Theta lifts and local Maass forms (Q395045)

\textit{K. Bringmann} et al. have recently introduced a class of modular objects [``Locally harmonic Maass forms and the kernel of the Shintani lift'', Preprint, \url{arXiv:1206.1100}], called locally harmonic Maass forms, which satisfy negative weight modularity and are annihilated almost everywhere by the hyperbolic Laplacian. The relevance of these forms comes from their applications, ranging from special values of derivatives of \(L\)-functions (see [\textit{J. Bruinier} and \textit{K. Ono}, Ann. Math. (2) 172, No. 3, 2135--2181 (2010; Zbl 1244.11046)] for example), to Zagier's duality, partition theory (for instance [\textit{G. E. Andrews}, Invent. Math. 169, No. 1, 37--73 (2007; Zbl 1214.11116)] among others) and physics (see [\textit{T. Eguchi} et al., Exp. Math. 20, No. 1, 91--96 (2011; Zbl 1266.58008)]).NEWLINENEWLINEThe main result of the paper is to realize locally harmonic Maass forms as theta-lifts of harmonic weak Maass forms. This can be seen as a generalization to this context of the classical Shimura-Shintani-Waldspurger correspondence for elliptic modular forms.