Jacobi Maaß forms

Article

Abstract

In this paper, we give a new definition for the space of non-holomorphic Jacobi Maaß forms (denoted by J k,m nh ) of weight k∈ℤ and index m∈ℕ as eigenfunctions of a degree three differential operator \(\mathcal{C}^{k,m}\) . We show that the three main examples of Jacobi forms known in the literature: holomorphic, skew-holomorphic and real-analytic Eisenstein series, are contained in J k,m nh . We construct new examples of cuspidal Jacobi Maaß forms F f of weight k∈2ℤ and index 1 from weight k−1/2 Maaß forms f with respect to Γ0(4) and show that the map f F f is Hecke equivariant. We also show that the above map is compatible with the well-known representation theory of the Jacobi group. In addition, we show that all of J k,m nh can be “essentially” obtained from scalar or vector valued half integer weight Maaß forms.

Keywords

Jacobi forms Maass forms Jacobi group Automorphic representation 

Mathematics Subject Classification (2000)

11F50 11F37 

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Copyright information

© Mathematisches Seminar der Universität Hamburg and Springer 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OklahomaNormanUSA

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