Intersections of higher weight cycles and modular forms
Gordon, B. Brent
Compositio Mathematica, Tome 89 (1993), p. 1-44 / Harvested from Numdam
@article{CM_1993__89_1_1_0,
     author = {Gordon, B. Brent},
     title = {Intersections of higher weight cycles and modular forms},
     journal = {Compositio Mathematica},
     volume = {89},
     year = {1993},
     pages = {1-44},
     mrnumber = {1248890},
     zbl = {0820.11030},
     language = {en},
     url = {http://dml.mathdoc.fr/item/CM_1993__89_1_1_0}
}
Gordon, B. Brent. Intersections of higher weight cycles and modular forms. Compositio Mathematica, Tome 89 (1993) pp. 1-44. http://gdmltest.u-ga.fr/item/CM_1993__89_1_1_0/

[A] E. Artin: Geometric Algebra, Wiley Interscience, New York, 1957. | MR 82463 | Zbl 0077.02101

[Bo1] A. Borel: Density properties for certain subgroups of semisimple groups without compact components, Ann. of Math. 72, (1960), 179-188. | MR 123639 | Zbl 0094.24901

[Bo2] A. Borel: Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963) 111-132. | MR 146301 | Zbl 0116.38603

[BHC] A. Borel and Harish-Chandra: Arithmetic subgroups of algebraic groups Ann. of Math. 75 (1962) 485-535 | MR 147566 | Zbl 0107.14804

[Car] H. Cartan: Formes modulaire expose 4, Functions Automorphes Sem. Henri Cartan, 10e annee 1957/58, Ecole Normal Superieure, Paris, 1958 | Numdam

[Cas] J.W.S. Cassels: Rational Quadratic Forms, Academic Press, London, New York, San Francisco, 1978. | MR 522835 | Zbl 0395.10029

[Ch] C.C. Chevalley: The Algebraic Theory of Spinors, Columbia Univ. Press, Morningside Heights, NY, 1954. | MR 60497 | Zbl 0057.25901

[Cg1] J.W. Cogdell: Arithmetic cycles on Picard modular surfaces and modular forms of Nebentypus, J. reine angew. Math. 357 (1985) 115-137. | MR 783537 | Zbl 0547.10028

[Cg2] J. Cogdell: The Weil representation and cycles on Picard modular surfaces, preprint

[D1] P. Deligne: Travaux de Shimura, Seminaire Bourbaki, vol. 1970/71. Exposes 382-399, Lect. Notes Math. 244, Springer-Verlag Berlin, Heidelberg, New York, 1971, pp. 123-165. | Numdam | MR 498581 | Zbl 0225.14007

[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. | MR 546620 | Zbl 0437.14012

[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. | MR 546622 | Zbl 0449.10022

[E] M. Eichler: Quadratische Formen und Orthogonale Gruppen, Springer-Verlag, Berlin, Heidelberg, New York, 1974. | MR 351996 | Zbl 0277.10017

[EZ] M. Eichler and D. Zagier: The Theory of Jacobi Forms, Progress in Math. 55, Birkhauser, Boston, Basel, Stuttgart 1985. | MR 781735 | Zbl 0554.10018

[F] E. Freitag: Siegelsche Modulfunktionen, Springer-Verlag, Berlin, Heidelberg, New York, 1983. | MR 871067 | Zbl 0498.10016

[Go1] B.B. Gordon: Intersections of higher weight cycles over quaternionic modular surfaces and modular forms of Nebentypus, Bull. AMS. 14 (1986) 293-298. | MR 828829 | Zbl 0593.14003

[Go2] B.B. Gordon: Algebraically defined subspaces in the cohomology of a Kuga fiber variety, Pacifc J. Math. 131 (1988) 261-276. | MR 922218 | Zbl 0615.14026

[GH] P. Griffiths and J. Harris: Principles of Algebraic Geometry, John Wiley and Sons, New York, 1978. | MR 507725 | Zbl 0408.14001

[Gr] A. Grothendieck: Standard conjectures on algebraic cycles, Algebraic Geometry: Papers Presented at the Bombay Colloquium, 1969, Oxford Univ. Press, Oxford, 1969. | MR 268189 | Zbl 0201.23301

[HLR] G. Harder, R. Langlands and M. Rapoport: Algebraische Zyklen auf Hilbert Blumenthal-Flächen, J. reine u. angew. Math. 366 (1986) 53-120. | MR 833013 | Zbl 0575.14004

