Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-05T21:37:59.824Z Has data issue: false hasContentIssue false
  • English
  • Français

Formal Fourier Jacobi expansions and special cycles of codimension two

Part of: Computational number theory Arithmetic algebraic geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  06 August 2015

Martin Westerholt-Raum
Martin Westerholt-Raum*
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, D-53111, Bonn, Germany email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that formal Fourier Jacobi expansions of degree two are Siegel modular forms. As a corollary, we deduce modularity of the generating function of special cycles of codimension two, which were defined by Kudla. A second application is the proof of termination of an algorithm to compute Fourier expansions of arbitrary Siegel modular forms of degree two. Combining both results enables us to determine relations of special cycles in the second Chow group.

Keywords

special cycles of codimension twoformal Fourier Jacobi expansionscomputing Siegel modular forms

MSC classification

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties
Secondary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11F30: Fourier coefficients of automorphic forms 11Y40: Algebraic number theory computations
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 12 , December 2015 , pp. 2187 - 2211
Copyright
© The Author 2015 

References

Andrianov, A., Modular descent and the Saito–Kurokawa conjecture, Invent. Math. 53 (1979), 267–280.CrossRefGoogle Scholar
Aoki, H., Estimating Siegel modular forms of genus 2 using Jacobi forms, J. Math. Kyoto Univ. 40 (2000), 581–588.Google Scholar
Bergström, J., Faber, C. and van der Geer, G., Siegel modular forms of genus 2 and level 2: cohomological computations and conjectures, Int. Math. Res. Not. IMRN 2008 (2008), doi: 10.1093/imrn/rnn100.CrossRefGoogle Scholar
Borcherds, R., The Gross–Kohnen–Zagier theorem in higher dimensions, Duke Math. J. 97 (1999), 219–233.CrossRefGoogle Scholar
Borcherds, R., Correction to: ‘The Gross–Kohnen–Zagier theorem in higher dimensions’ [Duke Math. J. 97 (1999), 219–233; MR 1682249 (2000f:11052)], Duke Math. J. 105 (2000), 183–184.Google Scholar
Bruinier, J., Two applications of the curve lemma for orthogonal groups, Math. Nachr. 274–275 (2004), 19–31.CrossRefGoogle Scholar
Bruinier, J., On the converse theorem for Borcherds products, J. Algebra 397 (2014), 315–342, doi: 10.1016/j.jalgebra.2013.08.034.CrossRefGoogle Scholar
Bruinier, J. H., van der Geer, G., Harder, G. and Zagier, D. B., The 1-2-3 of modular forms, in Lectures from the summer school on modular forms and their applications held in Nordfjordeid, June 2004, Universitext, ed. Ranestad, K. (Springer, Berlin, 2008).Google Scholar
Chow, W.-L., On equivalence classes of cycles in an algebraic variety, Ann. of Math. (2) 64 (1956), 450–479.CrossRefGoogle Scholar
Eichler, M. and Zagier, D., The theory of Jacobi forms (Birkhäuser, Boston, MA, 1985).CrossRefGoogle Scholar
Faber, C., Chow rings of moduli spaces of curves. II. Some results on the Chow ring of M4, Ann. of Math. (2) 132 (1990), 421–449.CrossRefGoogle Scholar
Faber, C. and van der Geer, G., Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes. I, C. R. Math. Acad. Sci. Paris 338 (2004), 381–384.CrossRefGoogle Scholar
Faber, C. and van der Geer, G., Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes. II, C. R. Math. Acad. Sci. Paris 338 (2004), 467–470.CrossRefGoogle Scholar
Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero. I, Ann. of Math. (2) 79 (1964), 109–203.CrossRefGoogle Scholar
Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero. II, Ann. of Math. (2) 79 (1964), 205–326.CrossRefGoogle Scholar
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
