Intersections of higher weight cycles and modular forms

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1. [A] E. Artin: Geometric Algebra, Wiley Interscience, New York, 1957. Zbl0077.02101MR82463
2. [Bo1] A. Borel: Density properties for certain subgroups of semisimple groups without compact components, Ann. of Math.72, (1960), 179-188. Zbl0094.24901MR123639
3. [Bo2] A. Borel: Compact Clifford-Klein forms of symmetric spaces, Topology2 (1963) 111-132. Zbl0116.38603MR146301
4. [BHC] A. Borel and Harish-Chandra: Arithmetic subgroups of algebraic groupsAnn. of Math.75 (1962) 485-535 Zbl0107.14804MR147566
5. [Car] H. Cartan: Formes modulaire expose 4, Functions Automorphes Sem. Henri Cartan, 10e annee 1957/58, Ecole Normal Superieure, Paris, 1958 
6. [Cas] J.W.S. Cassels: Rational Quadratic Forms, Academic Press, London, New York, San Francisco, 1978. Zbl0395.10029MR522835
7. [Ch] C.C. Chevalley: The Algebraic Theory of Spinors, Columbia Univ. Press, Morningside Heights, NY, 1954. Zbl0057.25901MR60497
8. [Cg1] J.W. Cogdell: Arithmetic cycles on Picard modular surfaces and modular forms of Nebentypus, J. reine angew. Math.357 (1985) 115-137. Zbl0547.10028MR783537
9. [Cg2] J. Cogdell: The Weil representation and cycles on Picard modular surfaces, preprint 
10. [D1] P. Deligne: Travaux de Shimura, Seminaire Bourbaki, vol. 1970/71. Exposes 382-399, Lect. Notes Math.244, Springer-VerlagBerlin, Heidelberg, New York, 1971, pp. 123-165. Zbl0225.14007MR498581
11. [D2] P. Deligne: Varietes de Shimura: interpretation modulaire et techniques de construction de modeles canoniques, Automorphic Forms, Representations, and L-functions, Proc. Symp. Pure Math 33, part 2 Amer. Math. Soc., Providence, 1979, pp. 247-289. Zbl0437.14012MR546620
12. [D3] P. Deligne: Valeurs de fonctions L et periods d'integrales, Automorphic Forms, Representations, and L-functions, Proc. Symp. Pure Math 33, part 2, Amer. Math. Soc., Providence, 1979, pp 313-346. Zbl0449.10022MR546622
13. [E] M. Eichler: Quadratische Formen und Orthogonale Gruppen, Springer-Verlag, Berlin, Heidelberg, New York, 1974. Zbl0277.10017MR351996
14. [EZ] M. Eichler and D. Zagier: The Theory of Jacobi Forms, Progress in Math.55, Birkhauser, Boston, Basel, Stuttgart1985. Zbl0554.10018MR781735
15. [F] E. Freitag: Siegelsche Modulfunktionen, Springer-Verlag, Berlin, Heidelberg, New York, 1983. Zbl0498.10016MR871067
16. [Go1] B.B. Gordon: Intersections of higher weight cycles over quaternionic modular surfaces and modular forms of Nebentypus, Bull. AMS.14 (1986) 293-298. Zbl0593.14003MR828829
17. [Go2] B.B. Gordon: Algebraically defined subspaces in the cohomology of a Kuga fiber variety, Pacifc J. Math.131 (1988) 261-276. Zbl0615.14026MR922218
18. [GH] P. Griffiths and J. Harris: Principles of Algebraic Geometry, John Wiley and Sons, New York, 1978. Zbl0408.14001MR507725
19. [Gr] A. Grothendieck: Standard conjectures on algebraic cycles, Algebraic Geometry: Papers Presented at the Bombay Colloquium, 1969, Oxford Univ. Press, Oxford, 1969. Zbl0201.23301MR268189
20. [HLR] G. Harder, R. Langlands and M. Rapoport: Algebraische Zyklen auf Hilbert Blumenthal-Flächen, J. reine u. angew. Math.366 (1986) 53-120. Zbl0575.14004MR833013
21. [He] E. Hecke: Analytische Arithmetik der positiven quadratischen Formen, Mathematische Werke, Vanderhoeck und Ruprecht, Göttingen, 1970, pp. 789-918. MR104550
22. [Hi] F. Hirzebruch: Topological Methods in Algebraic Geometry, Grundlehren der mathematischen Wissenschaften131, Springer-Verlag, Berlin, Heidelberg, New York, 1978. Zbl0376.14001MR1335917
23. [HZ] F. Hirzebruch and D. Zagier: Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, Invent. Math.36 (1976) 57-113. Zbl0332.14009MR453649
24. [Ho] R. Howe: θ-series and invariant theory, Automorphic Forms, Representations, and L-functions, Proc. Symp. Pure Math 33, part 2, Amer. Math. Sac., Providence1979 pp. 275-285. Zbl0423.22016
