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EISENSTEIN COHOMOLOGY FOR ORTHOGONAL GROUPS AND THE SPECIAL VALUES OF L-FUNCTIONS FOR 
$\mathrm {GL}_1 \times \mathrm {O}(2n)$
Published online by Cambridge University Press: 28 July 2025
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.
MSC classification
Secondary:
11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations
11F70: Representation-theoretic methods; automorphic representations over local and global fields
11F75: Cohomology of arithmetic groups
22E50: Representations of Lie and linear algebraic groups over local fields
22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings
Information
- Type
- Research Article
- Information
- Journal of the Institute of Mathematics of Jussieu , Volume 24 , Issue 6 , November 2025 , pp. 2463 - 2522
- Copyright
- © The Author(s), 2025. Published by Cambridge University Press
Footnotes
In memory of Professor Günter Harder.
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