Almost Holomorphic Poincaré Series Corresponding to Products of Harmonic Siegel–Maass Forms

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This article investigates Poincaré series, particularly the study of Siegel–Poincaré series of degree 2 attached to products of terms of Fourier series of harmonic Siegel–Maass forms and holomorphic Siegel modular forms

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16 p.

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Bringmann, Kathrin; Richter, Olav K. & Westerholt-Raum, Martin August 12, 2016.

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This article investigates Poincaré series, particularly the study of Siegel–Poincaré series of degree 2 attached to products of terms of Fourier series of harmonic Siegel–Maass forms and holomorphic Siegel modular forms

Physical Description

16 p.

Notes

Abstract: We investigate Poincaré series, where we average products of terms of Fourier series of
real-analytic Siegel modular forms. There are some (trivial) special cases for which the
products of terms of Fourier series of elliptic modular forms and harmonic Maass forms
are almost holomorphic, in which case the corresponding Poincaré series are almost
holomorphic as well. In general, this is not the case. The main point of this paper is the
study of Siegel–Poincaré series of degree 2 attached to products of terms of Fourier
series of harmonic Siegel–Maass forms and holomorphic Siegel modular forms. We
establish conditions on the convergence and nonvanishing of such Siegel–Poincaré
series. We surprisingly discover that these Poincaré series are almost holomorphic
Siegel modular forms, although the product of terms of Fourier series of harmonic
Siegel–Maass forms and holomorphic Siegel modular forms (in contrast to the elliptic
case) is not almost holomorphic. Our proof employs tools from representation theory.
In particular, we determine some constituents of the tensor product of Harish-Chandra
modules with walls.

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  • Research in the Mathematical Sciences, 2016. New York, NY: Springer

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Publication Information

  • Publication Title: Resarch in the Mathematical Sciences
  • Volume: 3
  • Issue: 30
  • Pages: 1-16
  • Peer Reviewed: Yes

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UNT Scholarly Works

Materials from the UNT community's research, creative, and scholarly activities and UNT's Open Access Repository. Access to some items in this collection may be restricted.

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  • April 15, 2016

Accepted Date

  • August 12, 2016

Creation Date

  • August 12, 2016

Added to The UNT Digital Library

  • Nov. 15, 2017, 11:13 a.m.

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Bringmann, Kathrin; Richter, Olav K. & Westerholt-Raum, Martin. Almost Holomorphic Poincaré Series Corresponding to Products of Harmonic Siegel–Maass Forms, article, August 12, 2016; New York, New York. (digital.library.unt.edu/ark:/67531/metadc1040564/: accessed October 16, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.