The Ring of Jacobi Forms

  • Martin Eichler
  • Don Zagier
Part of the Progress in Mathematics book series (PM, volume 55)


The object of this and the following section is to obtain as much information as possible about the algebraic structure of the set of Jacobi forms, in particular about
  1. i)

    the dimension of Jk,m (k,m fixed), i.e. the structure of this space as a vector space over ℂ;

  2. ii)

    the additive structure of \( {J_{*,m}} = \mathop \oplus \limits_k {J_{k,m}} \) (m fixed) as a module over the graded ring \( {M_*} = \mathop \oplus \limits_k {M_k} \) of ordinary modular forms;

  3. iii)

    the multiplicative structure of the bigraded ring \( {J_{*,*}} = \mathop \oplus \limits_{k,m} {J_{k,m}} \) of all Jacobi forms.

We will study only the case of forms on the full Jacobi group \( \Gamma _1^J \) (and usually only the case of forms of even weight), but many of the considerations could be extended to arbitrary Γ.


Modular Form Fourier Coefficient Eisenstein Series Cusp Form Trace Formula 
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Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Martin Eichler
    • 1
  • Don Zagier
    • 2
  1. 1.ArlesheimSwitzerland
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA

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