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Michael HARRIS defined the arithmetic automorphic periods for certaincuspidal representations of $GL_{n}$ over quadratic imaginary fields in hisCrelle paper 1997. He also showed that critical values of automorphicL-functions for $GL_{n}\times GL_{1}$ can be interpreted in terms of thesearithmetic automorphic periods. In the thesis, we generalize his results in twoways. Firstly, the arithmetic automorphic periods have been defined overgeneral CM fields. We also prove that these periods factorize as products oflocal periods over infinity places. Secondly, we show that critical values ofautomorphic $L$ functions for $GL_{n}\times GL_{n'}$ can be interpreted interms of these automorphic periods in many situations. Consequently we showthat the automorphic periods are functorial for automorphic induction andcyclic base change. We also define certain motivic periods if the motive isrestricted from a CM field to the field of rational numbers. We can calculateDeligne's period for tensor product of two such motives. We see directly thatour automorphic results are compatible with Deligne's conjecture for motives.

Special values of automorphic $L$-functions for $GL_{n}\times GL_{n'}$  over CM fields, factorization and functoriality of arithmetic automorphic  periods