Fourier coefficients of logarithmic vector-valued Poincaré series

In 1920, Ramanujan wrote to Hardy about his discovery of the mock theta functions. In the years since, there has been much work in understanding the transformation properties and asymptotic nature of these functions. Recently, Zwegers proved a relationship between mock theta functions and vector-valued modular forms, and Bringmann and Ono used the theory of Maass forms and Poincaré series to prove a conjecture of Andrews, yielding an exact formula for the coefficients of the f(q) mock theta function. Here we build upon these results, using the theory of vector-valued modular forms and Poincaré series to prove an exact formula for the coefficients of the ω(q) mock theta function.

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On coefficients of Poincaré series and single-valued periods of modular forms

Research in the Mathematical Sciences ◽

10.1007/s40687-020-00232-5 ◽

2020 ◽

Vol 7 (4) ◽

Author(s):

Tiago J. Fonseca

Keyword(s):

Modular Forms ◽

Fourier Coefficients ◽

Poincaré Series ◽

Series Expansions ◽

Poincare Series ◽

Integral Weight ◽

Periods Of Modular Forms

AbstractWe prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level $$\varGamma _0(N)$$ Γ 0 ( N ) and integral weight $$k\ge 2$$ k ≥ 2 coincides with the field generated by the single-valued periods of a certain motive attached to $$\varGamma _0(N)$$ Γ 0 ( N ) . This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of Brown and Acres–Broadhurst giving explicit series expansions for the single-valued periods of some modular forms. Our proof is based on Bringmann–Ono’s construction of harmonic lifts of Poincaré series.

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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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