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

Abstract

Michael HARRIS defined the arithmetic automorphic periods for certain cuspidal representations of $GL_{n}$ over quadratic imaginary fields in his Crelle paper 1997. He also showed that critical values of automorphic L-functions for $GL_{n}\times GL_{1}$ can be interpreted in terms of these arithmetic automorphic periods. In the thesis, we generalize his results in two ways. Firstly, the arithmetic automorphic periods have been defined over general CM fields. We also prove that these periods factorize as products of local periods over infinity places. Secondly, we show that critical values of automorphic $L$ functions for $GL_{n}\times GL_{n'}$ can be interpreted in terms of these automorphic periods in many situations. Consequently we show that the automorphic periods are functorial for automorphic induction and cyclic base change. We also define certain motivic periods if the motive is restricted from a CM field to the field of rational numbers. We can calculate Deligne's period for tensor product of two such motives. We see directly that our automorphic results are compatible with Deligne's conjecture for motives.