On the Fourier coefficients of Hilbert modular forms of half-integral weight over arbitrary algebraic number fields (Q357437)

This present article is one of a series of the author's works on an explicit version of Waldspurger-type formula for Hilbert-Maass modular forms of half-integral weight.NEWLINENEWLINEFor a modular form \(f\) of integral weight, \textit{J. L. Waldspurger} [J. Math. Pures Appl. (9) 60, 375--484 (1981; Zbl 0431.10015)] proved a proportionality relation between the central values of quadratic twists of \(L\)-function of \(f\) and the squares of Fourier coefficients of a modular form of half-integral weight corresponding to \(f\) by the Shimura correspondence. However, he did not give the coefficient of proportionality explicitly.NEWLINENEWLINEFor elliptic modular forms of level one, \textit{W. Kohnen} and \textit{D. Zagier} [Invent. Math. 64, 175--198 (1981; Zbl 0468.10015)] gave the constant explicitly by using theory of plus-spaces. \textit{W. Kohnen} [Math. Ann. 271, 237--268 (1985; Zbl 0542.10018); Glasg. Math. J. 30, No. 3, 285--291 (1988; Zbl 0659.10024)] generalized this formula to general level.NEWLINENEWLINEIn another direction, assuming multiplicity one property, \textit{G. Shimura} [Duke Math. J. 71, No. 2, 501--557 (1993; Zbl 0802.11017)] gave the explicit constant for Hilbert modular forms of half-integral weight. \textit{K. Khuri-Makdisi} [Duke Math. J. 84, No. 2, 399--452 (1996; Zbl 0859.11031)] generalized this theorem to non-holomorphic Hilbert modular forms (so-called Maass wave forms) of half-integral weight under multiplicity one assumption. By a similar method, the author of the present article obtained before an explicit constant for Maass wave forms of half-integral weight over imaginary quadratic fields [J. Reine Angew. Math. 526, 155--179 (2000; Zbl 0987.11033)]. NEWLINENEWLINEHere, Maass wave forms of half-integral weight over imaginary quadratic fields mean non-holomorphic adelic forms on the metaplectic group over the imaginary quadratic fields. The author generalized the explicit formula to the Hilbert-Maass modular forms of half-integral weight over arbitrary number fields [J. Number Theory 107, No. 1, 25--62 (2004; Zbl 1122.11029)].NEWLINENEWLINEThe present paper provides a correction of the miscalculations in Theorem 2.5 in [Zbl 1122.11029] and to give a more useful form (Theorem 1) of the main theorem in [Zbl 1122.11029]. The useful formula in Theorem 1 is a modification of the main theorem in [Zbl 1122.11029] by the product of Fourier coefficients of the given modular form \(f\) and its inversion.