Intersections of higher weight cycles and modular forms
Compositio Mathematica, Tome 89 (1993) no. 1, pp. 1-44. 
@article{CM_1993__89_1_1_0,
     author = {Gordon, B. Brent},
     title = {Intersections of higher weight cycles and modular forms},
     journal = {Compositio Mathematica},
     pages = {1--44},
     publisher = {Kluwer Academic Publishers},
     volume = {89},
     number = {1},
     year = {1993},
     mrnumber = {1248890},
     zbl = {0820.11030},
     language = {en},
     url = {http://www.numdam.org/item/CM_1993__89_1_1_0/}
}
TY  - JOUR
AU  - Gordon, B. Brent
TI  - Intersections of higher weight cycles and modular forms
JO  - Compositio Mathematica
PY  - 1993
SP  - 1
EP  - 44
VL  - 89
IS  - 1
PB  - Kluwer Academic Publishers
UR  - http://www.numdam.org/item/CM_1993__89_1_1_0/
LA  - en
ID  - CM_1993__89_1_1_0
ER  - 
%0 Journal Article
%A Gordon, B. Brent
%T Intersections of higher weight cycles and modular forms
%J Compositio Mathematica
%D 1993
%P 1-44
%V 89
%N 1
%I Kluwer Academic Publishers
%U http://www.numdam.org/item/CM_1993__89_1_1_0/
%G en
%F CM_1993__89_1_1_0
Gordon, B. Brent. Intersections of higher weight cycles and modular forms. Compositio Mathematica, Tome 89 (1993) no. 1, pp. 1-44. http://www.numdam.org/item/CM_1993__89_1_1_0/

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

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

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

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

[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 | Zbl

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

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

[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. | EuDML | Numdam | MR | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

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

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

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

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

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

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

[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. | EuDML | MR | Zbl

[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

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

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

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

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

[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 | Zbl

[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. | EuDML | Numdam | MR | Zbl

[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. | EuDML | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

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

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

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

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

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

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

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

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

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

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

[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 | Zbl

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

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

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

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

[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 | Zbl

[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. | EuDML | MR | Zbl

[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. | EuDML | MR | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

[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 | Zbl