Integral traces of singular values of weak Maass forms
For a holomorphic modular form, one can define traces of the form which are essentially the toric periods studied by \textit{J.-L. Waldspurger} [Compos. Math. 54, 173--242 (1985; Zbl 0567.10021)]. Waldspurger showed the square of the traces are the central values of base change \(L-\)functions. This together with his result on Shimura correspondence, implies the square of the traces are (up to a scalar multiple) the squares of Fourier coefficients of a holomorphic half integral weight form. \textit{W. Kohnen} [Math. Ann. 271, 237--268 (1985; Zbl 0542.10018)] proved that the traces are indeed (up to a scalar multiple) the Fourier coefficients of a holomorphic half integral weight form. The paper proved a corresponding result in the case of weakly holomorphic modular form. For a level 1 weakly holomorphic modular form, the authors used its traces as Fourier coefficients to construct a half integral weight weakly holomorphic form. This gives a Zagier lift, the negative weight analogue of Shintani lift. Moreover they showed that if the original integral weight form has integer coefficients, so does its Zagier lift.
scientific article; zbMATH DE number 5529338