l-independence for Compatible Systems of (mod l) Representations
Abstract
Let K be a number field. For any system of semisimple mod l Galois representations φl:GalK->GLN(Fl) arising from \'etale cohomology, there exists a finite normal extension L of K such that if we denote φl(GalK) and φl(GalL) by respectively l and γl for all l, and let Sl be the Fl-semisimple subgroup of GLN associated to γl (or 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 Sl->GLN 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 γl and Sl(Fl) are identical. Therefore, the composition factors of γl are finite simple groups of Lie type of characteristic l and cyclic groups. B(i) The total l-rank rkll of l is equal to the rank of Sl and is therefore independent of l. B(ii) The An-type l-rank rklAnl of l for n belonging to N\1,2,3,4,5,7,8 and the parity of (rklA4l)/4 are independent of l.