Integral presentations of GL(n)-automorphic L-functions over CM fields and applications

I will explain how to derive some new integral presentations for central values of GL(n)-automorphic L-functions over CM fields in terms of the Fourier-Whittaker expansions of certain projected cuspidal forms. Such integral presentations can be used to derive asymptotic nonvanishing results on average, and these results in turn have applications to Deligne’s rationality conjecture for automorphic motives over CM fields via recent work(s) of Grobner, Harris, and Lin.