Monodromy of four dimensional irreducible compatible systems of Q

2022/08/08 by Chun Yin Hui, Hui, Chun Yin · 1 citation
Mathematics · #11F22 #11F70 #11F80 #20G05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2208.04004

openalex publication_date 2022/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F be a totally real field and n≤ 4 a natural number. We study the monodromy groups of any n-dimensional strictly compatible system \ρ_λ\_λ of λ-adic representations of F with distinct Hodge-Tate numbers such that ρ_λ0 is irreducible for some λ₀. When F=ℚ, n=4, and ρ_λ0 is fully symplectic, the following assertions are obtained. (i) The representation ρ_λ is fully symplectic for almost all λ. (ii) If in addition the similitude character μ_λ0 of ρ_λ0 is odd, then the system \ρ_λ\_λ is potentially automorphic and the residual image ρ_λ(Gal_ℚ) has a subgroup conjugate to Sp₄(\mathbbF_ℓ) for almost all λ.

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• Rectangular representations and λ-independence of algebraic monodromy groups

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