Construction of vector-valued modular integrals and vector-valued mock modular forms

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• 作者：Jose Gimenez ; Tobias Mühlenbruch ; Wissam Raji
• 关键词：Modular forms ; Fourier coefficients ; Modular integrals ; Supplementary function
• 刊名：The Ramanujan Journal
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：41
• 期：1-3
• 页码：51-114
• 全文大小：900 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Number Theory
  Field Theory and Polynomials
  Combinatorics
  Fourier Analysis
  Functions of a Complex Variable
  
• 出版者：Springer U.S.
• ISSN：1572-9303
• 卷排序：41

文摘

It is known that, given a vector-valued modular form of negative weight, its Fourier coefficients can be calculated based on the principal part of the form. In this paper we start with an arbitrary principal part and complete the Fourier expansion using the calculation. We show that the so-obtained function is a vector-valued modular integral of negative weight on the full modular group. Next, we construct the supplementary function associated to a vector-valued modular cusp form of positive weight. The constructions are inspired by the construction of Eichler integrals by Knopp. We conclude with a comparison of these forms and their integrals to vector-valued weak harmonic Maass forms.