25. [K] K. Kodaira: On Kähler varieties of restricted type, Ann. of Math.60 (1954) 28-48. Zbl0057.14102MR68871
26. [Kd1] S.S. Kudla: Intersection numbers for quotients of the comples 2-ball and Hilbert modular forms, Invent. Math.47 (1978) 189-208. Zbl0399.10030MR501929
27. [Kd2] S.S. Kudla: Theta-functions and Hilbert modular forms, Nagoya Math. J.69 (1978) 97-106. Zbl0371.10021MR466025
28. [KM1] S.S. Kudla and J. Millson: Harmonic differentials and closed geodesics on a Riemann surface, Invent. Math.54 (1979) 193-211. Zbl0429.30038MR553218
29. [KM2] S.S. Kudla and J. Millson: The Poincaré dual of a geodesic algebraic curve in a quotient of the 2-ball, Canad. J. Math.33 (1979) 485-499. Zbl0506.32013MR617638
30. [KM3] S.S. Kudla and J. Millson: Geodesic cycles and the Wiel representation, I: quotients of hyperbolic space and Siegel modular forms, Compos. Math.45 (1982) 207-271. Zbl0495.10016MR651982
31. [KM4] S.S. Kudla and J. Millson: The theta correspondence and harmonic forms, I, Math. Ann.274 (1986) 352-378; II Math. Ann.277 (1987) 267-314. Zbl0618.10022MR886423
32. [KM5] S.S. Kudla and J. Millson: Tubes, cohomology with growth conditions and an application to the theta correspondence, Canad. J. Math.40 (1988) 1-37. Zbl0652.10021MR928212
33. [KM6] S.S. Kudla and J. Millson: Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables, preprint. Zbl0722.11026MR1079646
34. [Kg1] M. Kuga: Fibre Varieties over a Symmetric Space whose Fibres are Abelian Varieties, vols. I and II, Univ. of Chicago, Chicago, 1964 
35. [Kg2] M. Kuga: Fibre varieties over symmetric space whose fibres are abelian varieties, Proc. of the U.S.-Japan Seminar in Differential Geometry, Kyoto, Japan, 1965, Nippon Hyoronsha, 1966, 72-81. Zbl0173.48901MR217224
36. [Kg3] M. Kuga: Algebraic cycles in gtfabv, J. Fac. Sci Univ. Tokyo, Sec. IA, Math. 29 (1982) 13-29. Zbl0491.14003MR657869
37. [KS] M. Kuga and G. Shimura: On the zeta function of a fiber variety whose fibres are abelian varieties, Ann. of Math.82 (1965) 487-539. Zbl0166.16801MR184942
38. [L] A. Lascoux: Syzygies des varietes determinantales, Advances in Math.30 (1978) 202-237. Zbl0394.14022MR520233
39. [LV] G. Lion and M. Vergne: The Weil representation, Maslov index and theta series, Progress in Math.6, Birkhauser, Boston, BaselStuttgart, 1980. Zbl0444.22005MR573448
40. [McD] I.G. MacDonald:Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1979. Zbl0487.20007MR553598
41. [MS] Y. Matsushima and G. Shimura: On the cohomology groups attached to certain vector valued differential forms on the product of the upper half planes, Ann. of Math. (2) 78 (1963) 417-449. Zbl0141.38704MR155340
42. [Mi1] J. Millson: Higher weight cycles, J. Fac. Sci. Univ. Tokyo, sec. IA, Math.28 (1981) 465-479. Zbl0509.22011MR656031
43. [Mi2] J. Millson: Cycles and harmonic forms on locally symmetric spaces, Canad. Math. Bull.28 (1985) 3-38. Zbl0575.10022MR778258
44. [Mi3] J. Millson: Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables, Theta Functions, Bowdoin, 1987, Proc. of Symp. in Pure Math. 49, part 2, Amer. Math. Soc., Providence, 1989 pp. 129-142. Zbl0681.32024MR1013170
45. [MT] G.D. Mostow and T. Tamagawa: On the compactness of arithmetically defined homogeneous spaces, Ann. of Math.76 (1962) 440-463. Zbl0196.53201MR141672
46. [Mu1] D. Mumford: Families of abelian varieties, Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math. 9, Amer. Math. Soc., Providence, 1966, pp. 347-351. Zbl0199.24601MR206003
47. [Mu2] D. Mumford: A note on Shimura's paper "Discontinuous groups and abelian varieties", Math. Ann.181 (1969) 345-351. Zbl0169.23301MR248146
48. [O] T. Oda: On modular forms associated with indefinite quadratic forms of signature (2, n - 2), Math. Ann.231 (1977) 97-144. Zbl0346.10013MR466026
49. [PS] I.I. Piatetski-Shapiro: Automorphic Forms and the Geometry of Classical Domains, Gordon and Breach, New York, 1969. 
