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# Forum Mathematicum

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Volume 31, Issue 5

# A construction of residues of Eisenstein series and related square-integrable classes in the cohomology of arithmetic groups of low k-rank

Neven Grbac
• Corresponding author
• Faculty of Engineering, Juraj Dobrila University of Pula, Zagrebačka 30, HR-52100 Pula, Croatia
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• Other articles by this author:
/ Joachim Schwermer
• Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria; and Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
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• Other articles by this author:
Published Online: 2019-06-13 | DOI: https://doi.org/10.1515/forum-2019-0029

## Abstract

The cohomology of an arithmetic congruence subgroup of a connected reductive algebraic group defined over a number field is captured in the automorphic cohomology of that group. The residual Eisenstein cohomology is by definition the part of the automorphic cohomology represented by square-integrable residues of Eisenstein series. The existence of residual Eisenstein cohomology classes depends on a subtle combination of geometric conditions (coming from cohomological reasons) and arithmetic conditions in terms of analytic properties of automorphic L-functions (coming from the study of poles of Eisenstein series). Hence, there are almost no unconditional results in the literature regarding the very existence of non-trivial residual Eisenstein cohomology classes. In this paper, we show the existence of certain non-trivial residual cohomology classes in the case of the split symplectic, and odd and even special orthogonal groups of rank two, as well as the exceptional group of type ${\mathrm{G}}_{2}$, defined over a totally real number field. The construction of cuspidal automorphic representations of ${\mathrm{GL}}_{2}$ with prescribed local and global properties is decisive in this context.

MSC 2010: 11F75; 11F67; 11F70; 22E40; 22E55

## References

• [1]

J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Ann. of Math. Stud. 120, Princeton University, Princeton, 1989. Google Scholar

• [2]

A. I. Badulescu, Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations. With an appendix by N. Grbac, Invent. Math. 172 (2008), no. 2, 383–438.

• [3]

A. I. Badulescu and D. Renard, Unitary dual of $\mathrm{GL}\left(n\right)$ at Archimedean places and global Jacquet–Langlands correspondence, Compos. Math. 146 (2010), no. 5, 1115–1164. Google Scholar

• [4]

A. Borel and W. Casselman, ${L}^{2}$-cohomology of locally symmetric manifolds of finite volume, Duke Math. J. 50 (1983), no. 3, 625–647. Google Scholar

• [5]

A. Borel, J.-P. Labesse and J. Schwermer, On the cuspidal cohomology of S-arithmetic subgroups of reductive groups over number fields, Compos. Math. 102 (1996), no. 1, 1–40. Google Scholar

• [6]

A. Borel and J. Tits, Compléments à l’article: “Groupes réductifs”, Publ. Math. Inst. Hautes Études Sci. 41 (1972), 253–276.

• [7]

D. Bump, Automorphic Forms and Representations, Cambridge Stud. Adv. Math. 55, Cambridge University, Cambridge, 1997. Google Scholar

• [8]

C. J. Bushnell and G. Henniart, The Local Langlands Conjecture for $\mathrm{GL}\left(2\right)$, Grundlehren Math. Wiss. 335, Springer, Berlin, 2006. Google Scholar

• [9]

C. Chevalley, Deux théorèmes d’arithmétique, J. Math. Soc. Japan 3 (1951), 36–44.

• [10]

L. Clozel, On the cuspidal cohomology of arithmetic subgroups of $\mathrm{SL}\left(2n\right)$ and the first Betti number of arithmetic 3-manifolds, Duke Math. J. 55 (1987), no. 2, 475–486. Google Scholar

• [11]

J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro and F. Shahidi, Functoriality for the classical groups, Publ. Math. Inst. Hautes Études Sci. 99 (2004), 163–233.

