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$\ell$-independence for compatible systems of (mod $\ell$) representations

Part of: Representation theory of groups Discontinuous groups and automorphic forms (Co)homology theory

Published online by Cambridge University Press:  02 March 2015

Chun Yin Hui
Chun Yin Hui*
Affiliation:
University of Luxembourg, Mathematics Research Unit, 6 rue Richard Coudenhove-Kalergi, L-1359, Luxembourg email [email protected], [email protected]
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Abstract

Let $K$ be a number field. For any system of semisimple mod $\ell$ Galois representations $\{{\it\phi}_{\ell }:\text{Gal}(\bar{\mathbb{Q}}/K)\rightarrow \text{GL}_{N}(\mathbb{F}_{\ell })\}_{\ell }$ arising from étale cohomology (Definition 1), there exists a finite normal extension $L$ of $K$ such that if we denote ${\it\phi}_{\ell }(\text{Gal}(\bar{\mathbb{Q}}/K))$ and ${\it\phi}_{\ell }(\text{Gal}(\bar{\mathbb{Q}}/L))$ by $\bar{{\rm\Gamma}}_{\ell }$ and $\bar{{\it\gamma}}_{\ell }$, respectively, for all $\ell$ and let $\bar{\mathbf{S}}_{\ell }$ be the $\mathbb{F}_{\ell }$-semisimple subgroup of $\text{GL}_{N,\mathbb{F}_{\ell }}$ associated to $\bar{{\it\gamma}}_{\ell }$ (or $\bar{{\rm\Gamma}}_{\ell }$) by Nori’s theory [On subgroups of$\text{GL}_{n}(\mathbb{F}_{p})$, Invent. Math. 88 (1987), 257–275] for sufficiently large $\ell$, then the following statements hold for all sufficiently large $\ell$.

A(i) The formal character of $\bar{\mathbf{S}}_{\ell }{\hookrightarrow}\text{GL}_{N,\mathbb{F}_{\ell }}$ (Definition 1) is independent of $\ell$ and equal to the formal character of $(\mathbf{G}_{\ell }^{\circ })^{\text{der}}{\hookrightarrow}\text{GL}_{N,\mathbb{Q}_{\ell }}$, where $(\mathbf{G}_{\ell }^{\circ })^{\text{der}}$ is the derived group of the identity component of $\mathbf{G}_{\ell }$, the monodromy group of the corresponding semi-simplified $\ell$-adic Galois representation ${\rm\Phi}_{\ell }^{\text{ss}}$.

A(ii) The non-cyclic composition factors of $\bar{{\it\gamma}}_{\ell }$ and $\bar{\mathbf{S}}_{\ell }(\mathbb{F}_{\ell })$ are identical. Therefore, the composition factors of $\bar{{\it\gamma}}_{\ell }$ are finite simple groups of Lie type of characteristic $\ell$ and are cyclic groups.

B(i) The total $\ell$-rank $\text{rk}_{\ell }\bar{{\rm\Gamma}}_{\ell }$ of $\bar{{\rm\Gamma}}_{\ell }$ (Definition 14) is equal to the rank of $\bar{\mathbf{S}}_{\ell }$ and is therefore independent of $\ell$.

B(ii) The $A_{n}$-type $\ell$-rank $\text{rk}_{\ell }^{A_{n}}\bar{{\rm\Gamma}}_{\ell }$ of $\bar{{\rm\Gamma}}_{\ell }$ (Definition 14) for $n\in \mathbb{N}\setminus \{1,2,3,4,5,7,8\}$ and the parity of $(\text{rk}_{\ell }^{A_{4}}\bar{{\rm\Gamma}}_{\ell })/4$ are independent of $\ell$.

Keywords

Galois representationsℓ-independencebig Galois imageétale cohomology

MSC classification

Primary: 11F80: Galois representations 14F20: Étale and other Grothendieck topologies and (co)homologies 20D05: Finite simple groups and their classification
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 7 , July 2015 , pp. 1215 - 1241
Copyright
© The Author 2015 

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