Arithmetic vector bundles and automorphic forms on Shimura varieties, II

Michael Harris

Compositio Mathematica (1986)

  • Volume: 60, Issue: 3, page 323-378
  • ISSN: 0010-437X

How to cite


Harris, Michael. "Arithmetic vector bundles and automorphic forms on Shimura varieties, II." Compositio Mathematica 60.3 (1986): 323-378. <>.

author = {Harris, Michael},
journal = {Compositio Mathematica},
keywords = {automorphic vector bundles; automorphic forms on Shimura varieties; rationality of the Fourier-Jacobi coefficients},
language = {eng},
number = {3},
pages = {323-378},
publisher = {Martinus Nijhoff Publishers},
title = {Arithmetic vector bundles and automorphic forms on Shimura varieties, II},
url = {},
volume = {60},
year = {1986},

AU - Harris, Michael
TI - Arithmetic vector bundles and automorphic forms on Shimura varieties, II
JO - Compositio Mathematica
PY - 1986
PB - Martinus Nijhoff Publishers
VL - 60
IS - 3
SP - 323
EP - 378
LA - eng
KW - automorphic vector bundles; automorphic forms on Shimura varieties; rationality of the Fourier-Jacobi coefficients
UR -
ER -


  1. [1] W.L. Baily, Jr and A. Borel: Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math.84 (1966) 442-528. Zbl0154.08602MR216035
  2. [5] P. Deligne: Travaux de Shimura. Sém. Bourbaki23 (1970/71) No. 389. Zbl0225.14007MR498581
  3. [6] P. Deligne: Variétés de Shimura: Interprétation modulaire, et techniques de construction des modèles canoniques. Proc. Symp. Pure Math.XXXIII (1979) Part 2, 247-290. Zbl0437.14012MR546620
  4. [11] P.B. Garrett: Arithmetic and structure of automorphic forms on bounded symmetric domains. Am. J. Math., 105 (1983) 1171-1194; II, Ibid., 1195-1216. Zbl0527.10025MR714773
  5. [12] M. Harris: Eisenstein series on Shimura varieties. Ann. of Math., 119 (1984). Zbl0589.10030MR736560
  6. [13] M. Harris: Arithmetic vector bundles on Shimura varieties, in A utomorphic Forms of Several Variables, Progress in Math. (Boston: Birkhäuser-Verlag) (1984). Zbl0571.10031MR763013
  7. [19] G. Shimura: On canonical models of arithmetic quotients of bounded symmetric domains I, II. Ann. of Math.91 (1970) 144-222; 92 (1970) 528-549. Zbl0237.14010MR257031
  8. [20] G. Shimura: On certain reciprocity laws for theta functions and modular forms. Acta Math.141 (1978) 35-71. Zbl0402.10030MR491518
  9. [21] G. Shimura: The arithmetic of automorphic forms with respect to a unitary group. Ann. of Math.107 (1978) 569-605. Zbl0409.10016MR563087
  10. [23] G. Shimura: Automorphic forms and the periods of abelian varieties. J. Math. Soc. Japan31 (1979) 561-592. Zbl0456.10015MR535097
  11. [24] G Shimura: The arithmetic of certain zeta functions and automorphic forms on orthogonal groups. Ann. of Math.111 (1980) 313-375. Zbl0438.12003MR569074
  12. [26] A. Ash; D. Mumford, M. Rapoport, and Y-S. Tai: Smooth Compactification of Locally Symmetric Varieties, Brookline, Mass.: Math. Sci. Press (1975). Zbl0334.14007MR457437
  13. [27] W.L. Baily, Jr.: On the Fourier coefficients of certain Eisenstein series on the adèle group. In: Number Theory, Algebraic Geometry, and Commutative Algebra: in Honor of Y. Akizuki, Tokyo: Kinokuniya (1973). Zbl0283.10017MR367320