• [12]

P. Deligne, La conjecture de Weil. I, Publ. Math. Inst. Hautes Études Sci. 43 (1974), 273–307.

• [13]

M. Dimitrov and D. Ramakrishnan, Arithmetic quotients of the complex ball and a conjecture of Lang, Doc. Math. 20 (2015), 1185–1205. Google Scholar

• [14]

J. Franke, Harmonic analysis in weighted ${L}_{2}$-spaces, Ann. Sci. Éc. Norm. Supér. (4) 31 (1998), no. 2, 181–279. Google Scholar

• [15]

J. Franke, A topological model for some summand of the Eisenstein cohomology of congruence subgroups, Eisenstein Series and Applications, Progr. Math. 258, Birkhäuser, Boston (2008), 27–85. Google Scholar

• [16]

J. Franke and J. Schwermer, A decomposition of spaces of automorphic forms, and the Eisenstein cohomology of arithmetic groups, Math. Ann. 311 (1998), no. 4, 765–790.

• [17]

S. Gelbart and H. Jacquet, A relation between automorphic representations of $\mathrm{GL}\left(2\right)$ and $\mathrm{GL}\left(3\right)$, Ann. Sci. Éc. Norm. Supér. (4) 11 (1978), no. 4, 471–542. Google Scholar

• [18]

R. Godement and H. Jacquet, Zeta Functions of Simple Algebras, Lecture Notes in Math. 260, Springer, Berlin, 1972. Google Scholar

• [19]

N. Grbac, Correspondence between the residual spectra of rank two split classical groups and their inner forms, Functional Analysis IX, Various Publ. Ser. (Aarhus) 48, University of Aarhus, Aarhus (2007), 44–57. Google Scholar

• [20]

N. Grbac, On the residual spectrum of split classical groups supported in the Siegel maximal parabolic subgroup, Monatsh. Math. 163 (2011), no. 3, 301–314.

• [21]

N. Grbac, The Franke filtration of the spaces of automorphic forms supported in a maximal proper parabolic subgroup, Glas. Mat. Ser. III 47(67) (2012), no. 2, 351–372. Google Scholar

• [22]

N. Grbac and H. Grobner, The residual Eisenstein cohomology of ${\mathrm{Sp}}_{4}$ over a totally real number field, Trans. Amer. Math. Soc. 365 (2013), no. 10, 5199–5235. Google Scholar

• [23]

N. Grbac and J. Schwermer, An exercise in automorphic cohomology—the case ${\mathrm{GL}}_{2}$ over a quaternion algebra, Arithmetic Geometry and Automorphic Forms, Adv. Lect. Math. (ALM) 19, International Press, Somerville (2011), 209–252. Google Scholar

• [24]

N. Grbac and J. Schwermer, On residual cohomology classes attached to relative rank one Eisenstein series for the symplectic group, Int. Math. Res. Not. IMRN 2011 (2011), no. 7, 1654–1705. Google Scholar

• [25]

N. Grbac and J. Schwermer, Eisenstein series, cohomology of arithmetic groups, and automorphic L-functions at half-integral arguments, Forum Math. 26 (2014), no. 6, 1635–1662. Google Scholar

• [26]

H. Grobner, Automorphic forms, cohomology and CAP representations. The case ${\mathrm{GL}}_{2}$ over a definite quaternion algebra, J. Ramanujan Math. Soc. 28 (2013), no. 1, 19–48. Google Scholar

• [27]

H. Grobner, Residues of Eisenstein series and the automorphic cohomology of reductive groups, Compos. Math. 149 (2013), no. 7, 1061–1090.

• [28]

H. Grobner, A cohomological injectivity result for the residual automorphic spectrum of ${\mathrm{GL}}_{n}$, Pacific J. Math. 268 (2014), no. 1, 33–46. Google Scholar

• [29]

G. Harder, Eisenstein cohomology of arithmetic groups. The case ${\mathrm{GL}}_{2}$, Invent. Math. 89 (1987), no. 1, 37–118. Google Scholar

• [30]

H. Jacquet and R. P. Langlands, Automorphic Forms on $\mathrm{GL}\left(2\right)$, Lecture Notes in Math. 114, Springer, Berlin, 1970. Google Scholar

• [31]

H. Jacquet and J. A. Shalika, A non-vanishing theorem for zeta functions of ${\mathrm{GL}}_{n}$, Invent. Math. 38 (1976/77), no. 1, 1–16. Google Scholar

• [32]

H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. I, Amer. J. Math. 103 (1981), no. 3, 499–558.