Hulek, K., Nef divisors on moduli spaces of abelian varieties, in Complex analysis and algebraic geometry (de Gruyter, Berlin, 2000), 255–274.Google Scholar
Ibukiyama, T. and Kyomura, R., A generalization of vector valued Jacobi forms, Osaka J. Math. 48 (2011), 783–808.Google Scholar
Ibukiyama, T., Poor, C. and Yuen, D., Jacobi forms that characterize paramodular forms, Abh. Math. Semin. Univ. Hambg. 83 (2013), 111–128, doi: 10.1007/s12188-013-0078-y.CrossRefGoogle Scholar
Igusa, J.-I., On the graded ring of theta-constants, Amer. J. Math. 86 (1964), 219–246.CrossRefGoogle Scholar
Kawamata, Y., A generalization of Kodaira–Ramanujam’s vanishing theorem, Math. Ann. 261 (1982), 43–46.CrossRefGoogle Scholar
Kudla, S., Algebraic cycles on Shimura varieties of orthogonal type, Duke Math. J. 86 (1997), 39–78.CrossRefGoogle Scholar
Kudla, S. and Millson, J., The theta correspondence and harmonic forms. I, Math. Ann. 274 (1986), 353–378.CrossRefGoogle Scholar
Kudla, S. and Millson, J., The theta correspondence and harmonic forms. II, Math. Ann. 277 (1987), 267–314.CrossRefGoogle Scholar
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
Lecomte, F., Rigidité des groupes de Chow, Duke Math. J. 53 (1986), 405–426.CrossRefGoogle Scholar
Maass, H., Über eine Spezialschar von Modulformen zweiten Grades, Invent. Math. 52 (1979), 95–104.CrossRefGoogle Scholar
Maass, H., Über eine Spezialschar von Modulformen zweiten Grades. II, Invent. Math. 53 (1979), 249–253.CrossRefGoogle Scholar
Maass, H., Über eine Spezialschar von Modulformen zweiten Grades. III, Invent. Math. 53 (1979), 255–265.CrossRefGoogle Scholar
Manni, R. S., Modular forms of the fourth degree, in Classification of irregular varieties (Trento, 1990), Lecture Notes in Mathematics, vol. 1515 (Springer, Berlin, 1992), 106–111.CrossRefGoogle Scholar
Mason, G. and Marks, C., Structure of the module of vector-valued modular forms, J. Lond. Math. Soc. (2) 82 (2010), 32–48.Google Scholar
Mumford, D., Towards an enumerative geometry of the moduli space of curves, in Arithmetic and geometry, Vol. II, Progress in Mathematics, vol. 36 (Birkhäuser, Boston, MA, 1983), 271–328.CrossRefGoogle Scholar
Nishino, T., Function theory in several complex variables, Translations of Mathematical Monographs, vol. 193 (American Mathematical Society, Providence, RI, 2001); translated from the 1996 Japanese original by Norman Levenberg and Hiroshi Yamaguchi.Google Scholar
Poor, C., Formal series of Jacobi forms, Talk at Explicit theory of automorphic forms, applications and computations, CIRM, May 2011.Google Scholar
Raum, M., Computing genus 1 Jacobi forms, Math. Comp., to appear, arXiv:1212.1834.Google Scholar
Schwermer, J., Geometric cycles, arithmetic groups and their cohomology, Bull. Amer. Math. Soc. (N.S.) 47 (2010), 187–279.CrossRefGoogle Scholar
Skoruppa, N.-P., Jacobi forms of critical weight and Weil representations, in Modular forms on Schiermonnikoog (Cambridge University Press, Cambridge, 2008), 239–266.Google Scholar
Tsushima, R., On the spaces of Siegel cusp forms of degree two, Amer. J. Math. 104 (1982), 843–885.CrossRefGoogle Scholar
Tsushima, R., An explicit dimension formula for the spaces of generalized automorphic forms with respect to Sp(2, Z), Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), 139–142.CrossRefGoogle Scholar
Viehweg, E., Vanishing theorems, J. reine angew. Math. 335 (1982), 1–8.Google Scholar
Zagier, D., Sur la conjecture de Saito–Kurokawa (d’après H. Maass), in Seminar on number theory, Paris 1979–80, Progress in Mathematics, vol. 12 (Birkhäuser, Boston, MA, 1981), 371–394.Google Scholar
Zhang, S., Heights of Heegner cycles and derivatives of L-series, Invent. Math. 130 (1997), 99–152.CrossRefGoogle Scholar
Zhang, W., Modularity of generating functions of special cycles on Shimura varieties, PhD thesis, Columbia University, New York (2009).Google Scholar