Abstract
We discuss mock automorphic forms from the point of view of representation theory, that is, mock automorphic forms obtained from weak harmonic Maaß forms giving rise to nontrivial \((\mathfrak g,K)\)-cohomology. We consider the possibility of replacing the ‘holomorphic’ condition with a “cohomological” one when generalizing to general reductive groups. Such a candidate for replacement allows for growing Fourier coefficients, in contrast to automorphic forms under the Miatello-Wallach conjecture. In the second part, we provide an overview of the connection with BPS black hole counts as a physical motivation for studying mock automorphic forms.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Notes
This is simply to guarantee the existence of a Whittaker model; we expect that this condition can be relaxed.
References
W. Duke, “Almost a century of answering the question: what is a mock theta function?,” Notices Amer. Math. Soc. 61 (11), 1314–1320 (2014).
K. Ono, “Unearthing the visions of a master: harmonic Maass forms and number theory,” in Current Developments in Mathematics, 2008 (Int. Press, Somerville, MA, 2009), pp. 347–454.
A. Folsom, “Perspectives on mock modular forms,” J. Number Theory 176, 500–540 (2017).
D. Zagier, “Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann),” in Astérisque (326) (2009), Séminaire Bourbaki (2010), Exp. no. 986, vii–viii, Vol. 2007/2008, pp. 143–164.
J. Bruinier and J. Funke, “On two geometric theta lifts,” Duke Math. J. 125 (1), 45–90 (2004).
A. Dabholkar, S. Murthy and D. Zagier, Quantum Black Holes, Wall Crossing, and Mock Modular Forms, Preprint: ar**v:1208.4074 (2012).
S. Zwegers, Mock Theta Functions, Ph.D. thesis, Utrecht University (2002).
K. Bringmann and S. Kudla, “A classification of harmonic Maass forms,” Math. Ann. 370 (3–4), 1729–1758 (2018).
R. Schulze–Pillot, “Weak Maaß forms and (\(\mathfrak g\),K)-modules,” Ramanujan J 26 (3), 437–445 (2011).
M. Westerholt–Raum, “Harmonic Weak Siegel Maass Forms,” I. Int. Math. Res. Not., No. 5, 1442–1472 (2018).
S. Kudla, M. Rapoport, and T. Yang, Modular Forms and Special Cycles on Shimura Curves, in Annals of Mathematics Studies (Princeton University Press, Princeton, NJ, 2006), Vol. 161.
K.–W. Lan, “Higher Koecher’s principle,” Math. Res. Lett. 23 (1), 163–199 (2016).
R. Miatello and N. R. Wallach, “Automorphic forms constructed from Whittaker vectors,” J. Funct. Anal. 86 (2), 411–487 (1989).
S. D. Miller and T. Trinh, “On the nonexistence of automorphic eigenfunctions of exponential growth on \(SL(3,\mathbb Z) \backslash SL(3,\mathbb R)/SO(3,\mathbb R)\),” Res. Number Theory 5 (4), (2019), Paper No. 31.
S. D. Miller and G. Moore, “Landau–Siegel zeros and black hole entropy,” Asian J. Math. 4 (1), 183–211 (2000).
J. Buttcane and S. D. Miller, Weights, Raising and Lowering Operators, and K-Types for Automorphic Forms on \(SL (3, \mathbb R)\) , ar**v:1702.08851 (2017).
P. Fleig, H. Gustafsson, A. Kleinschmidt and D. Persson, Eisenstein Series and Automorphic Representations, in Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 2018), Vol. 176.
S. Kachru and A. Tripathy, BPS Jum** Loci and Special Cycles, Preprint: ar**v:1703.00455 (2017).
G. Moore, Attractors and Arithmetic, Preprint: ar**v:hep-th/9807056 (1998).
S. Kachru and A. Tripathy, “Black holes and Hurwitz class numbers,” International Journal of Modern Physics D 26.12 (2017).
D. Zagier, “Nombres de classes et formes modulaires de poids 3/2,” C. R. Acad. Sci. Paris Ser. A-B 281 (21), A883–A886 (1975).
N. Benjamin, S. Kachru, K. Ono, and L. Rolen, “Black holes and class groups,” Res. Math. Sci. 5 (4), 22 pp (2018), Paper No. 43.
D. Maulik and R. Pandharipande, “Gromov–Witten theory and Noether–Lefschetz theory,” in A Celebration of Algebraic Geometry, Clay Math. Proc. (Amer. Math. Soc., Providence, RI, 2013), Vol. 18, pp. 469–507.
R. Borcherds, “The Gross–Kohnen–Zagier theorem in higher dimensions,” Duke Math. J. 97 (2), 219–233 (1999).
S. Kachru and A. Tripathy, “BPS jum** loci are automorphic,” Communications in Mathematical Physics 360 (3), 919–933 (2018).
F. Hirzebruch and D. Zagier, “Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus,” Invent. Math. 36, 57–113 (1976).
S. Kudla and J. Millson, “Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables,” Inst. Hautes Études Sci. Publ. Math., No. 71, 121–172 (1990).
G. Moore, “Physical mathematics and the future,” in Vision Talk (Strings Conference, Princeton USA, 2014).
A. Weil, “Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen,” Math. Ann. 168, 149–156 (1967).
K. Imai, “Generalization of Hecke’s correspondence to Siegel modular forms,” Amer. J. Math. 102 (5), 903–936 (1980).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wong, T.A. A Note on Mock Automorphic Forms and the BPS Index. Math Notes 110, 273–282 (2021). https://doi.org/10.1134/S0001434621070294
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434621070294