Saito-Kurokawa lifting for odd weights (Q2712561)

There exists a lifting from modular forms of half integral weights on \(\Gamma_0(4)\) into Siegel modular forms of even integral weights of degree two called the Saito-Kurokawa lifting. \textit{W. Duke} and \textit{Ö. Imamoḡlu} [Int. Math. Res. Not. 1996, No. 7, 347-355 (1996; Zbl 0849.11039)] reconstructed this lifting with the help of the converse theorem of \textit{K. Imai} [Am. J. Math. 102, 903-936 (1980; Zbl 0447.10028)] and some results of \textit{S. Katok} and \textit{P. Sarnak} [Isr. J. Math. 84, 193-227 (1993; Zbl 0787.11016)]. In the paper the author establishes a kind of Saito-Kurokawa lifting in odd weight cases by using the method of Duke-Imamoḡlu. Indeed, for a positive odd integer \(k\), the author constructs two kinds of liftings from elliptic cusp forms of weight \(k-\frac{1}{2}\) to Siegel modular forms of degree two of weight \(k\) on the congruence subgroup \(\Gamma_0^{(2)}(4) \) with a non-trivial character mod \(4\) and studies a relationship of these two liftings. Also these liftings are studied by means of \textit{M. Eichler} and \textit{D. Zagier} [The theory of Jacobi forms (Prog. Math. 55), Birkhäuser, Basel (1985; Zbl 0554.10018)] where Jacobi forms are effectively used.