Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T13:51:20.174Z Has data issue: false hasContentIssue false
  • English
  • Français

Some constructions of modular forms for the Weil representation of SL2(ℤ)

Published online by Cambridge University Press:  11 January 2016

Nils R. Scheithauer
Nils R. Scheithauer*
Affiliation:
Fachbereich Mathematik, Technische Universität Darmstadt, 64289 Darmstadt, [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.

Modular forms for the Weil representation of SL2 (ℤ) play an important role in the theory of automorphic forms on orthogonal groups. In this paper we give some explicit constructions of these functions. As an application, we construct new examples of generalized Kac-Moody algebras whose denominator identities are holomorphic automorphic products of singular weight. They correspond naturally to the Niemeier lattices with root systems and to the Leech lattice.

Keywords

Primary 11F27Secondary 11F2217B65
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 220 , December 2015 , pp. 1 - 43
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

References

[1] Barnard, A. G., The singular theta correspondence, Lorentzian lattices and Borcherds-Kac-Moody algebras, Ph.D. dissertation, University of California, Berkeley, Berkeley, Calif., 2003. MR 2705173.Google Scholar
[2] Borcherds, R. E., Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405–444. MR 1172696. DOI 10.1007/BF01232032.CrossRefGoogle Scholar
[3] Borcherds, R. E., Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), 491–562. MR 1625724. DOI 10.1007/s002220050232.CrossRefGoogle Scholar
[4] Bosma, W., Cannon, J., and Playoust, C., “The Magma algebra system, I: The user language” in Computational Algebra and Number Theory (London, 1993), J. Symbolic Comput. 24, 1997, 235–265. MR 1484478. DOI 10.1006/jsco.1996.0125.Google Scholar
[5] Bruinier, J. H., Borcherds Products on O(2,l) and Chern Classes of Heegner Divisors, Lecture Notes in Math. 1780, Springer, Berlin, 2002. MR 1903920. DOI 10.1007/b83278.Google Scholar
[6] Conway, J. H. and Sloane, N. J. A., Sphere Packings, Lattices and Groups, 3rd ed., Grundlehren Math. Wiss. 290, Springer, New York, 1999. MR 1662447.Google Scholar
[7] Diamond, F. and Shurman, J., A First Course in Modular Forms, Graduate Texts in Math. 228, Springer, New York, 2005. MR 2112196.Google Scholar
[8] Harada, K. and Lang, M.-L., On some sublattices of the Leech lattice, Hokkaido Math. J. 19 (1990), 435–446. MR 1078499. DOI 10.14492/hokmj/1381517491.Google Scholar
[9] Nikulin, V. V., Integer symmetric bilinear forms and some of their geometric applications (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 111–177; English translation in Math. USSR Izv. 14 (1979), 103–167. MR 0525944.Google Scholar
[10] O'Meara, O. T., Introduction to Quadratic Forms, reprint of the 1973 edition, Classics Math., Springer, Berlin, 2000. MR 1754311. Google Scholar
[11] The PARI Group, Bordeaux, PARI/GP, version 2.3.4, 2008, http://pari.math.u-bordeaux.fr/ (accessed September 16, 2015).Google Scholar
[12] Scheithauer, N. R., On the classification of automorphic products and generalized Kac-Moody algebras, Invent. Math. 164 (2006), 641–678. MR 2221135. DOI 10.1007/s00222-006-0500-5.Google Scholar
[13] Scheithauer, N. R., The Weil representation of SL2(ℤ) and some applications, Int. Math. Res. Not. IMRN 2009, no. 8, 1488–1545. MR 2496771.CrossRefGoogle Scholar