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Weakly holomorphic modular forms and rank two hyperbolic Kac-Moody algebras


Authors: Henry H. Kim, Kyu-Hwan Lee and Yichao Zhang
Journal: Trans. Amer. Math. Soc. 367 (2015), 8843-8860
MSC (2010): Primary 11F22; Secondary 17B67, 11F41
DOI: https://doi.org/10.1090/S0002-9947-2015-06438-1
Published electronically: March 13, 2015
MathSciNet review: 3403073
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Abstract: In this paper, we compute basis elements of certain spaces of weight 0 weakly holomorphic modular forms and consider the integrality of Fourier coefficients of the modular forms. We use the results to construct automorphic correction of the rank $ 2$ hyperbolic Kac-Moody algebras $ \mathcal H(a)$, $ a=4,5,6$, through Hilbert modular forms explicitly given by Borcherds lifts of the weakly holomorphic modular forms. We also compute asymptotics of the Fourier coefficients as they are related to root multiplicities of the rank $ 2$ hyperbolic Kac-Moody algebras. This work is a continuation of an earlier work of the first and second authors, where automorphic correction was constructed for $ \mathcal H(a)$, $ a=3, 11, 66$.


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  • [1] Richard E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), no. 2, 405-444. MR 1172696 (94f:11030), https://doi.org/10.1007/BF01232032
  • [2] Richard E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491-562. MR 1625724 (99c:11049), https://doi.org/10.1007/s002220050232
  • [3] Jan Hendrik Bruinier, Hilbert modular forms and their applications, The 1-2-3 of modular forms, Universitext, Springer, Berlin, 2008, pp. 105-179. MR 2447162 (2009g:11057), https://doi.org/10.1007/978-3-540-74119-0_2
  • [4] Jan Hendrik Bruinier and Michael Bundschuh, On Borcherds products associated with lattices of prime discriminant, Ramanujan J. 7 (2003), no. 1-3, 49-61. Rankin memorial issues. MR 2035791 (2005a:11057), https://doi.org/10.1023/A:1026222507219
  • [5] Alex J. Feingold and Igor B. Frenkel, A hyperbolic Kac-Moody algebra and the theory of Siegel modular forms of genus $ 2$, Math. Ann. 263 (1983), no. 1, 87-144. MR 697333 (86a:17006), https://doi.org/10.1007/BF01457086
  • [6] Valery A. Gritsenko and Viacheslav V. Nikulin, Igusa modular forms and ``the simplest'' Lorentzian Kac-Moody algebras, Mat. Sb. 187 (1996), no. 11, 27-66.
  • [7] Valeri A. Gritsenko and Viacheslav V. Nikulin, Siegel automorphic form corrections of some Lorentzian Kac-Moody Lie algebras, Amer. J. Math. 119 (1997), no. 1, 181-224. MR 1428063 (98g:11056)
  • [8] Valeri A. Gritsenko and Viacheslav V. Nikulin, Automorphic forms and Lorentzian Kac-Moody algebras. I, Internat. J. Math. 9 (1998), no. 2, 153-199. MR 1616925 (99b:11040), https://doi.org/10.1142/S0129167X98000105
  • [9] Valeri A. Gritsenko and Viacheslav V. Nikulin, Automorphic forms and Lorentzian Kac-Moody algebras. II, Internat. J. Math. 9 (1998), no. 2, 201-275. MR 1616929 (99b:11041), https://doi.org/10.1142/S0129167X98000117
  • [10] V. A. Gritsenko and V. V. Nikulin, On the classification of Lorentzian Kac-Moody algebras, Uspekhi Mat. Nauk 57 (2002), no. 5(347), 79-138 (Russian, with Russian summary); English transl., Russian Math. Surveys 57 (2002), no. 5, 921-979. MR 1992083 (2004f:17034), https://doi.org/10.1070/RM2002v057n05ABEH000553
  • [11] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219 (92k:17038)
  • [12] Victor G. Kac and Dale H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. in Math. 53 (1984), no. 2, 125-264. MR 750341 (86a:17007), https://doi.org/10.1016/0001-8708(84)90032-X