• [33]

H. H. Kim, The residual spectrum of ${\mathrm{Sp}}_{4}$, Compos. Math. 99 (1995), no. 2, 129–151. Google Scholar

• [34]

H. H. Kim, The residual spectrum of ${\mathrm{G}}_{2}$, Canad. J. Math. 48 (1996), no. 6, 1245–1272. Google Scholar

• [35]

H. H. Kim, Residual spectrum of odd orthogonal groups, Int. Math. Res. Not. IMRN 2001 (2001), no. 17, 873–906.

• [36]

H. H. Kim and F. Shahidi, Symmetric cube L-functions for ${\mathrm{GL}}_{2}$ are entire, Ann. of Math. (2) 150 (1999), no. 2, 645–662. Google Scholar

• [37]

T. Konno, The residual spectrum of Sp(2), unpublished manuscript (1994).

• [38]

B. Kostant, Lie algebra cohomology and the generalized Borel–Weil theorem, Ann. of Math. (2) 74 (1961), 329–387.

• [39]

J.-P. Labesse and R. P. Langlands, L-indistinguishability for $\mathrm{SL}\left(2\right)$, Canad. J. Math. 31 (1979), no. 4, 726–785. Google Scholar

• [40]

J.-P. Labesse and J. Schwermer, On liftings and cusp cohomology of arithmetic groups, Invent. Math. 83 (1986), no. 2, 383–401.

• [41]

R. P. Langlands, Euler Products, Yale University, New Haven, 1971. Google Scholar

• [42]

R.P. Langlands, Letter to Armand Borel, 25. October 1972.

• [43]

R. P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math. 544, Springer, Berlin, 1976. Google Scholar

• [44]

R. P. Langlands, On the classification of irreducible representations of real algebraic groups, Representation Theory and Harmonic Analysis on Semisimple Lie Groups, Math. Surveys Monogr. 31, American Mathematical Society, Providence (1989), 101–170. Google Scholar

• [45]

J.-S. Li and J. Schwermer, Constructions of automorphic forms and related cohomology classes for arithmetic subgroups of ${\mathrm{G}}_{2}$, Compos. Math. 87 (1993), no. 1, 45–78. Google Scholar

• [46]

J.-S. Li and J. Schwermer, On the Eisenstein cohomology of arithmetic groups, Duke Math. J. 123 (2004), no. 1, 141–169.

• [47]

C. Mœglin and J.-L. Waldspurger, Le spectre résiduel de $\mathrm{GL}\left(n\right)$, Ann. Sci. Éc. Norm. Supér. (4) 22 (1989), no. 4, 605–674. Google Scholar

• [48]

C. Mœglin and J.-L. Waldspurger, Décomposition spectrale et séries d’Eisenstein, Progr. Math. 113, Birkhäuser, Basel, 1994. Google Scholar

• [49]

T. Oda and J. Schwermer, Mixed Hodge structures and automorphic forms for Siegel modular varieties of degree two, Math. Ann. 286 (1990), no. 1–3, 481–509.

