2020 Theses Doctoral

Comparison between motivic periods with Shalika periods

An, Yang

Let F/F^+ be a quadratic imaginary field extension of a totally real field F^+, and pi cong \tilde{\pi} otimes xi be a cuspidal automorphic representation of GL_n(AA_F) obtained from tilde{pi} by twisting a Hecke character xi. In the case of F^+ = QQ, Michael Harris defined arithmetic automorphic periods for certain tilde{pi} in his Crelle paper 1997, and showed that critical values of automorphic L-functions for pi can be interpreted in terms of these arithmetic automorphic periods. Lin Jie generalized his construction and results to the general totally real field F^+ in her thesis. On the other hand, for certain cuspidal representation Pi of GL_{2n}(F^+), which admits a Shalika model, Grobner and Raghuram related their critical values of L-functions to a non-zero complex number (called Shalika periods). We noticed that the automorphic induction AI(pi) of pi, considered by Harris and Lin, will automatically have a Shalika model, and by comparing common critical values of their identical L-functions, we relate the Shalika periods of AI(pi) with arithmetic automorphic periods of tilde{pi}. In the case F^+=QQ, this comparison will express each arithmetic automorphic period in terms of the corresponding Shalika periods.