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The residual Eisenstein cohomology of $ Sp_{4}$ over a totally real number field

Authors: Neven Grbac and Harald Grobner
Journal: Trans. Amer. Math. Soc. 365 (2013), 5199-5235
MSC (2010): Primary 11F75; Secondary 11F70, 11F55, 22E55
Published electronically: March 12, 2013
MathSciNet review: 3074371
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Abstract: Let $ G=Sp_4/k$ be the $ k$-split symplectic group of $ k$-rank 2, where $ k$ is a totally real number field. In this paper we compute the Eisenstein cohomology of $ G$ with respect to any finite-dimensional, irreducible, $ k$-rational representation $ E$ of $ G_\infty =R_{k/\mathbb{Q}}G(\mathbb{R})$, where $ R_{k/\mathbb{Q}}$ denotes the restriction of scalars from $ k$ to $ \mathbb{Q}$. This approach is based on the work of Schwermer regarding the Eisenstein cohomology for $ Sp_4/\mathbb{Q}$, Kim's description of the residual spectrum of $ Sp_4$, and the Franke filtration of the space of automorphic forms. In fact, taking the representation theoretic point of view, we write, for the group $ G$, the Franke filtration with respect to the cuspidal support, and give a precise description of the filtration quotients in terms of induced representations. This is then used as a prerequisite for the explicit computation of the Eisenstein cohomology. The special focus is on the residual Eisenstein cohomology. Under a certain compatibility condition for the coefficient system $ E$ and the cuspidal support, we prove the existence of non-trivial residual Eisenstein cohomology classes, which are not square-integrable, that is, represented by a non-square-integrable residue of an Eisenstein series.