50. [Sa1] I. Satake: Holomorphic embeddings of symmetric domains into a Siegel space, Amer. J. Math.87 (1965) 425-461. Zbl0144.08202MR196134
51. [Sa2] I. Satake: Clifford algebras and families of abelian varieties, Nagoya Math. J.27 (1966) 435-446, correctionsibid.31 (1968) 295-296. Zbl0154.20603MR210716
52. [Sa3] I. Satake: Symplectic representations of algebraic groups satisfying a certain analyticity condition, Acta Math.117 (1967) 215-299. Zbl0149.39004MR215859
53. [Sa4] I. Satake: Algebraic Structures of Symmetric Domains, Princeton Univ. Press, Princeton and Iwanami Shoten, Japan, 1980. Zbl0483.32017MR591460
54. [Shm1] G. Shimura: On analytic families of polarized abelian varieties and automorphic functions, Ann. of Math.78 (1963) 149-192. Zbl0142.05402MR156001
55. [Shm2] G. Shimura: On canonical models of arithmetic quotients of bounded symmetric domains, I, Ann. of Math. (2) 91 (1970) 144-222; II, ibid92 (1970) 528-549. Zbl0237.14010MR257031
56. [Shn] T. Shintani: On construction of homomorphic cusp forms of half-integral weight, Nagoya Math. J.58 (1975) 83-126. Zbl0316.10016MR389772
57. [Sp] T.A. Springer: Invariant Theory, Lecture Notes in Math.585Springer-Verlag, Berlin, Heidelberg, New York, 1977. Zbl0346.20020
58. [TT] D. Toledo and Y.-L.L. Tong: Duality and intersection theory in complex manifolds, I, Math. Ann.237 (1978) 41-77. Zbl0391.32008MR506654
59. [T1] Y.-L.L. Tong: Weighted intersection numbers on Hilbert modular surfaces, Comp. Math.38 (1979) 299-310. Zbl0409.10017MR535073
60. [T2] Y.-L.L. Tong: Some nonzero harmonic forms and their geometric duals, Theta Functions, Bowdoin, 1987. Proc. of Symp. in Pure Math. 49, part 2, Amer. Math. Soc., Providence, 1989, 151-164. Zbl0694.10030MR1013172
61. [TW1] Y.-L.L. Tong and S.P. Wang: Harmonic forms dual to geodesic cycles in quotients of SU(p, 1), Math. Ann258 (1982) 289-318. Zbl0466.58004MR649201
62. [TW2] Y.-L.L. Tong and S.P. Wang: Theta functions defined by geodesic cycles in quotients of SU( p, 1), Invent. Math.71 (1983) 467-499. Zbl0506.10024MR695901
63. [TW3] Y.-L.L. Tong and S.P. Wang: Correspondence of Hermitian modular forms to cycles associated to SU(p, 2), J. Diff Geom.18 (1983) 163-207. Zbl0559.10027MR697988
64. [TW4] Y.-L.L. Tong and S.P. Wang: Period integrals in noncompact quotients of SU(p, 1), Duke Math. J.52 (1985) 649-688. Zbl0582.10018MR808097
65. [TW5] Y.-L.L. Tong and S.P. Wang: Some nonzero cohomology of discrete groups, J. reine angew. Math.382 (1987) 85-144. Zbl0621.22012MR921168
66. [TW6] Y.-L.L. Tong and S.P. Wang: Construction of cohomology of discrete groups, Trans. Amer. Math. Soc.306 (1988) 735-763. Zbl0659.32026MR933315
67. [TW7] Y.-L.L. Tong and S.P. Wang: Geometric realization of discrete series for semisimple symmetric spaces, Invent. Math.96 (1989) 425-458. Zbl0685.22008MR989703
68. [Va] V.S. Varadarajan: Lie Groups, Lie Algebras, and their Representations, Prentice-Hall, Englewood Cliffs, NJ, 1974. Zbl0371.22001MR376938
69. [Vi] M.-F. Vigneras: Arithmetique des Algebres de Quaternions, Lecture Notes in Math.800Springer-Verlag, Berlin, Heidelberg, New York, 1980. Zbl0422.12008MR580949
70. [V1] N.J. Vilenkin: Special Functions and the Theory of Group Representations, Transl. of Math. Monographs22Amer. Math. Soc., Providence, 1968. Zbl0172.18404MR229863
71. [Wa] S.P. Wang: Correspondences of modular forms to cycles associated to O(p, q), J. Diff Geom.22 (1985) 151-213. Zbl0594.10021MR834276
72. [We] A. Weil: Sur certaines groupes d'operateurs unitaire, Acta Math.111 (1964) 143-211. Zbl0203.03305MR165033
73. [Wey] H. Weyl: The Classical Groups, Princeton University Press, Princeton, 1946. Zbl1024.20501MR1488158JFM65.0058.02
74. [Wh] G.W. Whitehead: Elements of Homotopy Theory, Springer-Verlag, New York, 1978. Zbl0406.55001MR516508
75. [Z1] D. Zagier: Modular forms associated to real quadratic fields, Invent. Math.30 (1975) 1-46. Zbl0308.10014MR382174
76. [Z2] D. Zagier: Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields, Modular Functions of One Variable, VI, Bonn, Lecture Notes in Math.627Springer-Verlag, Berlin, Heidelberg, New York, 1976. Zbl0372.10017MR485703
77. [Z3] D. Zagier: The Eichler-Selberg trace formula on SL 2(Z) appendix to Introduction to Modular Forms by S. Lang, Grundlehren der math. Wiss.222Springer-Verlag, Berlin, Heidelberg, New York, 1976. Zbl0344.10011MR480354

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