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Article: Duality and differential operators for harmonic Maass forms
Title  Duality and differential operators for harmonic Maass forms 

Authors  
Issue Date  2013 
Citation  Developments in Mathematics, 2013, v. 28, p. 85106 How to Cite? 
Abstract  Due to the graded ring nature of classical modular forms, there are many interesting relations between the coefficients of different modular forms. We discuss additional relations arising from Duality, Borcherds products, theta lifts. Using the explicit description of a lift for weakly holomorphic forms, we realize the differential operator ${D}^{k1} := {( \frac{1} {2\pi \mathrm{i}} \frac{\partial } {\partial z})}^{k1}$ acting on a harmonic Maass form for integers k > 2 in terms of ${\xi }_{2k} := 2\mathrm{i}{y}^{2k}\overline{ \frac{\partial } {\partial \overline{z}}}$ acting on a different form. Using this interpretation, we compute the image of D k − 1. We also answer a question arising in recent work on the padic properties of mock modular forms. Additionally, since such lifts are defined up to a weakly holomorphic form, we demonstrate how to construct a canonical lift from holomorphic modular forms to harmonic Maass forms. 
Persistent Identifier  http://hdl.handle.net/10722/192202 
ISSN  2015 SCImago Journal Rankings: 0.578 
DC Field  Value  Language 

dc.contributor.author  Bringmann, K  en_US 
dc.contributor.author  Kane, B  en_US 
dc.contributor.author  Rhoades, RC  en_US 
dc.date.accessioned  20131023T09:27:19Z   
dc.date.available  20131023T09:27:19Z   
dc.date.issued  2013  en_US 
dc.identifier.citation  Developments in Mathematics, 2013, v. 28, p. 85106  en_US 
dc.identifier.issn  13892177  en_US 
dc.identifier.uri  http://hdl.handle.net/10722/192202   
dc.description.abstract  Due to the graded ring nature of classical modular forms, there are many interesting relations between the coefficients of different modular forms. We discuss additional relations arising from Duality, Borcherds products, theta lifts. Using the explicit description of a lift for weakly holomorphic forms, we realize the differential operator ${D}^{k1} := {( \frac{1} {2\pi \mathrm{i}} \frac{\partial } {\partial z})}^{k1}$ acting on a harmonic Maass form for integers k > 2 in terms of ${\xi }_{2k} := 2\mathrm{i}{y}^{2k}\overline{ \frac{\partial } {\partial \overline{z}}}$ acting on a different form. Using this interpretation, we compute the image of D k − 1. We also answer a question arising in recent work on the padic properties of mock modular forms. Additionally, since such lifts are defined up to a weakly holomorphic form, we demonstrate how to construct a canonical lift from holomorphic modular forms to harmonic Maass forms.   
dc.language  eng  en_US 
dc.relation.ispartof  Developments in Mathematics  en_US 
dc.rights  This work is licensed under a Creative Commons AttributionNonCommercialNoDerivatives 4.0 International License.   
dc.title  Duality and differential operators for harmonic Maass forms  en_US 
dc.type  Article  en_US 
dc.description.nature  postprint   
dc.identifier.doi  10.1007/9781461440758_6  en_US 
dc.identifier.scopus  eid_2s2.084875795915  en_US 
dc.identifier.volume  28  en_US 
dc.identifier.spage  85  en_US 
dc.identifier.epage  106  en_US 