Eisenstein cohomology for orthogonal groups and the special values of $L$-functions for ${\mathrm{GL}}_1\times \mathrm{O}(2n)$

Abstract

For an even positive integer $n$, we study rank-one Eisenstein cohomology of the split orthogonal group $\mathrm{O}(2n+2)$ 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 ${\mathrm{GL}}_1\times \mathrm{O}(2n)$ 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.

arXiv comment57 pages, some modifications made to previous version

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