Nagoya Mathematical Journal

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

Nils R. Scheithauer

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Abstract

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 D122, E83 and to the Leech lattice.

Article information

Source
Nagoya Math. J., Volume 220 (2015), 1-43.

Dates
Received: 9 December 2013
Revised: 20 June 2014
Accepted: 13 August 2014
First available in Project Euclid: 1 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1448980485

Digital Object Identifier
doi:10.1215/00277630-3335405

Mathematical Reviews number (MathSciNet)
MR3429723

Zentralblatt MATH identifier
1344.11039

Subjects
Primary: 11F27: Theta series; Weil representation; theta correspondences
Secondary: 11F22: Relationship to Lie algebras and finite simple groups 17B65: Infinite-dimensional Lie (super)algebras [See also 22E65]

Keywords
modular forms for the Weil representation automorphic products generalized Kac–Moody algebras

Citation

Scheithauer, Nils R. Some constructions of modular forms for the Weil representation of $\operatorname{SL}_{2}(\mathbb{Z})$. Nagoya Math. J. 220 (2015), 1--43. doi:10.1215/00277630-3335405. https://projecteuclid.org/euclid.nmj/1448980485


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References

  • [1] A. G. Barnard, The singular theta correspondence, Lorentzian lattices and Borcherds-Kac-Moody algebras, Ph.D. dissertation, University of California, Berkeley, Berkeley, Calif., 2003.
  • [2] R. E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405–444.
  • [3] R. E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), 491–562.
  • [4] W. Bosma, J. Cannon, and C. Playoust, “The Magma algebra system, I: The user language” in Computational Algebra and Number Theory (London, 1993), J. Symbolic Comput. 24, 1997, 235–265.
  • [5] J. H. Bruinier, Borcherds Products on $O(2,l)$ and Chern Classes of Heegner Divisors, Lecture Notes in Math. 1780, Springer, Berlin, 2002.
  • [6] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Grundlehren Math. Wiss. 290, Springer, New York, 1999.
  • [7] F. Diamond and J. Shurman, A First Course in Modular Forms, Graduate Texts in Math. 228, Springer, New York, 2005.
  • [8] K. Harada and M.-L. Lang, On some sublattices of the Leech lattice, Hokkaido Math. J. 19 (1990), 435–446.
  • [9] V. V. Nikulin, 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.
  • [10] O. T. O’Meara, Introduction to Quadratic Forms, reprint of the 1973edition, Classics Math., Springer, Berlin, 2000.
  • [11] The PARI Group, Bordeaux, PARI/GP, version 2.3.4, 2008, http://pari.math.u-bordeaux.fr/ (accessed September 16, 2015).
  • [12] N. R. Scheithauer, On the classification of automorphic products and generalized Kac-Moody algebras, Invent. Math. 164 (2006), 641–678.
  • [13] N. R. Scheithauer, The Weil representation of $\operatorname{SL}_{2}(\mathbb{Z})$ and some applications, Int. Math. Res. Not. IMRN 2009, no. 8, 1488–1545.