Journal of Differential Geometry

Automorphic forms of $øverline\partial$-cohomology type as coherent cohomology classes

Michael Harris

Full-text: Open access

Article information

Source
J. Differential Geom., Volume 32, Number 1 (1990), 1-63.

Dates
First available in Project Euclid: 26 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1214445036

Digital Object Identifier
doi:10.4310/jdg/1214445036

Mathematical Reviews number (MathSciNet)
MR1064864

Zentralblatt MATH identifier
0711.14012

Subjects
Primary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]
Secondary: 17B56: Cohomology of Lie (super)algebras 22E46: Semisimple Lie groups and their representations 32L10: Sheaves and cohomology of sections of holomorphic vector bundles, general results [See also 14F05, 18F20, 55N30] 32N10: Automorphic forms

Citation

Harris, Michael. Automorphic forms of $øverline\partial$-cohomology type as coherent cohomology classes. J. Differential Geom. 32 (1990), no. 1, 1--63. doi:10.4310/jdg/1214445036. https://projecteuclid.org/euclid.jdg/1214445036


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References

  • [1] A. Andreotti and E. Vesentini, Carleman estimates for the Laplace-Beltramioperator on complex manifolds, Inst. Hautes Etudes Sci. Publ. Math. 25 (1965) 81-130.
  • [2] A. Ash, D. Mumford, M. Rapoport and Y-S. Tai, Smooth Compactification of locallysymmetric varieties, Math. Sci. Press, Brookline, MA, 1975.
  • [3] W. L. Baily, Jr. and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. 84 (1966) 442-528.
  • [4] D. Blasius, L. Clozel and D. Ramakrishnan, lgebricitedel action des operateurs de Hecke sur certaines formes de Maass Operateurs de Hecke et formes de Maass: application de laformule des traces, C. R. Acad. Sci. Paris 305 (1987) 705-708, 306 (1988) 59-62.
  • [5] D. Blasius, M. Harris and D. Ramakrishnan, to appear.
  • [6] A. Borel, Introduction to automorphicforms, Proc. Sympos. Pure Math., Vol. 9, Amer. Math. Soc, Providence, RI, 1966, 199-210.
  • [7] A. Borel, Introduction aux groupes arithmetiques, Hermann, Paris, 1969.
  • [8] A. Borel, Stable real cohomology of arithmetic groups, Ann. Sci. Ecole Norm. Sup.7 (1974) 235-272.
  • [9] A. Borel, Stable real cohomology of arithmetic groups.II, Manifolds and Lie Groups, Papers in honor of Y. Matsushima, Progress in Math., Vol. 14, Birkhaser, Boston, 1981, 21-55.
  • [10] A. Borel and H. Garland, Laplacian and the discrete spectrum of an arithmetic group, Amer. J. Math. 105 (1983) 309-335.
  • [11] A. Borel and H. Jacquet, Automorphicforms and automorphic representations, Proc. Sympos. Pure Math., Vol. 33, Part 1, Amer. Math. Soc, Providence, RI, 1979, 189-202.
  • [12] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Ann. of Math. Studies, No.94, Princeton University Press, Princeton, NJ, 1980.
  • [13] M. Borovoi, Langlands' conjecture concerning conjugation of Shimura varieties, Sci. Math. Sov. 3 (1984) 3-39.
  • [14] J.-L. Brylinski, Algebricite du quotient par un groupe discret d'un domaine de Siegel de 3eme espece, provenant d'un espace hermitien symetrique, manuscript, 1979.
  • [15] W. Casselman and M. S. Osborne, The n-cohomology of representationswith an infinitesimal character, Compositio Math. 31 (1975) 219-227.
  • [16] P. Deligne, Theoriede Hodge. II, Inst. Hautes Etudes Sci. Publ. Math. 40 (1971) 5-58.
  • [17] P. Deligne, Theoriede Hodge. II, Varietes de Shimura: Interpretation modulaire et techniques de construction de modeles canoniques, Proc. Sympos. Pure Math., Vol. 33, Part 2, Amer. Math. Soc, Providence, RI, 1979, 247-290.
  • [18] P. Deligne, Theoriede Hodge. II, Valeursde functions L et periodes d'integrales, Proc Sympos. Pure Math., Vol. 33, Part 2, Amer. Math. Soc, Providence, RI, 1979, 313-346.
  • [19] P. Deligne, J. S. Milne, A. Ogus and K.-Y. Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math., Vol. 900, Springer, Berlin, 1982.
  • [20] G. Faltings, On the cohomology of locallysymmetric hermitian spaces, Lecture Notes in Math., Vol. 1029, Springer, Berlin, 1984, 55-98.
  • [21] G. Faltings, Arithmetic varieties and rigidity (Sem. de Theorie des Nombres, Paris 1982-83), Progress in Math., Vol. 51, Boston, Birkhaser, 1984, 63-78.
  • [22] G. Faltings, Arithmetic theory of Siegel modularforms, Lecture Notes in Math., Vol. 1240, Springer, Berlin, 1987, 101-108.