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  • [Bor83] A. BOREL, Regularization Theorems in Lie algebra cohomology, Duke Math. J. 50 (1983), 605-623 MR 714820 (84h:17009)
  • [Bor74] A. BOREL, Stable real cohomology of arithmetic groups, Ann. Scient. Éc. Norm. Sup. 7 (1974), 235-272 MR 0387496 (52:8338)
  • [BJ] A. BOREL, H. JACQUET, Automorphic forms and automorphic representations, In: Automorphic Forms, Representations, and $ L$-functions, A. Borel, W. Casselman eds., Proc. Symp. Pure Maths. 33, Part I, Amer. Math. Soc., Providence, RI, 1979, pp. 189-202 MR 546598 (81m:10055)
  • [BW] A. BOREL, N. WALLACH, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, 2nd edition, Math. Surveys Monogr. 67, Amer. Math. Soc., Providence, RI, 2000 MR 1721403 (2000j:22015)
  • [Che] C. CHEVALLEY, Deux Théorèmes d'Arithmétique, J. Math. Soc. Japan 3 (1951), 36-44 MR 0044570 (13:440a)
  • [Fra98] J. FRANKE, Harmonic analysis in weighted $ L_2$-spaces, Ann. Scient. Éc. Norm. Sup. 31 (1998), 181-279 MR 1603257 (2000f:11065)
  • [Fra08] J. FRANKE, A topological model for some summand of the Eisenstein cohomology of congruence subgroups, In: Eisenstein series and applications, W.T. Gan, S.S. Kudla, Y. Tschinkel eds., Progr. Math. 258, Birkhäuser Boston, Boston, 2008, pp. 27-85 MR 2402680 (2009i:11069)
  • [FS] J. FRANKE, J. SCHWERMER, A decomposition of spaces of automorphic forms, and the Eisenstein cohomology of arithmetic groups, Math. Ann. 311 (1998), 765-790 MR 1637980 (99k:11077)
  • [Har87] G. HARDER, Eisenstein cohomology of arithmetic groups. The case GL$ _2$, Invent. Math. 89 (1987), 37-118 MR 892187 (89b:22018)
  • [Har93] G. HARDER, Eisensteinkohomologie und die Konstruktion gemischter Motive, Lecture Notes in Mathematics 1562, Springer-Verlag, Berlin, 1993 MR 1285354 (95g:11043)
  • [Har75] G. HARDER, On the cohomology of $ SL(2,\mathfrak{O})$, In: Lie Groups and their Representations, Proc. of the Summer School on Group Representations, I. M. Gelfand ed., Halsted and Wiley Press, London, 1975, pp. 139-150 MR 0425019 (54:12977)
  • [Har73] G. HARDER, On the cohomology of discrete arithmetically defined groups, In: Discrete Subgroups of Lie Groups and Applications to Moduli, Papers presented at the Bombay Colloquium, 1973, Oxford University Press, 1975, pp. 129-160 MR 0425018 (54:12976)
  • [Har90] G. HARDER, Some results on the Eisenstein cohomology of arithmetic subgroups of GL$ _n$, In: Cohomology of Arithmetic Groups and Automorphic Forms, J.-P. Labesse, J. Schwermer eds., Lecture Notes in Mathematics 1447, Springer-Verlag, Berlin, 1990, pp. 86-153 MR 1082964 (91j:11040)
  • [JL] H. JACQUET, R.P. LANGLANDS, Automorphic Forms on $ GL(2)$, Lecture Notes in Math. 114, Springer-Verlag, Berlin, New York, 1970 MR 0401654 (53:5481)
  • [KR] A. KEWENIG, T. RIEBAND, Geisterklassen im Bild der Borelabbildung für symplektische und orthogonale Gruppen, Diplomarbeit, Mathematisches Institut der Rheinisch Friedrich-Wilhelms-Universität Bonn, Bonn, 1997 (unpublished)
  • [Kim] H.H. KIM, The residual spectrum of $ Sp_4$, Compositio Math. 99 (1995), 129-151 MR 1351833 (97c:11056)
  • [Lan] R.P. LANGLANDS, On the Functional Equations Staisfied by Eisenstein Series, Lecture Notes in Math. 544, Springer-Verlag, Berlin-Heidelberg, New York, 1976 MR 0579181 (58:28319)
  • [LS] J.-S. LI, J. SCHWERMER, On the Eisenstein cohomology of arithmetic groups, Duke Math. J. 123 (2004), 141-169 MR 2060025 (2005h:11108)
  • [MW] C. MœGLIN, J.-L. WALDSPURGER, Spectral Decomposition and Eisenstein Series, Cambridge Tracts in Math. 113, Cambridge Univ. Press, Cambridge, 1995 MR 1361168 (97d:11083)
  • [OS] T. ODA, J. SCHWERMER, Mixed Hodge structures and automorphic forms for Siegel modular varieties of degree two, Math. Ann. 286 (1990), 481-509 MR 1032942 (90m:11072)
  • [Ram] D. RAMAKRISHNAN, Modularity of the Rankin-Selberg $ L$-series, and multiplicity one for $ SL(2)$, Annals of Math. 152 (2000), 45-111 MR 1792292 (2001g:11077)
  • [Sch83] J. SCHWERMER, Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen, Lecture Notes in Mathematics 988, Springer-Verlag, Berlin, 1983 MR 822473 (87i:22034)
  • [Sch86] J. SCHWERMER, On arithmetic quotients of the Siegel upper half space of degree two, Compositio Math. 58 (1986), 233-258 MR 844411 (87j:11040)
  • [Sch95] J. SCHWERMER, On Euler products and residual Eisenstein cohomology classes for Siegel modular varieties, Forum Math. 7 (1995), 1-28 MR 1307953 (96d:11062)
  • [Ser89] J.-P. SERRE, Abelian l-Adic Representations and Elliptic Curves, Advenced book classic series, Addison-Wesley Publishing Company, 1989 MR 1043865 (91b:11071)

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Additional Information

Neven Grbac
Affiliation: Department of Mathematics, University of Rijeka, Radmile Matejčić 2, HR-51000 Rijeka, Croatia

Harald Grobner
Affiliation: Fakultät für Mathematik, University of Vienna, Nordbergstrasse 15, A-1090 Wien, Austria

Keywords: Cohomology of arithmetic groups, Eisenstein cohomology, Eisenstein series, residue of Eisenstein series, automorphic forms, cuspidal automorphic representation, Franke filtration
Received by editor(s): September 15, 2010
Received by editor(s) in revised form: September 26, 2011, January 20, 2012, and January 24, 2012
Published electronically: March 12, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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