Fourier expansions with modular form coefficients

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Abstract

In this paper, we study the Fourier expansion where the coefficients are given as the evaluation of a sequence of modular forms at a fixed point in the upper half-plane. We show that for prime levels l for which the modular curve X0(l) is hyperelliptic (with hyperelliptic involution of the AtkinLehner type) then one can choose a sequence of weight k (any even integer) forms so that the resulting Fourier expansion is itself a meromorphic modular form of weight 2-k. These sequences have many interesting properties, for instance, the sequence of their first nonzero next-to-leading coefficient is equal to the terms in the Fourier expansion of a certain weight 2-k form. The results in the paper generalizes earlier work by Asai, Kaneko, and Ninomiya (for level one), and Ahlgren (for the cases where X0(l) has genus zero).

Original languageEnglish
Pages (from-to)1433-1446
Number of pages14
JournalInternational Journal of Number Theory
Volume5
Issue number8
DOIs
Publication statusPublished - Dec 2009
Externally publishedYes

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Keywords

  • Atkin-Lehner involution
  • Coefficient duality
  • Fourier expansions
  • Hyperelliptic modular curves
  • Meromorphic modular forms

ASJC Scopus subject areas

  • Algebra and Number Theory

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