On the Fourier Coefficients of the PoincarÉ Series

1972 ◽  
Vol s2-5 (4) ◽  
pp. 584-588
Author(s):  
Ernst A. Schwandt
Keyword(s):  
Fourier Coefficients ◽  
Poincaré Series ◽  
Poincare Series

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

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.

scholarly journals Differential operators on polar harmonic Maass forms and elliptic duality

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.

scholarly journals A NOTE ON FOURIER COEFFICIENTS OF POINCARÉ SERIES

Mathematika ◽  
2010 ◽  
Vol 57 (1) ◽  
pp. 31-40 ◽  
Author(s):  
Emmanuel Kowalski ◽  
Abhishek Saha ◽  
Jacob Tsimerman
Keyword(s):  
Fourier Coefficients ◽  
Poincaré Series ◽  
Poincare Series

scholarly journals Hyperbolic Fourier coefficients of Poincaré series

2016 ◽  
Vol 41 (1-3) ◽  
pp. 465-518
Author(s):  
Cormac O’Sullivan ◽  
Karen Taylor
Keyword(s):  
Fourier Coefficients ◽  
Poincaré Series ◽  
Poincare Series

scholarly journals Symplectic Kloosterman sums and Poincaré series

2021 ◽  
Author(s):  
Siu Hang Man
Keyword(s):  
Stationary Phase ◽  
Fourier Coefficients ◽  
Power Saving ◽  
Kloosterman Sums ◽  
Poincaré Series ◽  
Poincare Series

AbstractWe prove power-saving bounds for general Kloosterman sums on $${\text {Sp}}(4)$$ Sp ( 4 ) associated to all Weyl elements via a stratification argument coupled with p-adic stationary phase methods. We relate these Kloosterman sums to the Fourier coefficients of $${\text {Sp}}(4)$$ Sp ( 4 ) Poincare series.

scholarly journals EXACT FORMULAS FOR COEFFICIENTS OF JACOBI FORMS

2011 ◽  
Vol 07 (03) ◽  
pp. 825-833 ◽  
Author(s):  
KATHRIN BRINGMANN ◽  
OLAV K. RICHTER
Keyword(s):  
Fourier Coefficients ◽  
Theta Functions ◽  
Poincaré Series ◽  
Jacobi Forms ◽  
Mock Theta Functions ◽  
Explicit Formulas ◽  
Poincare Series ◽  
Real Analytic

In previous work, we introduced harmonic Maass–Jacobi forms. The space of such forms includes the classical Jacobi forms and certain Maass–Jacobi–Poincaré series, as well as Zwegers' real-analytic Jacobi forms, which play an important role in the study of mock theta functions and related objects. Harmonic Maass–Jacobi forms decompose naturally into holomorphic and non-holomorphic parts. In this paper, we give exact formulas for the Fourier coefficients of the holomorphic parts of harmonic Maass–Jacobi forms and, in particular, we obtain explicit formulas for the Fourier coefficients of weak Jacobi forms.

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