-independence for compatible systems of (mod ) representations. (2nd March 2015) - Hui, Chun Yin
Abstract : Let K be a number field. For any system of semisimple mod Galois representations {φ_:Gal(Q/K)→ GL_N(F_)}_ arising from étale cohomology (Definition 1), there exists a finite normal extension L of K such that if we denote φ_(Gal(Q/K)) and φ_(Gal(Q/L)) by Γ_ and γ_, respectively, for all and let S_ be the F_ -semisimple subgroup of GL_N, F_ associated to γ_ (or Γ_ ) by Nori's theory [ On subgroups of GL_n(F_p), Invent. Math.88 (1987), 257–275] for sufficiently large , then the following statements hold for all sufficiently large . A(i) The formal character of S_GL_N, F_ (Definition 1) is independent of and equal to the formal character of (G_^° )^derGL_N, Q_, where (G_^° )^der is the derived group of the identity component of G_, the monodromy group of the corresponding semi-simplified -adic Galois representation Φ_^ss . A(ii) The non-cyclic composition factors of γ_ and S_(F_) are identical. Therefore, the composition factors of γ_ are finite simple groups of Lie type of characteristic and are cyclic groups. B(i) The total -rank rk_Γ_ of Γ_ (Definition 14) is equal to the rank of S_ and is therefore independent of . B(ii) The A_n -type -rank rk_^A_nΓ_ of Γ_ (Definition 14) for nN{1, 2, 3, 4, 5, 7, 8} and the parity of (rk_^A_4Γ_)/4 are independent of .