ℓ-independence for compatible systems of (mod ℓ) representations

Abstract

© The Author 2015. Let K be a number field. For any system of semisimple mod ℓ Galois representations {φℓ: Gal(ℚ¯/K) → GL<inf>N</inf>(double-struck F<inf>ℓ</inf>)}<inf>ℓ</inf> arising from étale cohomology (Definition 1), there exists a finite normal extension L of K such that if we denote φ<inf>ℓ</inf> (Gal(ℚ¯/K)) and φ<inf>ℓ</inf>(Gal(ℚ¯/L)) by Γ¯<inf>ℓ</inf> and γ¯<inf>ℓ</inf>, respectively, for all ℓ and let S¯<inf>ℓ</inf> be the double-struck F<inf>ℓ</inf>-semisimple subgroup of GL<inf>N</inf>,<inf>double-struck Fℓ</inf> associated to γ¯<inf>ℓ</inf> (or Γ¯<inf>ℓ</inf>) by Nori's theory [On subgroups of GL<inf>n</inf>(double-struck F<inf>p</inf>), Invent. Math. 88 (1987), 257-275] for sufficiently large ℓ, then the following statements hold for all sufficiently large ℓ. A(i) The formal character of S¯<inf>ℓ</inf> → GL<inf>N</inf>,<inf>double-struck Fℓ</inf> (Definition 1) is independent of ℓ and equal to the formal character of (G˚<inf>ℓ</inf>)^der → GL<inf>N</inf>,<inf>ℚℓ</inf>, where (G˚<inf>ℓ</inf>)^der is the derived group of the identity component of G<inf>ℓ</inf>, the monodromy group of the corresponding semi-simplified ℓ-adic Galois representation Φ^SS<inf>ℓ</inf>. A(ii) The non-cyclic composition factors of γ¯<inf>ℓ</inf> and S¯<inf>ℓ</inf> (double-struck F<inf>ℓ</inf>) are identical. Therefore, the composition factors of γ¯<inf>ℓ</inf> are finite simple groups of Lie type of characteristic ℓ and are cyclic groups. B(i) The total ℓ-rank rk<inf>ℓ</inf>Γ¯<inf>ℓ</inf> of Γ¯<inf>ℓ</inf> (Definition 14) is equal to the rank of S¯<inf>ℓ</inf> and is therefore independent of ℓ. B(ii) The A<inf>n</inf>-type ℓ-rank rk^An<inf>ℓ</inf>Γ¯<inf>ℓ</inf> of Γ¯<inf>ℓ</inf> (Definition 14) for n ∈ ℕ\{1; 2; 3; 4; 5; 7; 8} and the parity of (rk^A4<inf>ℓ</inf>Γ¯<inf>ℓ</inf>)/4 are independent of ℓ.