Skip to main content Accessibility help
×
Home
Hostname: page-component-6f6fcd54b-tmjz7 Total loading time: 0.317 Render date: 2021-05-10T22:59:33.851Z Has data issue: true Feature Flags: {}

KUDLA’S MODULARITY CONJECTURE AND FORMAL FOURIER–JACOBI SERIES

Published online by Cambridge University Press:  01 October 2015

JAN HENDRIK BRUINIER
Affiliation:
Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstraße 7, D-64289 Darmstadt, Germany; bruinier@mathematik.tu-darmstadt.de
MARTIN WESTERHOLT-RAUM
Affiliation:
Chalmers tekniska högskola och Göteborgs Universitet, Institutionen för Matematiska vetenskaper, SE-412 96 Göteborg, Sweden; martin@raum-brothers.eu

Abstract

We prove modularity of formal series of Jacobi forms that satisfy a natural symmetry condition. They are formal analogs of Fourier–Jacobi expansions of Siegel modular forms. From our result and a theorem of Wei Zhang, we deduce Kudla’s conjecture on the modularity of generating series of special cycles of arbitrary codimension and for all orthogonal Shimura varieties.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

References

Andrianov, A. N., ‘Modular descent and the Saito–Kurokawa conjecture’, Invent. Math. 53(3) (1979), 267280.CrossRefGoogle Scholar
Aoki, H., ‘Estimating Siegel modular forms of genus 2 using Jacobi forms’, J. Math. Kyoto Univ. 40(3) (2000), 581588.CrossRefGoogle Scholar
Ash, A., Mumford, D., Rapoport, M. and Tai, Y.-S., ‘Smooth compactification of locally symmetric varieties’, in: Lie Groups: History, Frontiers and Applications, IV (Mathematical Science Press, Brookline, MA, 1975).Google Scholar
Blichfeldt, H. F., ‘The minimum value of quadratic forms, and the closest packing of spheres’, Math. Ann. 101(1) (1929), 605608.CrossRefGoogle Scholar
Borcherds, R. E., ‘The Gross–Kohnen–Zagier theorem in higher dimensions’, Duke Math. J. 97(2) (1999), 219233.CrossRefGoogle Scholar
Borel, A. and Wallach, N. R., Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Annals of Mathematics Studies, 94 (Princeton University Press and University of Tokyo Press, Princeton, NJ and Tokyo, 1980).Google Scholar
Braun, H., ‘Hermitian modular functions’, Ann. of Math. (2) 50(2) (1949), 827855.CrossRefGoogle Scholar
Bruinier, J. H., ‘Vector valued formal Fourier–Jacobi series’, Proc. Amer. Math. Soc. 143(2) (2015), 505512.CrossRefGoogle Scholar
Bruinier, J. H., van der Geer, G., Harder, G. and Zagier, D. B., The 1-2-3 of Modular Forms, Lectures from the Summer School on Modular Forms and their Applications held in Nordfjordeid, June 2004 (Universitext, Springer, Berlin, 2008).CrossRefGoogle Scholar
Eichler, M., ‘Über die Anzahl der linear unabhängigen Siegelschen Modulformen von gegebenem Gewicht’, Math. Ann. 213 (1975), 281291. erratum; ibid. 215 (1975), 195.CrossRefGoogle Scholar
Eichler, M. and Zagier, D. B., The Theory of Jacobi Forms, Progress in Mathematics, 55 (Birkhäuser Boston Inc., Boston, MA, 1985).CrossRefGoogle Scholar
Farkas, G., Grushevsky, S., Salvati Manni, R. and Verra, A., ‘Singularities of theta divisors and the geometry of A5’, J. Eur. Math. Soc. (JEMS) 16(9) (2014), 18171848.CrossRefGoogle Scholar
Freitag, E. and Kiehl, R., Étale Cohomology and the Weil Conjecture, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 13 (Springer, Berlin, 1988), Translated from the German by B. S. Waterhouse and W. C. Waterhouse, with an historical introduction by J. A. Dieudonné.CrossRefGoogle Scholar
Fritzsche, K. and Grauert, H., From Holomorphic Functions to Complex Manifolds, Graduate Texts in Mathematics, 213 (Springer, New York, 2002).CrossRefGoogle Scholar
Grauert, H. and Remmert, R., ‘Komplexe Räume’, Math. Ann. 136 (1958), 245318.CrossRefGoogle Scholar
Grushevsky, S., ‘Geometry of Ag and its compactifications’, in: Algebraic Geometry—Seattle 2005. Part 1, Proceedings of Symposia in Applied Mathematics, 80 (American Mathematical Society, Providence, RI, 2009), 193234.Google Scholar
