Year: 2025
Type: article
Abstract: For a Shimura variety $(G, X)$ ( G , X ) in the superrigid regime and neat level subgroup $K_{0}$ K 0 , we show that the canonical family of $\ell $ ℓ -adic representations associated to a number field point $y \in \mathrm{Sh}_{K_{0}}(G, X)(F)$ y ∈ Sh K 0 ( G , X ) ( F ) , $$ \left \{ \rho _{y, \ell } \colon \mathrm{Gal}(\overline{\mathbb{Q}}/F) \to G^{\mathrm{ad}}(\mathbb{Q}_{\ell }) \right \} _{\ell }, $$ { ρ y , ℓ : Gal ( Q ‾ / F ) → G ad ( Q ℓ ) } ℓ , form a compatible system of $G^{\mathrm{ad}}(\mathbb{Q}_{\ell })$ G ad ( Q ℓ ) -representations: there is an integer $N(y)$ N ( y ) such that for all $\ell $ ℓ , $\rho _{y, \ell }$ ρ y , ℓ is unramified away from $N(y) \ell $ N ( y ) ℓ , and for all $\ell \neq \ell '$ ℓ ≠ ℓ ′ and $v \nmid N(y)\ell \ell '$ v ∤ N ( y ) ℓ ℓ ′ , the semisimple parts of the conjugacy classes of $\rho _{y, \ell }(\mathrm{Frob}_{v})$ ρ y , ℓ ( Frob v ) and $\rho _{y, \ell '}(\mathrm{Frob}_{v})$ ρ y , ℓ ′ ( Frob v ) are (ℚ-rational and) equal. We deduce this from a stronger compatibility result for the canonical $G(\mathbb{Q}_{\ell })$
Source: Inventiones mathematicae
Authors: Christian Klevdal, Stefan Patrikis
Institutions: University of California San Diego, The Ohio State University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4413594890