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On The Image of Automorphic Galois Representations

Alireza Shavali

TL;DR

The work develops a framework of extra-twists (inner and outer) for GL_n automorphic representations and uses them to bound and describe the image of associated Galois representations. It proves a big-image theorem for GL_3 over totally real fields when the automorphic representation is of general type, showing openness of the image in a twisted SL_3 form determined by the extra-twists, and provides a conjectural generalization to GL_n together with a Mumford–Tate perspective via a hypothetical motive M_π. The approach blends cocycle twists, Lie-algebra comparisons, and Goursat-type arguments to translate automorphic symmetries into algebraic-group constraints, yielding precise descriptions of algebraic envelopes and Zariski closures of Galois images. It also recapitulates the GL_2 situation as a benchmark and formulates a global picture (via Azumaya algebras) that should govern p-adic realizations uniformly across primes under the Langlands and MT conjectures.

Abstract

In this paper, we study extra-twists for automorphic representations of $\mathrm{GL}_n$ and use them to give a precise description of the image of the Galois representations associated with regular algebraic cuspidal automorphic representations of $\mathrm{GL}_3$ over totally real fields. We also formulate a conjecture for the $\mathrm{GL}_n$-case and show how it follows from some standard conjectures in the Langlands program. Finally, assuming the existence of a motive associated with the representation, we study the relation of our constructions with the Mumford-Tate group.

On The Image of Automorphic Galois Representations

TL;DR

The work develops a framework of extra-twists (inner and outer) for GL_n automorphic representations and uses them to bound and describe the image of associated Galois representations. It proves a big-image theorem for GL_3 over totally real fields when the automorphic representation is of general type, showing openness of the image in a twisted SL_3 form determined by the extra-twists, and provides a conjectural generalization to GL_n together with a Mumford–Tate perspective via a hypothetical motive M_π. The approach blends cocycle twists, Lie-algebra comparisons, and Goursat-type arguments to translate automorphic symmetries into algebraic-group constraints, yielding precise descriptions of algebraic envelopes and Zariski closures of Galois images. It also recapitulates the GL_2 situation as a benchmark and formulates a global picture (via Azumaya algebras) that should govern p-adic realizations uniformly across primes under the Langlands and MT conjectures.

Abstract

In this paper, we study extra-twists for automorphic representations of and use them to give a precise description of the image of the Galois representations associated with regular algebraic cuspidal automorphic representations of over totally real fields. We also formulate a conjecture for the -case and show how it follows from some standard conjectures in the Langlands program. Finally, assuming the existence of a motive associated with the representation, we study the relation of our constructions with the Mumford-Tate group.

Paper Structure

This paper contains 12 sections, 24 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Extra-Twists for ${\mathrm{GL}}_n$
  3. Inner and outer twists
  4. Constructing forms of algebraic groups using 1-cocycles
  5. Image of Galois Representations with Extra-Twists
  6. Extra-twists and Galois representations
  7. The Lie algebra of the image
  8. Image of Automorphic Galois Representations
  9. The ${\mathrm{GL}}_2$-case
  10. The ${\mathrm{GL}}_3$-case
  11. The ${\mathrm{GL}}_n$-case
  12. Relations to the Mumford-Tate Group

Key Result

Theorem 1.1

Let $F=E^\Gamma$ be the field fixed by all extra-twists of $\pi$. Then there exists a finite extension $L/K$ and a semi-simple algebraic group $H_p$ defined over $F_p:=F\otimes_{{\mathbb Q}}{\mathbb Q}_p$ which is a form of $\mathrm{SL}_3$ (constructed using the extra-twists), such that $\rho_{\pi,p

Theorems & Definitions (60)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • ...and 50 more