Monodromy and irreducibility of type $A_1$ automorphic Galois representations
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| Format: | Preprint |
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2024
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| author | Hui, Chun-Yin Lee, Wonwoong |
| author_facet | Hui, Chun-Yin Lee, Wonwoong |
| contents | Let $K$ be a totally real field and $π$ be a regular algebraic polarized cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_K)$. Let $\{ρ_{π,λ}:\mathrm{Gal}_K\to\mathrm{GL}_n(\overline E_λ)\}_λ$ be the compatible system of Galois representations attached to $π$ and denote by $\mathbf G_λ$ the algebraic monodromy group of $ρ_{π,λ}$. Suppose there exists $λ_0$ such that (a) $ρ_{π,λ_0}$ is irreducible; (b) $\mathbf G_{λ_0}$ is connected and of type $A_1$; and (c) the tautological representation of $\mathbf G_{λ_0}$ is of a certain type. We prove that $\bullet$ $\mathbf G_{λ,\mathbb C}\subset\mathrm{GL}_{n, \mathbb C}$ is independent of $λ$; $\bullet$ $ρ_{π,λ}$ is irreducible for all $λ$, and residually irreducible for almost all $λ$. Moreover, if $K=\mathbb Q$ or $n$ is odd, we prove that the same conclusions hold without the assumption that $π$ is polarized. We also prove that if $K=\mathbb Q$, then the compatible system $\{ρ_{π,λ}\}_λ$ is constructed from certain two-dimensional modular compatible systems up to twist. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_12566 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monodromy and irreducibility of type $A_1$ automorphic Galois representations Hui, Chun-Yin Lee, Wonwoong Number Theory 11F80, 11F70, 11F22, 20G05 Let $K$ be a totally real field and $π$ be a regular algebraic polarized cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_K)$. Let $\{ρ_{π,λ}:\mathrm{Gal}_K\to\mathrm{GL}_n(\overline E_λ)\}_λ$ be the compatible system of Galois representations attached to $π$ and denote by $\mathbf G_λ$ the algebraic monodromy group of $ρ_{π,λ}$. Suppose there exists $λ_0$ such that (a) $ρ_{π,λ_0}$ is irreducible; (b) $\mathbf G_{λ_0}$ is connected and of type $A_1$; and (c) the tautological representation of $\mathbf G_{λ_0}$ is of a certain type. We prove that $\bullet$ $\mathbf G_{λ,\mathbb C}\subset\mathrm{GL}_{n, \mathbb C}$ is independent of $λ$; $\bullet$ $ρ_{π,λ}$ is irreducible for all $λ$, and residually irreducible for almost all $λ$. Moreover, if $K=\mathbb Q$ or $n$ is odd, we prove that the same conclusions hold without the assumption that $π$ is polarized. We also prove that if $K=\mathbb Q$, then the compatible system $\{ρ_{π,λ}\}_λ$ is constructed from certain two-dimensional modular compatible systems up to twist. |
| title | Monodromy and irreducibility of type $A_1$ automorphic Galois representations |
| topic | Number Theory 11F80, 11F70, 11F22, 20G05 |
| url | https://arxiv.org/abs/2407.12566 |