Abstract
We classify the simple even lattices of square free level and signature \((2,n), n \ge 4\). A lattice is called simple if the space of cusp forms of weight \(1+n/2\) for the dual Weil representation of the lattice is trivial. For a simple lattice every formal principal part obeying obvious conditions is the principal part of a vector valued modular form. Using this, we determine all holomorphic Borcherds products of singular weight (arising from vector valued modular forms with non-negative principal part) for the simple lattices. We construct the corresponding vector valued modular forms by eta products and compute expansions of the automorphic products at different cusps.
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References
Borcherds, R.E.: Automorphic forms on \(O_{s+2,2}(R)\) and infinite products. Invent. Math. 120, 161–213 (1995)
Borcherds, R.E.: Automorphic forms with singularities on Grassmannians. Invent. Math. 132(3), 491–562 (1998)
Borcherds, R.E.: The Gross–Kohnen–Zagier theorem in higher dimensions. Duke Math. J. 97(2), 219–233 (1999)
Borcherds, R.E.: Reflection groups of Lorentzian lattices. Duke Math. J. 104(2000), 319–366 (2000)
Bruinier, J.H.: Borcherds Products on \(O(2, l)\) and Chern Classes of Heegner Divisors. Springer (2001)
Bruinier, J.H.: On the rank of Picard groups of modular varieties attached to orthogonal groups. Compos. Math. 133(1), 49–63 (2002)
Bruinier, J.H., Kuss, M.: Eisenstein series attached to lattices and modular forms on orthogonal groups. Manuscr. Math. 106, 443–45 (2001)
Conway, J., Sloane, N.J.: Sphere Packings, Lattices and Groups, 3rd edn. Springer (1999)
Freitag, E.: Dimension formulae for vector valued modular forms. http://www.rzuser.uni-heidelberg.de/~t91/preprints/riemann-roch.pdf (2012)
Hagemeier, H.: Automorphe Produkte singulären Gewichts. PhD thesis (2010).
Opitz, S.: Diskriminantenformen und vektorwertige Modulformen. Master’s thesis (2013).
Scheithauer, N.R.: Twisting the fake monster super algebra. Adv. Math. 164, 325–348 (2001)
Scheithauer, N.R.: On the classification of automorphic products and generalized Kac–Moody algebras. Invent. Math. 164, 641–678 (2006)
Scheithauer, N.R.: The Weil representation of \(\text{ SL }_{2}({\mathbb{Z}})\) and some applications. Int. Math. Res. Not. 2009(8), 1488–1545 (2009)
Scheithauer, N.R.: Some constructions of modular forms for the Weil representation of \(\text{ SL }_{2}({\mathbb{Z}})\). Nagoya Math. J. (to appear)
Acknowledgments
The work of this paper extends the doctoral thesis of the second author [10]. The second author likes to thank the supervisor of this thesis, Jan Bruinier, for his support and encouragement. Moreover, we thank Nils Scheithauer who supervised the master’s theses of the first and the third author and suggested the topic of this paper. Without his constant support and effort this work would not have been possible. We also like to thank Sebastian Opitz who performed the necessary computer calculations in the determination of the simple lattices in section 8.
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Dittmann, M., Hagemeier, H. & Schwagenscheidt, M. Automorphic products of singular weight for simple lattices. Math. Z. 279, 585–603 (2015). https://doi.org/10.1007/s00209-014-1383-6
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DOI: https://doi.org/10.1007/s00209-014-1383-6