Harmonic Maass-Jacobi forms with singularities and a theta-like decomposition

Bringmann, Kathrin; Raum, Martin; Richter, Olav

doi:10.1090/S0002-9947-2015-06418-6

Harmonic Maass-Jacobi forms with singularities and a theta-like decomposition

Authors: Kathrin Bringmann, Martin Raum and Olav K. Richter
Journal: Trans. Amer. Math. Soc. 367 (2015), 6647-6670
MSC (2010): Primary 11F50; Secondary 11F60, 11F55, 11F37, 11F30, 11F27
DOI: https://doi.org/10.1090/S0002-9947-2015-06418-6
Published electronically: January 15, 2015
MathSciNet review: 3356950
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Real-analytic Jacobi forms play key roles in different areas of mathematics and physics, but a satisfactory theory of such Jacobi forms has been lacking. In this paper, we fill this gap by introducing a space of harmonic Maass-Jacobi forms with singularities which includes the real-analytic Jacobi forms from Zwegers's PhD thesis. We provide several structure results for the space of such Jacobi forms, and we employ Zwegers's $\widehat {\mu }$ -functions to establish a theta-like decomposition.

[1] Rolf Berndt and Ralf Schmidt, Elements of the representation theory of the Jacobi group, Progress in Mathematics, vol. 163, Birkhäuser Verlag, Basel, 1998. MR 1634977
[2] Kathrin Bringmann and Amanda Folsom, Almost harmonic Maass forms and Kac-Wakimoto characters, J. Reine Angew. Math. 694 (2014), 179–202. MR 3259042, https://doi.org/10.1515/crelle-2012-0102
[3] Kathrin Bringmann and R. Olivetto, Kac-Wakimoto characters and non-holomorphic Jacobi forms, preprint, 2013.
[4] Kathrin Bringmann, Martin Raum, and Olav K. Richter, Kohnen’s limit process for real-analytic Siegel modular forms, Adv. Math. 231 (2012), no. 2, 1100–1118. MR 2955204, https://doi.org/10.1016/j.aim.2012.06.003
[5] Kathrin Bringmann and Olav K. Richter, Zagier-type dualities and lifting maps for harmonic Maass-Jacobi forms, Adv. Math. 225 (2010), no. 4, 2298–2315. MR 2680205, https://doi.org/10.1016/j.aim.2010.03.033
[6] Jan Hendrik Bruinier and Jens Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45–90. MR 2097357, https://doi.org/10.1215/S0012-7094-04-12513-8
[7] C. Conley and M. Raum, Harmonic Maaß-Jacobi forms of degree with higher rank indices, preprint, 2011.
[8] A. Dabholkar, S. Murthy, and D. Zagier, Quantum black holes, wall crossing, and mock modular forms, to appear in Cambridge Monographs on Mathematical Physics.
[9] Tohru Eguchi, Hirosi Ooguri, and Yuji Tachikawa, Notes on the 𝐾3 surface and the Mathieu group 𝑀₂₄, Exp. Math. 20 (2011), no. 1, 91–96. MR 2802725, https://doi.org/10.1080/10586458.2011.544585
[10] Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735
[11] Lothar Göttsche, Hiraku Nakajima, and K\B{o}ta Yoshioka, Instanton counting and Donaldson invariants, J. Differential Geom. 80 (2008), no. 3, 343–390. MR 2472477
[12] Lothar Göttsche and Don Zagier, Jacobi forms and the structure of Donaldson invariants for 4-manifolds with 𝑏₊=1, Selecta Math. (N.S.) 4 (1998), no. 1, 69–115. MR 1623706, https://doi.org/10.1007/s000290050025
[13] Masanobu Kaneko and Don Zagier, A generalized Jacobi theta function and quasimodular forms, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 165–172. MR 1363056, https://doi.org/10.1007/978-1-4612-4264-2_6
[14] Toshiya Kawai and K\B{o}ta Yoshioka, String partition functions and infinite products, Adv. Theor. Math. Phys. 4 (2000), no. 2, 397–485. MR 1838446, https://doi.org/10.4310/ATMP.2000.v4.n2.a7
[15] Anatoly Libgober, Elliptic genera, real algebraic varieties and quasi-Jacobi forms, Topology of stratified spaces, Math. Sci. Res. Inst. Publ., vol. 58, Cambridge Univ. Press, Cambridge, 2011, pp. 95–120. MR 2796409
[16] Andreas Malmendier and Ken Ono, 𝑆𝑂(3)-Donaldson invariants of ℂℙ² and mock theta functions, Geom. Topol. 16 (2012), no. 3, 1767–1833. MR 2967063, https://doi.org/10.2140/gt.2012.16.1767
[17] Jan Manschot, Stability and duality in 𝒩=2 supergravity, Comm. Math. Phys. 299 (2010), no. 3, 651–676. MR 2718927, https://doi.org/10.1007/s00220-010-1104-x
[18] Toshio Nishino, Function theory in several complex variables, Translations of Mathematical Monographs, vol. 193, American Mathematical Society, Providence, RI, 2001. Translated from the 1996 Japanese original by Norman Levenberg and Hiroshi Yamaguchi. MR 1818167
[19] G. Oberdieck, A Serre derivative for even weight Jacobi forms, preprint, 2012.
[20] Ameya Pitale, Jacobi Maaßforms, Abh. Math. Semin. Univ. Hambg. 79 (2009), no. 1, 87–111. MR 2541345, https://doi.org/10.1007/s12188-008-0013-9
[21] M. Raum, Dual weights in the theory for harmonic Siegel modular forms, Ph.D. thesis, University of Bonn, Germany, 2012.
[22] Nils-Peter Skoruppa, Developments in the theory of Jacobi forms, Automorphic functions and their applications (Khabarovsk, 1988) Acad. Sci. USSR, Inst. Appl. Math., Khabarovsk, 1990, pp. 167–185. MR 1096975
[23] Nils-Peter Skoruppa, Explicit formulas for the Fourier coefficients of Jacobi and elliptic modular forms, Invent. Math. 102 (1990), no. 3, 501–520. MR 1074485, https://doi.org/10.1007/BF01233438
[24] Nils-Peter Skoruppa, Jacobi forms of critical weight and Weil representations, Modular forms on Schiermonnikoog, Cambridge Univ. Press, Cambridge, 2008, pp. 239–266. MR 2512363, https://doi.org/10.1017/CBO9780511543371.013
[25] Don Zagier, Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann), Astérisque 326 (2009), Exp. No. 986, vii–viii, 143–164 (2010). Séminaire Bourbaki. Vol. 2007/2008. MR 2605321
[26] S. Zwegers, Mock theta functions, Ph.D. thesis, Universiteit Utrecht, The Netherlands, 2002.
[27] S. Zwegers, Multivariable Appell functions, preprint, 2010.