  • [23] P. B. Garrett and M. Harris, Special values of tripleproduct L-functions, in preparation.
  • [24] M. Harris, Special values of zeta functions attached to Siegel modularforms, Ann. Sci. Ecole Norm. Sup. 14 (1981) 77-120.
  • [25] M. Harris, Arithmetic vector bundles and automorphicforms on Shimura varieties. I, Invent. Math. 82 (1985) 151-189; II, Compositio Math. 60 (1986) 323-378.
  • [26] M. Harris, Formes automorphes "geometriques" non-holomorphes: Problemes d'arithmeticite (Sem de Theorie des Nombres, Paris 1984-85), Progress in Math., Vol. 63, Birkhaser, Boston, 1986, 109-129.
  • [27] M. Harris, Functorial properties of toroidal compactifications of locally symmetric varieties, Proc. London Math. Soc, to appear.
  • [28] M.Harris and D.H.Phong, Cohomologie de Dolbeault a croissancel ogarithmique a I'infini, C. R. Acad. Sci. Paris 302 (1986) 307-310.
  • [29] H. Hecht and W. Schmid, On integrable representations of a semisimple Lie group, Math. Ann. 220 (1976), 147-149.
  • [30] G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat, Toroidal embeddings, Lecture Notes in Math., Vol. 339, Springer, Berlin, 1973.
  • [31] A. W. Knapp, Representation theory of semisimple groups, Princeton University Press, Princeton, NJ, 1986.
  • [32] D. Knutson, Algebraicspaces, Lecture Notes in Math., Vol. 203, Springer, Berlin, 1971.
  • [33] R. P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Math., Vol. 544, Springer, Berlin, 1976.
  • [34] J. S. Milne, The action of an automorphism of C on a Shimura variety and its special points, Progress in Math., Vol. 35, Birkhaser, Boston, 1983, 234-265.
  • [35] J. S. Milne, Automorphic vector bundles on connected Shimura varieties, Invent. Math. 92 (1988) 91-128.
  • [36] D. Mumford, Hirzebruch's proportionality theorem in the non-compact case, Invent. Math. 42 (1977) 239-272.
  • [37] M. S. Narasimhan and K. Okamoto, An analogue of the Borel-Weil-Bott theorem for Hermitian symmetric pairs of non-compacttype, Ann. of Math. (2) 91 (1970) 486-511.
  • [38] T. Oda, Hodge structures attached to geometric automorphic forms, in Automorphic Forms and Number Theory, Advanced Stud. Pure Math., Vol. 7, North-Holland, Amsterdam, 1985, 223-276.
  • [39] K. Okamoto and H. Ozeki, On square integrable d-bar cohomology classes attached to Hermitian symmetric spaces, Osaka J. Math. 4 (1967) 85-94.
  • [40] J. Repka, Tensor products of unitary representations of SL2 (E), Amer. J. Math. 100 (1978) 747-774.
  • [41] I. Satake, Theory of spherical functions on reductivealgebraic groups over p-adicfields, Inst. Hautes Etudes Sci. Publ. Math. 18 (1963) 1-69.
  • [42] W. Schmid, On a conjecture of Langlands, Ann. of Math. (2) 93 (1971) 1-42.
  • [43] W. Schmid, On a conjecture of Langlands, L2 cohomology and the discrete series, Ann. of Math. (2) 103 (1976) 375-394.
  • [44] J.-P.Serre, Geometrie algebrique et geometrie analytique, Ann. Inst. Fourier (Grenoble) 6 (1956) 1-42.
  • [45] J.-P.Serre, Corps locaux, Hermann, Paris, 1962.
  • [46] G. Shimura, Moduli of abelian varieties and number theory, Proc. Sympos. Pure Math., Vol. 9, Amer. Math. Soc, Providence, RI, 1966, 312-332.
  • [47] G. Shimura, On canonical models of arithmetic quotients of boundedsymmetric domains. I, II, Ann. of Math. (2) 91 (1970) 144-222; 92 (1970) 528-549.
  • [48] G. Shimura, On certain reciprocity laws for theta functions and modular forms, Acta Math. 141 (1978) 35-71.
  • [49] G. Shimura, Automorphic forms and the periods of abelian varieties, J. Math. Soc. Japan 31 (1979) 561-592.
  • [50] G. Shimura, On certain zeta functions attached to two Hilbert modular forms. I, II, Ann. of Math. (2) 114 (1981), 127-164; 569-607.
  • [51] J. Tits, Reductive groups over local fields, Proc. Sympos. Pure Math., Vol. 33, Part 1, Amer. Math. Soc, Providence, RI, 1979, 29-69.
  • [52] D. A. Vogan, Jr., Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984) 141-187.
  • [53] D. A. Vogan, Jr. and G. Zuckerman, Unitary representations with non-zerocohomology, Compositio Math. 53 (1984) 51-90.
  • [54] N. Wallach, On the constant term of a square integrable automorphic form, Operator Algebras and Group Representations, Vol. II (Neptun, 1980), Monographs Stud. Math. 18 (1984) 227-237.
  • [55] F. L. Williams, The n-cohomology of limits of discreteseries, preprint.
  • [56] F. L. Williams, Unitary representations and a general vanishing theorem for (0, r)-cohomology, preprint.