Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg

, Volume 79, Issue 1, pp 87-111

First online:

Jacobi Maaß forms

  • Ameya PitaleAffiliated withDepartment of Mathematics, University of Oklahoma Email author 

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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.


Jacobi forms Maass forms Jacobi group Automorphic representation

Mathematics Subject Classification (2000)

11F50 11F37