[He] E. Hecke: Analytische Arithmetik der positiven quadratischen Formen, Mathematische Werke, Vanderhoeck und Ruprecht, Göttingen, 1970, pp. 789-918. | MR 104550

[Hi] F. Hirzebruch: Topological Methods in Algebraic Geometry, Grundlehren der mathematischen Wissenschaften 131, Springer-Verlag, Berlin, Heidelberg, New York, 1978. | MR 1335917 | Zbl 0376.14001

[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. | MR 453649 | Zbl 0332.14009

[Ho] R. Howe: θ-series and invariant theory, Automorphic Forms, Representations, and L-functions, Proc. Symp. Pure Math 33, part 2, Amer. Math. Sac., Providence 1979 pp. 275-285. | Zbl 0423.22016

[K] K. Kodaira: On Kähler varieties of restricted type, Ann. of Math. 60 (1954) 28-48. | MR 68871 | Zbl 0057.14102

[Kd1] S.S. Kudla: Intersection numbers for quotients of the comples 2-ball and Hilbert modular forms, Invent. Math. 47 (1978) 189-208. | MR 501929 | Zbl 0399.10030

[Kd2] S.S. Kudla: Theta-functions and Hilbert modular forms, Nagoya Math. J. 69 (1978) 97-106. | MR 466025 | Zbl 0371.10021

[KM1] S.S. Kudla and J. Millson: Harmonic differentials and closed geodesics on a Riemann surface, Invent. Math. 54 (1979) 193-211. | MR 553218 | Zbl 0429.30038

[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. | MR 617638 | Zbl 0506.32013

[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. | Numdam | MR 651982 | Zbl 0495.10016

[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. | MR 886423 | Zbl 0618.10022

[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. | MR 928212 | Zbl 0652.10021

[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. | Numdam | MR 1079646 | Zbl 0722.11026

[Kg1] M. Kuga: Fibre Varieties over a Symmetric Space whose Fibres are Abelian Varieties, vols. I and II, Univ. of Chicago, Chicago, 1964

[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. | MR 217224 | Zbl 0173.48901

[Kg3] M. Kuga: Algebraic cycles in gtfabv, J. Fac. Sci Univ. Tokyo, Sec. IA, Math. 29 (1982) 13-29. | MR 657869 | Zbl 0491.14003

[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. | MR 184942 | Zbl 0166.16801

[L] A. Lascoux: Syzygies des varietes determinantales, Advances in Math. 30 (1978) 202-237. | MR 520233 | Zbl 0394.14022

[LV] G. Lion and M. Vergne: The Weil representation, Maslov index and theta series, Progress in Math. 6, Birkhauser, Boston, Basel Stuttgart, 1980. | MR 573448 | Zbl 0444.22005

[McD] I.G. Macdonald:Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1979. | MR 553598 | Zbl 0487.20007

[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. | MR 155340 | Zbl 0141.38704

[Mi1] J. Millson: Higher weight cycles, J. Fac. Sci. Univ. Tokyo, sec. IA, Math. 28 (1981) 465-479. | MR 656031 | Zbl 0509.22011

[Mi2] J. Millson: Cycles and harmonic forms on locally symmetric spaces, Canad. Math. Bull. 28 (1985) 3-38. | MR 778258 | Zbl 0575.10022

[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. | MR 1013170 | Zbl 0681.32024

[MT] G.D. Mostow and T. Tamagawa: On the compactness of arithmetically defined homogeneous spaces, Ann. of Math. 76 (1962) 440-463. | MR 141672 | Zbl 0196.53201

[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. | MR 206003 | Zbl 0199.24601

[Mu2] D. Mumford: A note on Shimura's paper "Discontinuous groups and abelian varieties", Math. Ann. 181 (1969) 345-351. | MR 248146 | Zbl 0169.23301

[O] T. Oda: On modular forms associated with indefinite quadratic forms of signature (2, n - 2), Math. Ann. 231 (1977) 97-144. | MR 466026 | Zbl 0346.10013

[PS] I.I. Piatetski-Shapiro: Automorphic Forms and the Geometry of Classical Domains, Gordon and Breach, New York, 1969. | Zbl 0196.09901

[Sa1] I. Satake: Holomorphic embeddings of symmetric domains into a Siegel space, Amer. J. Math. 87 (1965) 425-461. | MR 196134 | Zbl 0144.08202

[Sa2] I. Satake: Clifford algebras and families of abelian varieties, Nagoya Math. J. 27 (1966) 435-446, corrections ibid. 31 (1968) 295-296. | MR 210716 | Zbl 0154.20603

