Eisenstein Cohomology for GL(N) and the special values of Rankin-Selberg L-functions over a totally imaginary number field

TL;DR

This paper establishes rationality results for ratios of critical values of Rankin-Selberg L-functions over totally imaginary fields by analyzing Eisenstein cohomology for GL(N), extending previous work from totally real fields.

Contribution

It generalizes Eisenstein cohomology methods to totally imaginary fields, revealing the impact of the base field's structure on L-function rationality.

Findings

Rationality of ratios of critical L-values over imaginary fields.

Extension of Eisenstein cohomology techniques from real to imaginary fields.

Identification of the influence of the base field's nature on cohomological properties.

Abstract

Rationality results are proved for the ratios of critical values of Rankin-Selberg L-functions of GL(n) x GL(n') over a totally imaginary field F, by studying rank-one Eisenstein cohomology for the group GL(N)/F, where N = n+n', generalizing the methods and results of previous work with Guenter Harder where the base field was totally real. In contrast to the totally real situation, the internal structure of the totally imaginary base field has a delicate effect on the rationality results.