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
$\ell$
.