  14. [28] D. Blasius: Arithmetic of monomial relations between the periods of abelian varieties. Thesis, Princeton University (1981). 
  15. [29] A. Borel: L2-cohomology and intersection cohomology of certain arithmetic varieties. In B. Srinivasan and J. Sally (eds) Emmy Noether in Bryn Mawr. New York: Springer-Verlag (1983) 119-131. Zbl0541.14015MR713796
  16. [30] A. Borel and J. Tits: Groupes Rèductifs. Publ. Math. I.H.E.S.27 (1965) 55-151. Zbl0145.17402MR207712
  17. [31] J.L. Brylinski: "1-motifs" et formes automorphes. In Journées Automorphes, Publ. Math. de l'Univ. ParisVII15 (1983). Zbl0565.14001MR723182
  18. [32] C.-L. Chai: Compactifications of Siegel Moduli Schemes, Cambridge: Cambridge University Press (1985). Zbl0578.14009MR853543
  19. [33] D. Degeorge and N. Wallach: Limit formulas for multiplicities in L2(Γ). Ann. of Math.107 (1978) 133-150. Zbl0397.22007
  20. [34] P. Deligne: Espaces Hermitiens symétriques, notes from a course at I.H.E.S. 
  21. [35] P. Deligne and M. Rapoport: Les schémas de modules de courbes elliptiques. Lecture Notes in Math349 (1972) 143-316. Zbl0281.14010MR337993
  22. [36] P. Garrett: Modular curves on arithmetic quotients. Duke Math. J.49 (1982) 633-654. Zbl0501.10025MR672500
  23. [37] M. Harris: Special values of zeta functions attached to Siegel modular forms. Ann. Sci. de l'E.N.S.14 (1981) 77-120. Zbl0465.10022MR618732
  24. [38] M. Harris: Maass operators and Eisenstein series. Math. Ann.258 (1981) 135-144. Zbl0455.10018MR641820
  25. [39] R. Indik: Fourier coefficients of non-holomorphic Eisenstein series on a tube domain associated to an orthogonal group, to appear. 
  26. [40] H.P. Jakobsen and M. Vergne: Restrictions and expansions of holomorphic representations, J. Fun. Analysis34, (1979) 29-53. Zbl0433.22011MR551108
  27. [41] N. Katz: p-adic L-functions for CM fields. Inv. Math.59 (1978) 199-297. Zbl0417.12003MR513095
  28. [42] S.S. Kudla: On certain arithmetic automorphic forms for SU(1, q). Inv. Math.52 (1979) 1-26. Zbl0408.10016MR532744
  29. [43] H. Maass: Siegel's Modular Forms and Dirichlet Series, Lecture Notes in Math., 216 (1971). Zbl0224.10028
  30. [44] D. Mumford: On the equations defining abelian varieties: I, II, III, Inv. Math.1 (1966) 287-354; 3 (1967) 75-135, 215-244. Zbl0219.14024MR204427
  31. [45] T. Oda: On modular forms associated with indefinite quadratic forms of signature (2, n - 2), Math. Ann.231 (1977) 97-144. Zbl0346.10013MR466026
  32. [46] T. Oda: Hodge structures of Shimura varieties attached to the unit groups of quaternion algebras, in Advanced Studies in Pare Mathematics 2: Galois Groups and Their Representations, 15-36Tokyo: Kinokuniya Press (1983). Zbl0594.14003MR732459
  33. [47] I.I. Piatetskii-Shapiro: Geometry of Classical Domains and the Theory of Automorphic Functions, New York, Gordon and Breach (1969). 
  34. [48] I.I. Piatetskii-Shapiro and S. Rallis: L-functions for the classical groups, notes on a seminar at I.A.S. prepared by J. Cogdell (1984). 
  35. [49] I. Satake: Algebraic Structures of Symmetric Domains, Princeton University Press (1980). Zbl0483.32017MR591460
  36. [50] W. Schmid: On a conjecture of Langlands. Ann. of Math.93 (1971) 1-42. Zbl0291.43013MR286942
  37. [51] I. Schur: Dissertation, Berlin (1901). 
