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On irreducibility of six-dimensional compatible systems of $\mathbb{Q}$

Boyi Dai

TL;DR

The paper proves that a 6-dimensional $E$-rational regular strictly compatible system over $ ext{Q}$ cannot have widespread reducibility: if one member is irreducible, then all but finitely many are irreducible. It distinguishes between non–Lie-irreducible and Lie-irreducible cases, using induction/restriction arguments, density results, and potential automorphy (BLGGT14) to propagate irreducibility and preserve semisimple types across $ ext{λ}$. For pure essentially self-dual regular systems, the authors show a finite decomposition into components that can be extended to strictly compatible subsystems, aligning with conjectures about big image and automorphy. The approach relies on $ ext{λ}$-independence of monodromy groups, semisimple reductions, and automorphy lifting to establish structural rigidity across the whole family, with implications for understanding the arithmetic of high-dimensional Galois representations.

Abstract

We study irreducibility of 6-dimensional strictly compatible systems of $\mathbb{Q}$ with distinct Hodge-Tate weights. We prove if one of the representations is irreducible, then all but finitely many of them are irreducible.

On irreducibility of six-dimensional compatible systems of $\mathbb{Q}$

TL;DR

The paper proves that a 6-dimensional -rational regular strictly compatible system over cannot have widespread reducibility: if one member is irreducible, then all but finitely many are irreducible. It distinguishes between non–Lie-irreducible and Lie-irreducible cases, using induction/restriction arguments, density results, and potential automorphy (BLGGT14) to propagate irreducibility and preserve semisimple types across . For pure essentially self-dual regular systems, the authors show a finite decomposition into components that can be extended to strictly compatible subsystems, aligning with conjectures about big image and automorphy. The approach relies on -independence of monodromy groups, semisimple reductions, and automorphy lifting to establish structural rigidity across the whole family, with implications for understanding the arithmetic of high-dimensional Galois representations.

Abstract

We study irreducibility of 6-dimensional strictly compatible systems of with distinct Hodge-Tate weights. We prove if one of the representations is irreducible, then all but finitely many of them are irreducible.

Paper Structure

This paper contains 19 sections, 24 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compatible systems
  4. $\lambda$-independence
  5. Lie-irreduciblility
  6. Essential self-duality and oddness
  7. Automorphic Galois representations
  8. Semisimple reduction
  9. The proof
  10. The non Lie-irreducible case
  11. $[K:\mathbb{Q}]=6$
  12. $[K:\mathbb{Q}]=3$
  13. $[K:\mathbb{Q}]=2$
  14. The Lie-irreducible case
  15. Cases (a) and (g)
  16. ...and 4 more sections

Key Result

Theorem 1.1

Given an elliptic curve $E$ over a number field $K$. Consider compatible system $\{\rho_{\ell}\}$ and mod $\ell$ compatible system $\{\overline{\rho}_{\ell}\}$ of $\mathrm{Gal}_K$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 37 more