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

Authors  
Issue Date  2016 
Citation  Research in the Mathematical Sciences, How to Cite? 
Abstract  In 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 halfplane 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 nonmeromorphic modular forms, known as polar harmonic Maass forms, which are meromorphic whenever the principal parts condition is satisfied. 
Persistent Identifier  http://hdl.handle.net/10722/227319 
DC Field  Value  Language 

dc.contributor.author  Bringmann, K   
dc.contributor.author  Kane, BR   
dc.date.accessioned  20160718T09:09:46Z   
dc.date.available  20160718T09:09:46Z   
dc.date.issued  2016   
dc.identifier.citation  Research in the Mathematical Sciences,   
dc.identifier.uri  http://hdl.handle.net/10722/227319   
dc.description.abstract  In 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 halfplane 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 nonmeromorphic modular forms, known as polar harmonic Maass forms, which are meromorphic whenever the principal parts condition is satisfied.   
dc.language  eng   
dc.relation.ispartof  Research in the Mathematical Sciences   
dc.title  A problem of Petersson about weight 0 meromorphic modular forms   
dc.type  Article   
dc.identifier.email  Kane, BR: bkane@hku.hk   
dc.identifier.authority  Kane, BR=rp01820   
dc.identifier.hkuros  258909   