Locally harmonic Maass forms and periods of meromorphic modular forms

Abstract In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a q-hypergeometric series.

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K. Bringmann, A. Folsom, K. Ono, L. Rolen: “Harmonic Maass Forms and Mock Modular Forms: Theory and Applications”.

Jahresbericht der Deutschen Mathematiker-Vereinigung ◽

10.1365/s13291-018-00196-0 ◽

2019 ◽

Vol 121 (4) ◽

pp. 291-296

Author(s):

Martin Raum

Keyword(s):

Modular Forms ◽

Maass Forms ◽

Mock Modular Forms ◽

Harmonic Maass Forms

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Harmonic Maass Forms and Mock Modular Forms: Theory and Applications

10.1090/coll/064 ◽

2017 ◽

Cited By ~ 41

Author(s):

Kathrin Bringmann ◽

Amanda Folsom ◽

Ken Ono ◽

Larry Rolen

Keyword(s):

Modular Forms ◽

Maass Forms ◽

Mock Modular Forms ◽

Harmonic Maass Forms

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Differential operators on polar harmonic Maass forms and elliptic duality

The Quarterly Journal of Mathematics ◽

10.1093/qmath/haz009 ◽

2019 ◽

Vol 70 (4) ◽

pp. 1181-1207

Author(s):

Kathrin Bringmann ◽

Paul Jenkins ◽

Ben Kane

Keyword(s):

Modular Forms ◽

Fourier Coefficients ◽

Differential Operators ◽

Poincaré Series ◽

Maass Forms ◽

Poincare Series ◽

Weakly Holomorphic Modular Forms ◽

Integral Weight ◽

Harmonic Maass Forms ◽

Holomorphic Modular Forms

Abstract In this paper, we study polar harmonic Maass forms of negative integral weight. Using work of Fay, we construct Poincaré series which span the space of such forms and show that their elliptic coefficients exhibit duality properties which are similar to the properties known for Fourier coefficients of harmonic Maass forms and weakly holomorphic modular forms.

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Real-dihedral harmonic Maass forms and CM-values of Hilbert modular functions

Compositio Mathematica ◽

10.1112/s0010437x15007770 ◽

2016 ◽

Vol 152 (6) ◽

pp. 1159-1197

Author(s):

Yingkun Li

Keyword(s):

Modular Forms ◽

Fourier Coefficients ◽

Hilbert Modular Forms ◽

Algebraic Numbers ◽

Modular Functions ◽

Maass Forms ◽

Stark's Conjecture ◽

Hilbert Modular ◽

Harmonic Maass Forms ◽

The Individual

In this paper, we study real-dihedral harmonic Maass forms and their Fourier coefficients. The main result expresses the values of Hilbert modular forms at twisted CM 0-cycles in terms of these Fourier coefficients. This is a twisted version of the main theorem in Bruinier and Yang [CM-values of Hilbert modular functions, Invent. Math. 163 (2006), 229–288] and provides evidence that the individual Fourier coefficients are logarithms of algebraic numbers in the appropriate real-quadratic field. From this result and numerical calculations, we formulate an algebraicity conjecture, which is an analogue of Stark’s conjecture in the setting of harmonic Maass forms. Also, we give a conjectural description of the primes appearing in CM-values of Hilbert modular functions.

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Cycle integrals of meromorphic modular forms and coefficients of harmonic Maass forms

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2020.124898 ◽

2021 ◽

Vol 497 (2) ◽

pp. 124898

Author(s):

C. Alfes–Neumann ◽

Kathrin Bringmann ◽

J. Males ◽

M. Schwagenscheidt

Keyword(s):

Modular Forms ◽

Maass Forms ◽

Harmonic Maass Forms

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q-series and weight 3/2 Maass forms

Compositio Mathematica ◽

10.1112/s0010437x09004072 ◽

2009 ◽

Vol 145 (03) ◽

pp. 541-552 ◽

Cited By ~ 9

Author(s):

Kathrin Bringmann ◽

Amanda Folsom ◽

Ken Ono

Keyword(s):

Recent Work ◽

Modular Forms ◽

Hypergeometric Series ◽

Basic Hypergeometric Series ◽

Maass Forms ◽

Type Series ◽

Harmonic Maass Forms

AbstractDespite the presence of many famous examples, the precise interplay between basic hypergeometric series and modular forms remains a mystery. We consider this problem for canonical spaces of weight 3/2 harmonic Maass forms. Using recent work of Zwegers, we exhibit forms that have the property that their holomorphic parts arise from Lerch-type series, which in turn may be formulated in terms of the Rogers–Fine basic hypergeometric series.

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