Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg

, Volume 79, Issue 1, pp 87–111

Jacobi Maaß forms

Article

DOI: 10.1007/s12188-008-0013-9

Cite this article as:
Pitale, A. Abh. Math. Semin. Univ. Hambg. (2009) 79: 87. doi:10.1007/s12188-008-0013-9

Abstract

In this paper, we give a new definition for the space of non-holomorphic Jacobi Maaß forms (denoted by Jk,mnh) 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 Jk,mnh. We construct new examples of cuspidal Jacobi Maaß forms Ff 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 fFf 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 Jk,mnh can be “essentially” obtained from scalar or vector valued half integer weight Maaß forms.

Keywords

Jacobi formsMaass formsJacobi groupAutomorphic representation

Mathematics Subject Classification (2000)

11F5011F37

Copyright information

© Mathematisches Seminar der Universität Hamburg and Springer 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OklahomaNormanUSA