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On coefficients, potentially abelian quotients, and residual irreducibility of compatible systems

Gebhard Böckle, Chun-Yin Hui

TL;DR

The article develops a canonical construction of algebraic monodromy groups for semisimple $E$-rational compatible systems and proves that their maximal abelian and potentially abelian quotients are independent of the chosen place $\lambda$. It shows how to descend coefficients to a finite extension $E'/E$ so that the system becomes strongly $E'$-rational with split monodromy, and it establishes density-one residual irreducibility results for the irreducible factors of the $\,\lambda$-adic realizations. A CM-motive perspective via the motivic Galois group $\mathbf{M}_K^{\mathrm{CM}}$ is developed to describe potentially abelian representations, yielding lambda-independence results for the abelian content. The work culminates in two main theorems: a lambda-independent description of abelian/pab quotients and a scrutiny of residual irreducibility for almost all $\lambda$, extending PSW-type results and enabling uniform control across $\ell$-adic realizations.

Abstract

Let $\{ρ_λ:G_K\rightarrow GL_n(\overline E_λ)\}$ be a semisimple E-rational compatible system of a number field K. In a first step, building upon the theory of pseudocharacters [Ro96],[Ch14], we attach to each $ρ_λ$ an algebraic monodromy group $G_λ$ defined over $E_λ$ and also prove that the compatible system can be descended to a strongly E'-rational compatible system $\{ρ_{λ'}: G_K\rightarrow GL_n(E'_{λ'})\}$ for some finite extension E'/E. Secondly, we demonstrate that the maximal potentially abelian quotient of $G_λ$ is independent of $λ$ in a strong sense. Finally, as an application, we generalize a result of Patrikis--Snowden--Wiles on residual irreducibility of compatible systems.

On coefficients, potentially abelian quotients, and residual irreducibility of compatible systems

TL;DR

The article develops a canonical construction of algebraic monodromy groups for semisimple -rational compatible systems and proves that their maximal abelian and potentially abelian quotients are independent of the chosen place . It shows how to descend coefficients to a finite extension so that the system becomes strongly -rational with split monodromy, and it establishes density-one residual irreducibility results for the irreducible factors of the -adic realizations. A CM-motive perspective via the motivic Galois group is developed to describe potentially abelian representations, yielding lambda-independence results for the abelian content. The work culminates in two main theorems: a lambda-independent description of abelian/pab quotients and a scrutiny of residual irreducibility for almost all , extending PSW-type results and enabling uniform control across -adic realizations.

Abstract

Let be a semisimple E-rational compatible system of a number field K. In a first step, building upon the theory of pseudocharacters [Ro96],[Ch14], we attach to each an algebraic monodromy group defined over and also prove that the compatible system can be descended to a strongly E'-rational compatible system for some finite extension E'/E. Secondly, we demonstrate that the maximal potentially abelian quotient of is independent of in a strong sense. Finally, as an application, we generalize a result of Patrikis--Snowden--Wiles on residual irreducibility of compatible systems.
Paper Structure (27 sections, 29 theorems, 72 equations)

This paper contains 27 sections, 29 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. $E$-compatible systems and algebraic monodromy groups
  3. Potentially abelian quotients and $\lambda$-independence
  4. Descent of coefficients and residual irreducibility
  5. Organization of the article
  6. Basic definitions and constructions
  7. $E$-rationality
  8. Weil restriction of scalars
  9. Extension and restriction of scalars of compatible systems
  10. Coefficients of algebraic monodromy groups and compatible systems
  11. Algebraic monodromy groups under $E$-rationality
  12. Proof of Theorem \ref{['coeffd']}
  13. Abelian $\ell$-adic representations
  14. Local algebraicity
  15. Serre groups $\mathbf{S}_\mathfrak{m}$ and representations
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $K$ be a number field and $\{\rho_\lambda: \operatorname{Gal}_K\to \mathrm{GL}_n(\overline{E}_\lambda)\}_{\Pi_E}$ be a semisimple $E$-rational compatible system of $K$ with $\{\mathbf{G}_\lambda\}_{\Pi_E}$ as the system of algebraic monodromy groups defined over $E_\lambda$ (Definition def:Glamb

Theorems & Definitions (53)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 3.1
  • Remark 3.2
  • ...and 43 more