l-independence for Compatible Systems of (mod l) Representations

Let K be a number field. For any system of semisimple mod l Galois representations {ϕ_l:Gal_K->GL_N(F_l)} arising from étale cohomology, there exists a finite normal extension L of K such that if we denote ϕ_l(Gal_K) and ϕ_l(Gal_L) by respectively Γ_l and γ_l for all l, and let S_l be the F_l-semisimple subgroup of GL_N associated to γ_l (or Γ_l) by Nori [No87] for all sufficiently large l, then the following statements hold for all sufficiently large l: A(i) The formal character of S_l->GL_N is independent of l and is equal to the formal character of the tautological representation of the derived group of the identity component of the monodromy group of the corresponding semi-simplified l-adic Galois representation. A(ii) The non-cyclic composition factors of γ_l and S_l(F_l) are identical. Therefore, the composition factors of γ_l are finite simple groups of Lie type of characteristic l and cyclic groups. B(i) The total l-rank rk_lΓ_l of Γ_l is equal to the rank of S_l and is therefore independent of l. B(ii) The A_n-type l-rank rk_l^{A_n}Γ_l of Γ_l for n belonging to N\{1,2,3,4,5,7,8} and the parity of (rk_l^{A_4}Γ_l)/4 are independent of l.

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