  38. [52] G. Shimura: Modules des variétés abéliennes polarisées et fonctions modulaires: I, II, III, In Sém. Cartan1957/58, exposés 18-20, New York: Benjamin (1967). 
  39. [53] G. Shimura: On some arithmetic properties of modular forms of one and several variables. Ann. of Math.102 (1975) 491-515. Zbl0327.10028MR491519
  40. [54] G. Shimura: On the Fourier coefficients of modular forms of several variables. Göttinger Nachr. Akad. Wiss. Nr.17 (1975) 261-268. Zbl0332.32024MR485706
  41. [55] G. Shimura: The special values of the zeta functions associated with cusp forms. Comm. Pure Appl. Math.29 (1976) 783-804. Zbl0348.10015MR434962
  42. [56] G. Shimura: On certain zeta functions attached to two Hilbert modular forms: I, II. Ann. of Math.114 (1981) 127-164, 569-607. Zbl0486.10021
  43. [57] G. Shimura: Arithmetic of differential operators on symmetric domains. Duke Math. J.48 (1981) 813-843. Zbl0487.10021MR782579
  44. [58] G. Shimura: Algebraic relations between critical values of zeta functions and inner products. Am. J. Math.105 (1983) 253-285. Zbl0518.10032MR692113
  45. [59] G. Shimura: On Eisenstein series, Duke Math. J.50 (1983) 417-476. Zbl0519.10019MR705034
  46. [60] L.-C. Tsao: The rationality of the Fourier coefficients of certain Eisenstein series on tube domains. Compositio Math.32 (1976) 225-291. Zbl0346.10014MR419367
  47. [61] M. Vergne and H. Rossi: Analytic continuation of holomorphic discrete series. A cta Math.136 (1976) 1-59. Zbl0356.32020MR480883
  48. [62] N. Wallach: Analytic continuation of the discrete series II. Trans. A.M.S.251 (1979) 19-37. Zbl0419.22018MR531967
  49. [63] J.-L. Waldspurger: Sur les coefficients de Fourier des formes modulaires de poids demi-en-tier. J. Math. pures et appl.60 (1981) 375-484. Zbl0431.10015MR646366
  50. [64] J.-L. Waldspurger: Correspondances de Shimura et quaternions, preprint (1983). Zbl0724.11026MR1103429
  51. [65] J. Wolf: Fine structure of Hermitian symmetric spaces. In: Symmetric Spaces Short Lectures, pp. 271-357, New York: Dekker (1972). Zbl0257.32014MR404716
  52. [66] J. Wolf and A. Kóranyi: Generalized Cayley transformations of bounded symmetric domains. Am. J. Math.87 (1965) 899-939. Zbl0137.27403MR192002
  53. [67] S. Zucker: L2-cohomology of warped products and arithmetic groups. Inv. Math.70 (1982) 169-218. Zbl0508.20020MR684171
  54. [68] S. Zucker: L2-cohomology and intersection homology of locally symmetric varieties, II, preprint (1983). Zbl0624.14014
  55. [69] P. Deligne: Théorie de Hodge, II, Publ. Math. I.H.E.S.40 (1971) 5-58. Zbl0219.14007MR498551
  56. [70] L. Clozel: On limit multiplicities of discrete series representations in spaces of automorphic formsInv. Math.83 (1986) 265-284. Zbl0582.22012MR818353
  57. [71] G. Faltings: Arithmetische Kompaktifizierung des Modulraums der abelschen Varietäten. In Arbeitstagung Bonn1984, Lect. Notes in Math.1111 (1985) 321-383. Zbl0597.14036MR797429
  58. [72] M. Harris: Arithmetic vector bundles and automorphic forms on Shimura varieties, I. Inv. Math.82 (1985) 151-189. Zbl0598.14019MR808114
  59. [73] M. Harris: Automorphic forms of discrete type as coherent cohomology classes (preprint). MR492234

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