Hostname: page-component-cb9f654ff-c75p9 Total loading time: 0 Render date: 2025-08-24T12:50:58.335Z Has data issue: false hasContentIssue false
  • English
  • Français

EISENSTEIN COHOMOLOGY FOR ORTHOGONAL GROUPS AND THE SPECIAL VALUES OF L-FUNCTIONS FOR $\mathrm {GL}_1 \times \mathrm {O}(2n)$

Part of: Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  28 July 2025

Chandrasheel Bhagwat Open the ORCID record for Chandrasheel Bhagwat [Opens in a new window]  and
A. Raghuram
Chandrasheel Bhagwat*
Affiliation:
https://ror.org/028qa3n13Indian Institute of Science Education and Research , Dr. Homi Bhabha Road, Pashan, Pune 411008, INDIA
A. Raghuram
Affiliation:
Department of Mathematics, https://ror.org/03qnxaf80Fordham University at Lincoln Center , New York, NY 10023, USA (araghuram@fordham.edu)
*
cbhagwat@iiserpune.ac.in
Get access Rights & Permissions [Opens in a new window]

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

Primary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
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 , First View , pp. 1 - 60
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

In memory of Professor Günter Harder.

References

Arthur, J (2013) The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, vol. 61. American Mathematical Society Colloquium Publications. Providence, RI: American Mathematical Society.Google Scholar
Atobe, H and Gan, WT (2017) On the local Langlands correspondence and Arthur conjecture for even orthogonal groups. Representation Theory 21, 354–415.10.1090/ert/504CrossRefGoogle Scholar
Balasubramanyam, B and Raghuram, A (2017) Special values of adjoint L-functions and congruences for automorphic forms on GL(n) over a number field. Amer. J. Math. 139(3), 641–679.10.1353/ajm.2017.0017CrossRefGoogle Scholar
Bernstein, IN and Zelevinskii, AV (1976) Representations of the group $\mathrm{GL}(\mathrm{n},\mathrm{F})$ , where $\mathrm{F}$ is a local non-Archimedean field. Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, Translation in Russian mathematical Surveys 31(3), 5–70.Google Scholar
Bhagwat, C and Raghuram, A (2017) Special values of L-functions for orthogonal groups. C. R. Math. Acad. Sci. Paris 355(3), 263–267.10.1016/j.crma.2017.01.016CrossRefGoogle Scholar
Borel, A (1979) Automorphic $\mathrm{L}$ -functions. In Automorphic Forms, Representations and $L$ -functions, vol XXXIII. Proc. Sympos. Pure Math. Providence, RI: Amer. Math. Soc., Part 2, 27–61.Google Scholar
Borel, A and J-P, Serre (1973) Corners and arithmetic groups. Commen. Math. Helvetici 48, 436–491.10.1007/BF02566134CrossRefGoogle Scholar
Borel, A and Wallach, N (2000) Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. Providence, RI: American Mathematical Society.10.1090/surv/067CrossRefGoogle Scholar
Casselman, W and Shahidi, F (1998) On irreducibility of standard modules for generic representations. Ann. Sci. École Norm. Sup. (4) 31(4) 561–589.10.1016/S0012-9593(98)80107-9CrossRefGoogle Scholar
Chen, S-Y (2024) Period relations between the Betti–Whittaker periods for $GL(n)$ under duality. Int. Math. Res. Not. IMRN 2024(15), 11032–11063.10.1093/imrn/rnae103CrossRefGoogle Scholar
Clozel, L (1990) Motifs et formes automorphes: applications du principe de fonctorialité. In Automorphic Forms, Shimura Varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), vol. 10. Perspect. Math. Boston, MA: Academic Press, 77–159.Google Scholar
Deligne, P (1979) Valeurs de fonctions $L$ et périodes d’intégrales (French), vol. XXXIII. With an appendix by N. Koblitz and Ogus, A.. Proc. Sympos. Pure Math., Providence, RI: Amer. Math. Soc., Part 2, 313–346.Google Scholar
Deligne, P and Raghuram, A (2025) Motives, periods, and functoriality. Tunis. J. Math. 7(1), 131–165.10.2140/tunis.2025.7.131CrossRefGoogle Scholar
Ginzburg, D, Jiang, D and Soudry, D (2015) On CAP representations for even orthogonal groups I: A correspondence of unramified representations. Chin. Ann. Math. Ser. B 36(4), 485–522.10.1007/s11401-015-0916-6CrossRefGoogle Scholar
Harder, G (1987) Eisenstein cohomology of arithmetic groups. The case $\mathrm{GL}_2$ . Invent. Math. 89(1), 37–118.10.1007/BF01404673CrossRefGoogle Scholar