[Sa3] I. Satake: Symplectic representations of algebraic groups satisfying a certain analyticity condition, Acta Math. 117 (1967) 215-299. | MR 215859 | Zbl 0149.39004

[Sa4] I. Satake: Algebraic Structures of Symmetric Domains, Princeton Univ. Press, Princeton and Iwanami Shoten, Japan, 1980. | MR 591460 | Zbl 0483.32017

[Shm1] G. Shimura: On analytic families of polarized abelian varieties and automorphic functions, Ann. of Math. 78 (1963) 149-192. | MR 156001 | Zbl 0142.05402

[Shm2] G. Shimura: On canonical models of arithmetic quotients of bounded symmetric domains, I, Ann. of Math. (2) 91 (1970) 144-222; II, ibid 92 (1970) 528-549. | MR 257031 | Zbl 0237.14010

[Shn] T. Shintani: On construction of homomorphic cusp forms of half-integral weight, Nagoya Math. J. 58 (1975) 83-126. | MR 389772 | Zbl 0316.10016

[Sp] T.A. Springer: Invariant Theory, Lecture Notes in Math. 585 Springer-Verlag, Berlin, Heidelberg, New York, 1977. | Zbl 0346.20020

[TT] D. Toledo and Y.-L.L. Tong: Duality and intersection theory in complex manifolds, I, Math. Ann. 237 (1978) 41-77. | MR 506654 | Zbl 0391.32008

[T1] Y.-L.L. Tong: Weighted intersection numbers on Hilbert modular surfaces, Comp. Math. 38 (1979) 299-310. | Numdam | MR 535073 | Zbl 0409.10017

[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. | MR 1013172 | Zbl 0694.10030

[TW1] Y.-L.L. Tong and S.P. Wang: Harmonic forms dual to geodesic cycles in quotients of SU(p, 1), Math. Ann 258 (1982) 289-318. | MR 649201 | Zbl 0466.58004

[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. | MR 695901 | Zbl 0506.10024

[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. | MR 697988 | Zbl 0559.10027

[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. | MR 808097 | Zbl 0582.10018

[TW5] Y.-L.L. Tong and S.P. Wang: Some nonzero cohomology of discrete groups, J. reine angew. Math. 382 (1987) 85-144. | MR 921168 | Zbl 0621.22012

[TW6] Y.-L.L. Tong and S.P. Wang: Construction of cohomology of discrete groups, Trans. Amer. Math. Soc. 306 (1988) 735-763. | MR 933315 | Zbl 0659.32026

[TW7] Y.-L.L. Tong and S.P. Wang: Geometric realization of discrete series for semisimple symmetric spaces, Invent. Math. 96 (1989) 425-458. | MR 989703 | Zbl 0685.22008

[Va] V.S. Varadarajan: Lie Groups, Lie Algebras, and their Representations, Prentice-Hall, Englewood Cliffs, NJ, 1974. | MR 376938 | Zbl 0371.22001

[Vi] M.-F. Vigneras: Arithmetique des Algebres de Quaternions, Lecture Notes in Math. 800 Springer-Verlag, Berlin, Heidelberg, New York, 1980. | MR 580949 | Zbl 0422.12008

[V1] N.J. Vilenkin: Special Functions and the Theory of Group Representations, Transl. of Math. Monographs 22 Amer. Math. Soc., Providence, 1968. | MR 229863 | Zbl 0172.18404

[Wa] S.P. Wang: Correspondences of modular forms to cycles associated to O(p, q), J. Diff Geom. 22 (1985) 151-213. | MR 834276 | Zbl 0594.10021

[We] A. Weil: Sur certaines groupes d'operateurs unitaire, Acta Math. 111 (1964) 143-211. | MR 165033 | Zbl 0203.03305

[Wey] H. Weyl: The Classical Groups, Princeton University Press, Princeton, 1946. | JFM 65.0058.02 | MR 1488158 | Zbl 1024.20502 | Zbl 1024.20501

[Wh] G.W. Whitehead: Elements of Homotopy Theory, Springer-Verlag, New York, 1978. | MR 516508 | Zbl 0406.55001

[Z1] D. Zagier: Modular forms associated to real quadratic fields, Invent. Math. 30 (1975) 1-46. | MR 382174 | Zbl 0308.10014

[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. 627 Springer-Verlag, Berlin, Heidelberg, New York, 1976. | MR 485703 | Zbl 0372.10017

[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. 222 Springer-Verlag, Berlin, Heidelberg, New York, 1976. | MR 480354 | Zbl 0344.10011