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Spans of special cycles of codimension less than 5

  • Martin Raum EMAIL logo
Veröffentlicht/Copyright: 11. Juni 2014

Abstract

We show that the span of special cycles in the r-th Chow group of a Shimura variety of orthogonal type is finite dimensional, if r<5. As our main tool, we develop the theory of Jacobi forms with rational index MMatN().

Funding statement: The author is supported by the ETH Zurich Postdoctoral Fellowship Program and by the Marie Curie Actions for People COFUND Program.

Acknowledgements

The author thanks the referee for helpful comments.

References

[1] A. Andrianov, Modular descent and the Saito–Kurokawa conjecture, Invent. Math. 53 (1979), no. 3, 267–280. 10.1007/BF01389767Suche in Google Scholar

[2] R. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491–562. 10.1007/s002220050232Suche in Google Scholar

[3] R. Borcherds, The Gross–Kohnen–Zagier theorem in higher dimensions, Duke Math. J. 97 (1999), no. 2, 219–233. 10.1215/S0012-7094-99-09710-7Suche in Google Scholar

[4] R. Borcherds, Correction to “The Gross–Kohnen–Zagier theorem in higher dimensions” [Duke Math. J. 97 (1999), no. 2, 219–233], Duke Math. J. 105 (2000), no. 1, 183–184.. 10.1215/S0012-7094-00-10519-4Suche in Google Scholar

[5] M. Cheng and J. Duncan, The largest Mathieu group and (mock) automorphic forms, String-Math 2011, Proc. Sympos. Pure Math. 85, American Mathematical Society, Providence (2012), 53–82. 10.1090/pspum/085/1374Suche in Google Scholar

[6] W.-L. Chow, On equivalence classes of cycles in an algebraic variety, Ann. of Math. (2) 64 (1956), 450–479. 10.1142/9789812776921_0022Suche in Google Scholar

[7] A. Dabholkar and D. Gaiotto, Spectrum of CHL dyons from genus-two partition function, J. High Energy Phys. 2007 (2007), no. 12, Paper No. 87. 10.1088/1126-6708/2007/12/087Suche in Google Scholar

[8] A. Dabholkar, S. Murthy and D. Zagier, Quantum black holes, wall crossing, and mock modular forms, preprint (2012), https://arxiv.org/abs/1208.4074. Suche in Google Scholar

[9] E. Hecke, Neue Herleitung der Klassenzahlrelationen von Hurwitz und Kronecker, Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl. 1926 (1926), 244–249. Suche in Google Scholar

[10] F. Hirzebruch and D. B. Zagier, Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, Invent. Math. 36 (1976), 57–113. 10.1007/978-3-642-61711-9_23Suche in Google Scholar

[11] S. Kudla, Algebraic cycles on Shimura varieties of orthogonal type, Duke Math. J. 86 (1997), no. 1, 39–78. 10.1215/S0012-7094-97-08602-6Suche in Google Scholar

[12] H. Maass, Über eine Spezialschar von Modulformen zweiten Grades, Invent. Math. 52 (1979), no. 1, 95–104. 10.1007/BF01389857Suche in Google Scholar

[13] H. Maass, Über eine Spezialschar von Modulformen zweiten Grades. II, Invent. Math. 53 (1979), no. 3, 249–253. 10.1007/BF01389765Suche in Google Scholar

[14] H. Maass, Über eine Spezialschar von Modulformen zweiten Grades. III, Invent. Math. 53 (1979), no. 3, 255–265. 10.1007/BF01389766Suche in Google Scholar

[15] D. Mumford, Tata lectures on theta. III, Progr. Math. 97, Birkhäuser-Verlag, Boston 1991. 10.1007/978-0-8176-4579-3Suche in Google Scholar

[16] V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238. Suche in Google Scholar

[17] R. Sekiguchi, On projective normality of Abelian varieties. II, J. Math. Soc. Japan 29 (1977), no. 4, 709–727. 10.2969/jmsj/02940709Suche in Google Scholar

[18] G. Shimura, On certain reciprocity-laws for theta functions and modular forms, Acta Math. 141 (1978), no. 1–2, 35–71. 10.1007/978-1-4612-2060-2_1Suche in Google Scholar

[19] N. Skoruppa, Jacobi forms of critical weight and Weil representations, Modular forms on Schiermonnikoog (Schiermonnikoog 2006), Cambridge University Press, Cambridge (2008), 239–266. 10.1017/CBO9780511543371.013Suche in Google Scholar

[20] D. Zagier, Sur la conjecture de Saito-Kurokawa (d’après H. Maass), Théorie des nombres. Séminaire Delange-Pisot-Poitou (Paris 1979–80), Progr. Math. 12, Birkhäuser-Verlag, Boston (1981), 371–394. Suche in Google Scholar

[21] W. Zhang, Modularity of generating functions of special cycles on Shimura varieties, Ph.D. thesis, Columbia University, New York 2009. Suche in Google Scholar

[22] C. Ziegler, Jacobi forms of higher degree, Abh. Math. Semin. Univ. Hambg. 59 (1989), 191–224. 10.1007/BF02942329Suche in Google Scholar

Received: 2013-04-04
Revised: 2013-11-21
Published Online: 2014-06-11
Published in Print: 2016-09-01

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