Eisenstein Cohomology for $\mathrm {GL}_N$ and the special values of Rankin–Selberg <i>L</i>-functions over a totally imaginary number field

Abstract This article presents new rationality results for the ratios of critical values of Rankin–Selberg L -functions of $\mathrm {GL}(n) \times \mathrm {GL}(n')$ over a totally imaginary field $F.$ The proof is based on a cohomological interpretation of Langlands’s contant term theorem via rank-one Eisenstein cohomology for the group $\mathrm {GL}(N)/F,$ where $N = n+n'.$ The internal structure of the totally imaginary base field has a delicate effect on the Galois equivariance properties of the critical values.