Compatibility of canonical ℓ-adic local systems on Shimura varieties

2023/03/07 by Christian Klevdal, Klevdal, Christian, Stefan Patrikis +1 · 1 citation
Mathematics · #11G18 #14G35 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2303.03863

openalex publication_date 2023/03/07 · openalex created_date 2023/03/10 · openalex updated_date 2026/07/28

Abstract

For a Shimura variety (G, X) in the superrigid regime and neat level subgroup K₀, we show that the canonical family of ℓ-adic representations associated to a number field point y ∈ Sh_K0(G, X)(F), \ ρ_{y, ℓ} \colon Gal(ℚ/F) → G^ad(ℚ_ℓ) \_ℓ, form a compatible system of G^ad(ℚ_ℓ)-representations: there is an integer N(y) such that for all ℓ, ρ_{y, ℓ} is unramified away from N(y) ℓ, and for all ℓ ≠ ℓ' and v \nmid N(y)ℓ ℓ', the semisimple parts of the conjugacy classes of ρ_{y, ℓ}(Frob_v) and ρ_{y, ℓ'}(Frob_v) are (ℚ-rational and) equal. We deduce this from a stronger compatibility result for the canonical G(ℚ_ℓ)-valued local systems on connected Shimura varieties inside Sh_K0(G, X). Our theorems apply in particular to Shimura varieties of non-abelian type and represent the first such independence-of-ℓ results in non-abelian type.

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• G-companions on algebraic stacks and applications to canonical ℓ-adic local systems on Shimura stacks

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