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Article: A problem of Petersson about weight 0 meromorphic modular forms

TitleA problem of Petersson about weight 0 meromorphic modular forms
Authors
Issue Date2016
Citation
Research in the Mathematical Sciences,  How to Cite?
AbstractIn this paper, we provide an explicit construction of weight 0 meromorphic modular forms. Following work of Petersson, we build these via Poincaré series. There are two main aspects of our investigation which differ from his approach. Firstly, the naive definition of the Poincaré series diverges and one must analytically continue via Hecke's trick. Hecke's trick is further complicated in our situation by the fact that the Fourier expansion does not converge everywhere due to singularities in the upper half-plane so it cannot solely be used to analytically continue the functions. To explain the second difference, we recall that Petersson constructed linear combinations from a family of meromorphic functions which are modular if a certain principal parts condition is satisfied. In contrast to this, we construct linear combinations from a family of non-meromorphic modular forms, known as polar harmonic Maass forms, which are meromorphic whenever the principal parts condition is satisfied.
Persistent Identifierhttp://hdl.handle.net/10722/227319

 

DC FieldValueLanguage
dc.contributor.authorBringmann, K-
dc.contributor.authorKane, BR-
dc.date.accessioned2016-07-18T09:09:46Z-
dc.date.available2016-07-18T09:09:46Z-
dc.date.issued2016-
dc.identifier.citationResearch in the Mathematical Sciences, -
dc.identifier.urihttp://hdl.handle.net/10722/227319-
dc.description.abstractIn this paper, we provide an explicit construction of weight 0 meromorphic modular forms. Following work of Petersson, we build these via Poincaré series. There are two main aspects of our investigation which differ from his approach. Firstly, the naive definition of the Poincaré series diverges and one must analytically continue via Hecke's trick. Hecke's trick is further complicated in our situation by the fact that the Fourier expansion does not converge everywhere due to singularities in the upper half-plane so it cannot solely be used to analytically continue the functions. To explain the second difference, we recall that Petersson constructed linear combinations from a family of meromorphic functions which are modular if a certain principal parts condition is satisfied. In contrast to this, we construct linear combinations from a family of non-meromorphic modular forms, known as polar harmonic Maass forms, which are meromorphic whenever the principal parts condition is satisfied.-
dc.languageeng-
dc.relation.ispartofResearch in the Mathematical Sciences-
dc.titleA problem of Petersson about weight 0 meromorphic modular forms-
dc.typeArticle-
dc.identifier.emailKane, BR: bkane@hku.hk-
dc.identifier.authorityKane, BR=rp01820-
dc.identifier.hkuros258909-

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