On a relation between certain -hypergeometric series and Maass waveforms
Authors:
Matthew Krauel, Larry Rolen and Michael Woodbury
Journal:
Proc. Amer. Math. Soc. 145 (2017), 543-557
MSC (2010):
Primary 11F03, 11F27
DOI:
https://doi.org/10.1090/proc/13246
Published electronically:
August 23, 2016
MathSciNet review:
3577859
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Abstract | References | Similar Articles | Additional Information
Abstract: In this paper, we answer a question of Li, Ngo, and Rhoades concerning a set of -series related to the
-hypergeometric series
from Ramanujan's Lost Notebook. Our results parallel a theorem of Cohen which says that
, along with its partner function
, encode the coefficients of a Maass waveform of eigenvalue
.
- [1] G. E. Andrews, Ramanujan's ``lost'' notebook. V. Euler's partition identity, Adv. Math. 61, 156-164 (1986).
- [2] George E. Andrews, Freeman J. Dyson, and Dean Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988), no. 3, 391–407. MR 928489, https://doi.org/10.1007/BF01388778
- [3] Kathrin Bringmann and Ben Kane, Multiplicative 𝑞-hypergeometric series arising from real quadratic fields, Trans. Amer. Math. Soc. 363 (2011), no. 4, 2191–2209. MR 2746680, https://doi.org/10.1090/S0002-9947-2010-05214-6
- [4] H. Cohen, 𝑞-identities for Maass waveforms, Invent. Math. 91 (1988), no. 3, 409–422. MR 928490, https://doi.org/10.1007/BF01388779
- [5] Daniel Corson, David Favero, Kate Liesinger, and Sarah Zubairy, Characters and 𝑞-series in ℚ(√2), J. Number Theory 107 (2004), no. 2, 392–405. MR 2072397, https://doi.org/10.1016/j.jnt.2004.03.002
- [6] J. Lewis and D. Zagier, Period functions and the Selberg zeta function for the modular group, The mathematical beauty of physics (Saclay, 1996) Adv. Ser. Math. Phys., vol. 24, World Sci. Publ., River Edge, NJ, 1997, pp. 83–97. MR 1490850
- [7] J. Lewis and D. Zagier, Period functions for Maass wave forms. I, Ann. of Math. (2) 153 (2001), no. 1, 191–258. MR 1826413, https://doi.org/10.2307/2661374
- [8] Jeremy Lovejoy, Overpartitions and real quadratic fields, J. Number Theory 106 (2004), no. 1, 178–186. MR 2049600, https://doi.org/10.1016/j.jnt.2003.12.014
- [9] Y. Li, H. Ngo, and C. Rhoades, Renormalization and quantum modular forms, part I: Maass wave forms, preprint, arXiv:1311.3043.
- [10] Don Zagier, Quantum modular forms, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010, pp. 659–675. MR 2757599
- [11] Sander P. Zwegers, Mock Maass theta functions, Q. J. Math. 63 (2012), no. 3, 753–770. MR 2967174, https://doi.org/10.1093/qmath/har020
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Additional Information
Matthew Krauel
Affiliation:
Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
Email:
mkrauel@math.uni-koeln.de
Larry Rolen
Affiliation:
Department of Mathematics, 212 McAllister Building, The Pennsylvania State University, University Park, Pennsylvania 16802
Email:
larryrolen@psu.edu
Michael Woodbury
Affiliation:
Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
Email:
woodbury@math.uni-koeln.de
DOI:
https://doi.org/10.1090/proc/13246
Received by editor(s):
December 28, 2015
Received by editor(s) in revised form:
March 15, 2016, and April 12, 2016
Published electronically:
August 23, 2016
Additional Notes:
The first author was supported by the European Research Council (ERC) Grant agreement n. 335220 - AQSER
The second author thanks the University of Cologne and the DFG for their generous support via the University of Cologne postdoc grant DFG Grant D-72133-G-403-151001011, funded under the Institutional Strategy of the University of Cologne within the German Excellence Initiative
Communicated by:
Kathrin Bringmann
Article copyright:
© Copyright 2016
American Mathematical Society