  • [13] Victor G. Kac and Minoru Wakimoto, Modular and conformal invariance constraints in representation theory of affine algebras, Adv. in Math. 70 (1988), no. 2, 156-236. MR 954660 (89h:17036), https://doi.org/10.1016/0001-8708(88)90055-2
  • [14] Seok-Jin Kang and Duncan J. Melville, Rank $ 2$ symmetric hyperbolic Kac-Moody algebras, Nagoya Math. J. 140 (1995), 41-75. MR 1369479 (97c:17039)
  • [15] Henry H. Kim and Kyu-Hwan Lee, Automorphic correction of the hyperbolic Kac-Moody algebra $ E_{10}$, J. Math. Phys. 54 (2013), no. 9, 091701, 13. MR 3135585, https://doi.org/10.1063/1.4820562
  • [16] Henry H. Kim and Kyu-Hwan Lee, Root multiplicities of hyperbolic Kac-Moody algebras and Fourier coefficients of modular forms, Ramanujan J. 32 (2013), no. 3, 329-352. MR 3130654, https://doi.org/10.1007/s11139-013-9474-2
  • [17] Henry H. Kim and Kyu-Hwan Lee, Rank 2 symmetric hyperbolic Kac-Moody algebras and Hilbert modular forms, J. Algebra 407 (2014), 81-104. MR 3197153, https://doi.org/10.1016/j.jalgebra.2014.03.003
  • [18] Joseph Lehner, Discontinuous groups and automorphic functions, Mathematical Surveys, No. VIII, American Mathematical Society, Providence, R.I., 1964. MR 0164033 (29 #1332)
  • [19] James Lepowsky and Robert V. Moody, Hyperbolic Lie algebras and quasiregular cusps on Hilbert modular surfaces, Math. Ann. 245 (1979), no. 1, 63-88. MR 552580 (81c:10030), https://doi.org/10.1007/BF01420431
  • [20] I. G. Macdonald, Affine root systems and Dedekind's $ \eta $-function, Invent. Math. 15 (1972), 91-143. MR 0357528 (50 #9996)
  • [21] Yves Martin, Multiplicative $ \eta $-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856. MR 1376550 (97d:11070), https://doi.org/10.1090/S0002-9947-96-01743-6
  • [22] Sebastian Mayer, Hilbert modular forms for the fields $ \mathbb{Q}(\sqrt 5)$, $ \mathbb{Q}(\sqrt {13})$, $ \mathbb{Q}(\sqrt {17})$, Dissertation (2007).
  • [23] William Stein, Modular forms, a computational approach, Graduate Studies in Mathematics, vol. 79, American Mathematical Society, Providence, RI, 2007. With an appendix by Paul E. Gunnells. MR 2289048 (2008d:11037)
  • [24] Jacob Sturm, On the congruence of modular forms, Number theory (New York, 1984-1985) Lecture Notes in Math., vol. 1240, Springer, Berlin, 1987, pp. 275-280. MR 894516 (88h:11031), https://doi.org/10.1007/BFb0072985
  • [25] Yichao Zhang, Zagier duality and integrality of Fourier coefficients for weakly holomorphic modular forms, to appear in J. Number Theory, arXiv: 1308.1037 (2013).
  • [26] Yichao Zhang, An isomorphism between scalar-valued modular forms and modular forms for Weil representations, to appear in Ramanujan J., arXiv:1307.4390 (2014).

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Additional Information

Henry H. Kim
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada – and – Korea Institute for Advanced Study, Seoul, Korea
Email: henrykim@math.toronto.edu

Kyu-Hwan Lee
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: khlee@math.uconn.edu

Yichao Zhang
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Address at time of publication: Department of Mathematics, The University of Arizona, Tucson, Arizona 85721
Email: yichao.zhang@uconn.edu, yichaozhang@math.arizona.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06438-1
Received by editor(s): September 19, 2013
Received by editor(s) in revised form: January 21, 2014, and April 2, 2014
Published electronically: March 13, 2015
Additional Notes: The first author was partially supported by an NSERC grant.
Article copyright: © Copyright 2015 American Mathematical Society

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