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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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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