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

We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level [Formula: see text] and integral weight [Formula: see text] coincides with the field generated by the single-valued periods of a certain motive attached to [Formula: see text] . This...

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Autor principal: Fonseca, Tiago J.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer International Publishing 2020
Materias:
Research
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7651428/
https://www.ncbi.nlm.nih.gov/pubmed/33195991
http://dx.doi.org/10.1007/s40687-020-00232-5
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author Fonseca, Tiago J.
author_facet Fonseca, Tiago J.
author_sort Fonseca, Tiago J.
collection PubMed
description We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level [Formula: see text] and integral weight [Formula: see text] coincides with the field generated by the single-valued periods of a certain motive attached to [Formula: see text] . 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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spelling pubmed-76514282020-11-12 On coefficients of Poincaré series and single-valued periods of modular forms Fonseca, Tiago J. Res Math Sci Research We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level [Formula: see text] and integral weight [Formula: see text] coincides with the field generated by the single-valued periods of a certain motive attached to [Formula: see text] . 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. Springer International Publishing 2020-11-05 2020 /pmc/articles/PMC7651428/ /pubmed/33195991 http://dx.doi.org/10.1007/s40687-020-00232-5 Text en © The Author(s) 2020 Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
spellingShingle Research
Fonseca, Tiago J.
On coefficients of Poincaré series and single-valued periods of modular forms
title On coefficients of Poincaré series and single-valued periods of modular forms
title_full On coefficients of Poincaré series and single-valued periods of modular forms
title_fullStr On coefficients of Poincaré series and single-valued periods of modular forms
title_full_unstemmed On coefficients of Poincaré series and single-valued periods of modular forms
title_short On coefficients of Poincaré series and single-valued periods of modular forms
title_sort on coefficients of poincaré series and single-valued periods of modular forms
topic Research
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7651428/
https://www.ncbi.nlm.nih.gov/pubmed/33195991
http://dx.doi.org/10.1007/s40687-020-00232-5
work_keys_str_mv AT fonsecatiagoj oncoefficientsofpoincareseriesandsinglevaluedperiodsofmodularforms

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