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Article: Duality and differential operators for harmonic Maass forms

TitleDuality and differential operators for harmonic Maass forms
Authors
Issue Date2013
Citation
Developments in Mathematics, 2013, v. 28, p. 85-106 How to Cite?
AbstractDue 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 ${D}^{k-1} := {( \frac{1} {2\pi \mathrm{i}} \frac{\partial } {\partial z})}^{k-1}$ acting on a harmonic Maass form for integers k > 2 in terms of ${\xi }_{2-k} := 2\mathrm{i}{y}^{2-k}\overline{ \frac{\partial } {\partial \overline{z}}}$ 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 p-adic 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 Identifierhttp://hdl.handle.net/10722/192202
ISSN
2015 SCImago Journal Rankings: 0.578

 

DC FieldValueLanguage
dc.contributor.authorBringmann, Ken_US
dc.contributor.authorKane, Ben_US
dc.contributor.authorRhoades, RCen_US
dc.date.accessioned2013-10-23T09:27:19Z-
dc.date.available2013-10-23T09:27:19Z-
dc.date.issued2013en_US
dc.identifier.citationDevelopments in Mathematics, 2013, v. 28, p. 85-106en_US
dc.identifier.issn1389-2177en_US
dc.identifier.urihttp://hdl.handle.net/10722/192202-
dc.description.abstractDue 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 ${D}^{k-1} := {( \frac{1} {2\pi \mathrm{i}} \frac{\partial } {\partial z})}^{k-1}$ acting on a harmonic Maass form for integers k > 2 in terms of ${\xi }_{2-k} := 2\mathrm{i}{y}^{2-k}\overline{ \frac{\partial } {\partial \overline{z}}}$ 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 p-adic 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.languageengen_US
dc.relation.ispartofDevelopments in Mathematicsen_US
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.titleDuality and differential operators for harmonic Maass formsen_US
dc.typeArticleen_US
dc.description.naturepostprint-
dc.identifier.doi10.1007/978-1-4614-4075-8_6en_US
dc.identifier.scopuseid_2-s2.0-84875795915en_US
dc.identifier.volume28en_US
dc.identifier.spage85en_US
dc.identifier.epage106en_US

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