cohomology of arithmetic groups, parabolic subgroups and the special values of l-functions on glₙ

let π be a cuspidal automorphic representation of glₙ(aq) with non-vanishing cohomology. under a certain local non-vanishing assumption we prove the rationality of the values of the automorphic l -function attached to π at critical points. conjecturally, any motivic l -function coincides with an l -function attached to an automorphic representation on glₙ , hence, our result corresponds to a conjecture of deligne on critical values of motivic l -functions.