Harder, G and Raghuram, A (2020) Eisenstein Cohomology for $\mathrm{GL}_N$ and the Special Values of Rankin–Selberg $L$ -functions, vol. 203. Annals of Math. Studies. Princeton, NJ: Princeton University Press.Google Scholar
Harish-Chandra, (1968) Automorphic Forms on Semi-simple Lie Groups, vol. 62. Lecture Notes in Mathematics, Springer-Verlag: Berlin-New York, x+138 pp.10.1007/BFb0098434CrossRefGoogle Scholar
Jacquet, H and Shalika, JA (1981) On Euler products and the classification of automorphic forms. II. Amer. J. Math. 103(4), 777–815.10.2307/2374050CrossRefGoogle Scholar
Jacquet, H and Shalika, JA (1976) A non-vanishing theorem for zeta functions of $\mathrm{G}{\mathrm{L}}_{\mathrm{n}}$ , Invent. Math. 38, 1–16.10.1007/BF01390166CrossRefGoogle Scholar
Kim, H (2004) Automorphic L-functions. In Lectures on Automorphic L-functions, vol. 20. Fields Inst. Monogr. Providence, RI: Amer. Math. Soc., 97–201.Google Scholar
Kim, W (2009) Square integrable representations and the standard module conjecture for general spin groups. Canad. J. Math. 61(3), 617–640.10.4153/CJM-2009-033-3CrossRefGoogle Scholar
Knapp, AW (1994) Local Langlands correspondence: the Archimedean case. In Motives vol. 55, Part 2. Proc. Sympos. Pure Math. Providence, RI: AMS, 393–410.10.1090/pspum/055.2/1265560CrossRefGoogle Scholar
Kostant, B (1961) Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. Math. 74(2), 329–387.10.2307/1970237CrossRefGoogle Scholar
Kutzko, PC (1977) Mackey’s theorem for nonunitary representations. Proc. Amer. Math. Soc. 64(1), 173–175.Google Scholar
Mœglin, C and J-L, Waldspurger (1989) Le spectre résiduel de GL(n). (French) [The residual spectrum of GL(n)]. Ann. Sci. École Norm. Sup. (4) 22(4), 605–674.10.24033/asens.1595CrossRefGoogle Scholar
Raghuram, A (2013) Comparison results for rertain periods of cusp forms on $\mathrm{GL}(2\mathrm{n})$ over a totally real number field. vol. 20. Ramanujan Math. Soc. Lect. Notes Ser. Mysore: Ramanujan Mathematical Society, 323–334.Google Scholar
Raghuram, A (2016) Critical values of Rankin–Selberg $L$ -functions for $\mathrm{GL}_n\times \mathrm{GL}_{n-1}$ and the symmetric cube $L$ -functions for $\mathrm{GL}_2$ . Forum Math. 28(3), 457–489.10.1515/forum-2014-0043CrossRefGoogle Scholar
Raghuram, A (2023) An arithmetic property of intertwining operators for $p$ -adic groups. Canad. J. Math. 75(1), 83–107.10.4153/S0008414X21000535CrossRefGoogle Scholar
Raghuram, A (2025) Eisenstein Cohomology for $\mathrm{GL}_N$ and the special values of Rankin–Selberg $L$ -functions over a totally imaginary field. Forum of Math. Sigma 13(E86), 1–68.10.1017/fms.2025.48CrossRefGoogle Scholar
Raghuram, A and Sarnobat, M (2018) Cohomological representations and functorial transfer from classical groups. In Cohomology of Arithmetic Groups, vol. 245. Springer Proc. Math. Stat. Cham: Springer, 157–176.10.1007/978-3-319-95549-0_6CrossRefGoogle Scholar
Raghuram, A and Shahidi, F (2008) On certain period relations for cusp forms on ${\mathrm{GL}}_{\mathrm{n}}$ . Int. Math. Res. Not. 2008, doi:10.1093/imrn/rnn077.Google Scholar
Shahidi, F (1985) Local coefficients as Artin factors for real groups. Duke Math. J. 52(4), 973–1007.10.1215/S0012-7094-85-05252-4CrossRefGoogle Scholar
Shahidi, F (2010) Eisenstein Series and Automorphic L-functions, vol. 58. American Mathematical Society Colloquium Publications. Providence, RI: American Mathematical Society.Google Scholar
Shimura, G (1977) On the periods of modular forms. Math. Ann. 229(3), 211–221.10.1007/BF01391466CrossRefGoogle Scholar
Shimura, G (1978) The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J. 45(3), 637–679.10.1215/S0012-7094-78-04529-5CrossRefGoogle Scholar
Soudry, D (2005) On Langlands functoriality from classical groups to ${\mathrm{GL}}_{\mathrm{n}}$ . Automorphic forms. I. Astérisque 298, 335–390.Google Scholar
Waldspurger, J-L (2003) La formule de Plancherel pour les groupes p-adiques (d’après Harish-Chandra). J. Inst. Math. Jussieu 2(2), 235–333.10.1017/S1474748003000082CrossRefGoogle Scholar
Wallach, N (1988) Real Reductive Groups. I, vol. 132. Pure and Applied Mathematics. Boston, MA: Academic Press, Inc.Google Scholar