$\ell$-independence for compatible systems of (mod $\ell$) representations. (2nd March 2015) - Hui, Chun Yin
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 $\ell$ .