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The Gross–Kohnen–Zagier theorem over totally real fields

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  10 August 2009

Xinyi Yuan ,
Shou-Wu Zhang  and
Wei Zhang
Xinyi Yuan
Affiliation:
Department of Mathematics, Columbia University, New York, NY 10027, USA (email: yxy@math.columbia.edu)
Shou-Wu Zhang
Affiliation:
Department of Mathematics, Columbia University, New York, NY 10027, USA (email: szhang@math.columbia.edu)
Wei Zhang
Affiliation:
Department of Mathematics, Columbia University, New York, NY 10027, USA (email: weizhang@math.columbia.edu)
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Abstract

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On Shimura varieties of orthogonal type over totally real fields, we prove a product formula and the modularity of Kudla’s generating series of special cycles in Chow groups.

Keywords

Shimura varietiesKudla cyclesSiegel modular forms

MSC classification

Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties 14G35: Modular and Shimura varieties
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 5 , September 2009 , pp. 1147 - 1162
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Borcherds, R., The Gross–Kohnen–Zagier theorem in higher dimensions, Duke Math. J. 97 (1999), 219–233.CrossRefGoogle Scholar
[2]Gross, B., Kohnen, W. and Zagier, D., Heegner points and derivatives of L-series. II, Math. Ann. 278 (1987), 497–562.CrossRefGoogle Scholar
[3]Gross, B. and Kudla, S., Heights and the central critical values of triple product L-functions, Compositio Math. 81 (1992), 143–209.Google Scholar
[4]Gross, B. and Zagier, D., Heegner points and derivatives of L-series, Invent. Math. 84 (1986), 225–320.CrossRefGoogle Scholar
[5]Hirzebruch, F. and Zagier, D., Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, Invent. Math. 36 (1976), 57–113.CrossRefGoogle Scholar
[6]Kudla, S., Algebraic cycles on Shimura varieties of orthogonal type, Duke Math. J. 86 (1997), 39–78.CrossRefGoogle Scholar
[7]Kudla, S., Special cycles and derivatives of Eisenstein series, in Heegner points and Rankin L-series, Mathematical Sciences Research Institute Publications, vol. 49 (Cambridge University Press, Cambridge, 2004), 243–270.CrossRefGoogle Scholar
[8]Kudla, S. and Millson, J., Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables, Publ. Math. Inst. Hautes Études Sci. 71 (1990), 121–172.CrossRefGoogle Scholar
[9]Kudla, S., Rapoport, M. and Yang, T., Modular forms and special cycles on Shimura curves, Annals of Mathematics Studies, vol. 161 (Princeton University Press, Princeton, NJ, 2006).CrossRefGoogle Scholar
[10]Serre, J.-P., A course in arithmetic, Graduate Texts in Mathematics, vol. 7 (Springer, New York, 1973).CrossRefGoogle Scholar
[11]Vogan, D. and Zuckerman, G., Unitary representations with nonzero cohomology, Compositio Math. 53 (1984), 51–90.Google Scholar
[12]Waldspurger, J.-L., Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), 375–484.Google Scholar
[13]Zagier, D., Modular points, modular curves, modular surfaces and modular forms, in Proc. Workshop Bonn 1984 (Bonn, 1984), Lecture Notes in Mathematics, vol. 1111 (Springer, Berlin, 1985), 225–248.Google Scholar
[14]Zhang, S.-W., Gross–Zagier formula for GL2, Asian J. Math. 5 (2001), 183–290.CrossRefGoogle Scholar
[15]Zhang, W., Modularity of generating functions of special cycles on Shimura varieties, PhD thesis, Columbia University (2009).Google Scholar