Special values of $L$-functions for orthogonal groups

In the 70's, Shimura proved rationality results for the critical values of Rankin-Selberg L-functions
attached to a pair of Hilbert modular forms, i.e., for $L$-functions for GL(2) x GL(2) over a totally
real field F. In this talk I will present some recent results, obtained in collaboration with Chandrasheel
Bhagwat, on the critical values of the degree 2n L-functions of SO(n,n) over $F$, when $n$ is an even
positive integer. Such rationality results are obtained by studying rank one Eisenstein cohomology for
the group SO(n+1,n+1) over $F$. The case $n=2$ essentially specializes to Shimura's results.