove -adic and residual big image results.
Paper Structure (17 sections, 38 theorems, 142 equations)

This paper contains 17 sections, 38 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some Automorphic Background
  4. Preliminary Lemmas
  5. Locally Algebraic Representations
  6. Irreducibility of Non-self-dual Representations
  7. The $(3,1)$-case
  8. The $(2,2)$-case
  9. Local-Global compatibility at $\ell=p$
  10. Lie Algebra Classification
  11. Extra-Twists for Reductive Groups
  12. Big Image Results
  13. Computing the Lie algebra
  14. Application to automorphic Galois representations
  15. Non-self-dual case
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $\pi_1$ and $\pi_2$ be isobaric automorphic representations of ${\mathrm{GL}}_2({\mathbb A}_K)$. Then the admissible representation $\pi_1 \boxtimes\pi_2$, defined locally using the local Langlands correspondence, is an isobaric automorphic representation.

Theorems & Definitions (77)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 67 more