Eisenstein Cohomology for Orthogonal Groups and the Special Values Of L-Functions for GL1×O(2n)

Abstract

For an even positive integer n, we study rank-one Eisenstein cohomology of the split orthogonal group O(2n+2) over a totally real number field F. This is used to prove a rationality result for the ratios of successive critical values of degree-2n Langlands L-functions associated to the group GL(1) x O(2n) over F. The case n = 2 specializes to classical results of Shimura on the special values of Rankin-Selberg L-functions attached to a pair of Hilbert modular forms.