Ibukiyama, T., Poor, C. and Yuen, D. S., ‘Jacobi forms that characterize paramodular forms’, Abh. Math. Semin. Univ. Hambg. 83(1) (2013), 111128.CrossRefGoogle Scholar
Kohnen, W., Krieg, A. and Sengupta, J., ‘Characteristic twists of a Dirichlet series for Siegel cusp forms’, Manuscripta Math. 87(4) (1995), 489499.CrossRefGoogle Scholar
Kohnen, W. and Skoruppa, N.-P., ‘A certain Dirichlet series attached to Siegel modular forms of degree two’, Invent. Math. 95(3) (1989), 541558.CrossRefGoogle Scholar
Krieg, A., Modular Forms on Half-Spaces of Quaternions, Lecture Notes in Mathematics, 1143 (Springer, Berlin, 1985).CrossRefGoogle Scholar
Kudla, S. S., ‘Algebraic cycles on Shimura varieties of orthogonal type’, Duke Math. J. 86(1) (1997), 3978.CrossRefGoogle Scholar
Kudla, S. S., ‘Special cycles and derivatives of Eisenstein series’, in: Heegner Points and Rankin L-Series, Mathematical Sciences Research Institute Publications, 49 (Cambridge University Press, Cambridge, 2004), 243270.CrossRefGoogle Scholar
Kudla, S. 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), 121172.CrossRefGoogle Scholar
Maass, H., ‘Über eine Spezialschar von Modulformen zweiten Grades’, Invent. Math. 52(1) (1979), 95104.CrossRefGoogle Scholar
Maass, H., ‘Über eine Spezialschar von Modulformen zweiten Grades II’, Invent. Math. 53(3) (1979), 249253.CrossRefGoogle Scholar
Maass, H., ‘Über eine Spezialschar von Modulformen zweiten Grades III’, Invent. Math. 53(3) (1979), 255265.CrossRefGoogle Scholar
Matsumura, H., Commutative Algebra, 2nd edn, Mathematics Lecture Note Series, 56 (Benjamin/Cummings Publishing Co., Reading, MA, 1980).Google Scholar
Namikawa, Y., Toroidal Compactification of Siegel Spaces, Lecture Notes in Mathematics, 812 (Springer, Berlin, 1980).CrossRefGoogle Scholar
Raum, M., ‘Formal Fourier Jacobi expansions and special cycles of codimension 2’, Compos. Math. (accepted), Preprint 2013, arXiv:1302.0880.Google Scholar
Runge, B., ‘Theta functions and Siegel–Jacobi forms’, Acta Math. 175(2) (1995), 165196.CrossRefGoogle Scholar
Salvati Manni, R., ‘Modular forms of the fourth degree’, in: Classification of Irregular Varieties (Trento, 1990), Lecture Notes in Mathematics, 1515 (Springer, Berlin, 1992), 106111.CrossRefGoogle Scholar
Shimura, G., ‘The arithmetic of automorphic forms with respect to a unitary group’, Ann. of Math. (2) 107(3) (1978), 569605.CrossRefGoogle Scholar
Siegel, C. L., ‘Die Modulgruppe in einer einfachen involutorischen Algebra’, in: Festschrift zur Feier des zweihundertjährigen Bestehens der Akademie der Wissenschaften in Göttingen, I. Math.-Phys. Kl. (Springer, Berlin, 1951), 157167.Google Scholar
Taïbi, O., ‘Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula’, Preprint, 2014, arXiv:14064247.Google Scholar
The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2014.Google Scholar
Wang, J., ‘Estimations on dimensions of spaces of Jacobi forms’, Sci. China A 42(2) (1999), 147153.CrossRefGoogle Scholar
Yuan, X., Zhang, S.-W. and Zhang, W., ‘The Gross–Kohnen–Zagier theorem over totally real fields’, Compos. Math. 145(5) (2009), 11471162.CrossRefGoogle Scholar
Zagier, D. B., ‘Sur la conjecture de Saito–Kurokawa (d’après H Maass)’, in: Seminar on Number Theory, Paris 1979–80, Progress of Mathematics, 12 (Birkhäuser, Boston, 1981), 371394.Google Scholar
Zhang, W., ‘Modularity of generating functions of special cycles on shimura varieties’, PhD Thesis, Columbia University, 2009.Google Scholar
Ziegler, C. D., ‘Jacobi forms of higher degree’, Abh. Math. Semin. Univ. Hambg. 59 (1989), 191224.CrossRefGoogle Scholar
You have Access
Open access

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

KUDLA’S MODULARITY CONJECTURE AND FORMAL FOURIER–JACOBI SERIES
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

KUDLA’S MODULARITY CONJECTURE AND FORMAL FOURIER–JACOBI SERIES
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

KUDLA’S MODULARITY CONJECTURE AND FORMAL FOURIER–JACOBI SERIES
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *