Jacobi Maaß forms

Abstract

In this paper, we give a new definition for the space of non-holomorphic Jacobi Maaß forms (denoted by J nh k,m ) of weight k ∈ Z and index m ∈ N as eigenfunctions of a degree three differential operator 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 nh k,m . 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 0 (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 nh k,m can be "essentially" obtained from scalar or vector valued half integer weight Maaß forms.

Jacobi forms over complex quadratic fields via the cubic Casimir operators

“…Our construction is of the same flavor as the method of Kohnen [14] (see also Section 4 of Pitale [17]). It is likely that our results can be extended to other complex quadratic fields, but for simplicity we restrict to the case that 3.1.…”

“…,λ (4) is an analog of Kohnen's [14] plus space, and in contrast to [14] and [17] the plus space condition here does not depend on k.…”

“…(3) It would be interesting to explore a Hecke theory for Jacobi forms over complex quadratic fields, which would allow one to investigate if the lift in (26) sends Hecke eigenforms to Hecke eigenforms. (4) The real-analytic Jacobi forms in [17] can essentially all be obtained from halfintegral weight Maass forms (see the Remark after Theorem 4.6 in [17]). We thank Ralf Schmidt for pointing out that an analogous statement should also hold in our setting here.…”

“…More generally, the MaassJacobi forms over the rationals in Berndt and Schmidt [1] and Pitale [17] are eigenfunctions of differential operators invariant under the action of the real Jacobi group. Automorphic forms over complex quadratic fields are clearly not holomorphic functions, and so one needs an analyticity condition to take the place of the Cauchy-Riemann differential equations.…”

“…We find that its center is the image of the center of the universal enveloping algebra of the Jacobi group, and is generated by a single cubic operator. This operator was used in [1] (page 82) to define Maass-Jacobi forms over the rationals, and such forms were studied in [17].…”