On Fourier coefficients of Maass wave forms of half integral weight belonging to Kohnen's spaces
Let \(f\) be an elliptic modular form of half-integral weight, which is a simultaneous Hecke eigenform. Let \(F\) correspond to \(f\) under the Shimura correspondence. \textit{J.-L. Waldspurger} [J. Math. Pure Appl. 60, 375-384 (1981; Zbl 0468.10014)] proved that the spares of the Fourier coefficients \(\alpha_f(n)\) at squarefree \(n\in \mathbb{N}\) are essentially proportional to the critical value of the zeta function attached to \(F\). The proportionality constant was derived later. In an earlier paper [J. Math. Soc. Japan 51, 615-730 (1999)] the author dealt with the analogous problem in some cases of modular forms \(f\) of half integral weight when the multiplicity is two. In the paper under review these results are generalized to the case of Maass wave forms of half integral weight belonging to Kohnen's space. The proof follows the same lines as in the earlier paper.
elliptic modular form of half-integral weight
scientific article; zbMATH DE number 1394756