• [50]

I. Piatetski-Shapiro, Work of Waldspurger, Lie Group Representations. II (College Park 1982/1983), Lecture Notes in Math. 1041, Springer, Berlin (1984), 280–302. Google Scholar

• [51]

J. Rohlfs and B. Speh, Pseudo-Eisenstein forms and the cohomology of arithmetic groups III: Residual cohomology classes, On Certain L-functions, Clay Math. Proc. 13, American Mathematical Society, Providence (2011), 501–523. Google Scholar

• [52]

R. D. Schafer, An Introduction to Nonassociative Algebras, Pure Appl. Math. 2, Academic Press, New York, 1966. Google Scholar

• [53]

J. Schwermer, Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen, Lecture Notes in Math. 988, Springer, Berlin, 1983. Google Scholar

• [54]

J. Schwermer, On arithmetic quotients of the Siegel upper half-space of degree two, Compos. Math. 58 (1986), no. 2, 233–258. Google Scholar

• [55]

J. Schwermer, Eisenstein series and cohomology of arithmetic groups: The generic case, Invent. Math. 116 (1994), no. 1–3, 481–511.

• [56]

J. Schwermer, On Euler products and residual Eisenstein cohomology classes for Siegel modular varieties, Forum Math. 7 (1995), no. 1, 1–28.

• [57]

F. Shahidi, On certain L-functions, Amer. J. Math. 103 (1981), no. 2, 297–355.

• [58]

F. Shahidi, Local coefficients as Artin factors for real groups, Duke Math. J. 52 (1985), no. 4, 973–1007.

• [59]

F. Shahidi, On the Ramanujan conjecture and finiteness of poles for certain L-functions, Ann. of Math. (2) 127 (1988), no. 3, 547–584.

• [60]

F. Shahidi, Third symmetric power L-functions for $\mathrm{GL}\left(2\right)$, Compos. Math. 70 (1989), no. 3, 245–273. Google Scholar

• [61]

F. Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series of $𝔭$-adic groups, Ann. of Math. (2) 132 (1990), 273–330. Google Scholar

• [62]

F. Shahidi, Twisted endoscopy and reducibility of induced representations for p-adic groups, Duke Math. J. 66 (1992), no. 1, 1–41.

• [63]

A. J. Silberger, Introduction to Harmonic Analysis on Reductive p-adic Groups, Math. Notes 23, Princeton University, Princeton, 1979. Google Scholar

• [64]

J. T. Tate, Fourier analysis in number fields, and Hecke’s zeta-functions, Algebraic Number Theory, Academic Press, London (1967), 305–347. Google Scholar

• [65]

D. Trotabas, Non annulation des fonctions L des formes modulaires de Hilbert au point central, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 1, 187–259.

• [66]

D. A. Vogan, Jr. and G. J. Zuckerman, Unitary representations with nonzero cohomology, Compos. Math. 53 (1984), no. 1, 51–90. Google Scholar

• [67]

C. Waldner, Geometric cycles and the cohomology of arithmetic subgroups of the exceptional group ${\mathrm{G}}_{2}$, J. Topol. 3 (2010), no. 1, 81–109. Google Scholar

• [68]

C. Waldner, Geometric cycles with local coefficients and the cohomology of arithmetic subgroups of the exceptional group ${\mathrm{G}}_{2}$, Geom. Dedicata 151 (2011), 9–25. Google Scholar

• [69]

J.-L. Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3 (1991), no. 3, 219–307. Google Scholar

• [70]

A. Weil, On a certain type of characters of the idèle-class group of an algebraic number-field, Proceedings of the International Symposium on Algebraic Number Theory, Science Council of Japan, Tokyo (1956), 1–7. Google Scholar

• [71]

S. Žampera, The residual spectrum of the group of type ${\mathrm{G}}_{2}$, J. Math. Pures Appl. (9) 76 (1997), no. 9, 805–835. Google Scholar

Revised: 2019-03-15

Published Online: 2019-06-13

Published in Print: 2019-09-01

Funding Source: Hrvatska Zaklada za Znanost

Award identifier / Grant number: 3628

Award identifier / Grant number: 9364

The first named author was supported in part by the Croatian Science Foundation (projects 3628 and 9364) and by the University of Rijeka (research grant 13.14.1.2.02). Both authors acknowledge the support obtained within the frame work of the Croatian-Austrian Scientific agreement (HR 17/2014).

Citation Information: Forum Mathematicum, Volume 31, Issue 5, Pages 1225–1263, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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