Remark on Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces. II
In [J. Math. Pures Appl. 60, 375--484 (1981; Zbl 0431.10015)] \textit{J.-L. Waldspurger} first found that the square of Fourier coefficients \(a(n)\) at a square-free integer \(n\) of modular forms \(f(z) =\sum^\infty_{n=1} a(n)e[nz]\) of half integral weight is essentially proportional to the central value of the zeta function at a certain integer attached to the modular form \(F\) if \(f\) corresponds to \(F\) by the Shimura correspondence \(\Psi\) and \(f\) is an eigenfunction of Hecke operators. \textit{W. Kohnen} and \textit{D. Zagier} [Invent. Math. 64, 175--198 (1981; Zbl 0468.10015), Math. Ann. 271, 237--268 (1985; Zbl 0542.10018)] determined explicitly the constant of the proportionality in the case of modular forms belonging to Kohnen's spaces \(S_{(2k+1)/2}(N,\chi)\) of weight \((2k+1)/2\) and of square-free level \(N\) with character \(\chi\) which is a subspace of \(S_{(2k+1)/2}(4N,\chi_1)\), where \(S_{(2k+1)/2}(4N,\chi_1)\) means the space of modular cusp forms of half integral weight given in [\textit{G. Shimura}, Ann. Math. (2) 97, 440--481 (1973; Zbl 0266.10022)]. Kohnen-Zagier treated the case where \(N =1\) (resp. \(N\) is an odd square-free integer and \(\chi\) is the trivial character of level \(N\)). In [Duke Math. J. 71, 501--557 (1993; Zbl 0802.11017)], \textit{G. Shimura} intended to generalize such formulas to the case of Hilbert modular forms \(f\) of half integral weight and succeeded in obtaining many general interesting formulas. Among these, some explicit and useful formulas about the proportionality constant were formulated under the assumption that \(f\) satisfies the multiplicity one theorem. For modular forms belonging to Kohnen's spaces \(S_{(2k+1)/2}(N,\chi)\), Shimura did not give the same explicit formula as that of Kohnen and Zagier. In Part I [see Zbl 1012.11035 below)] the author represented explicitly the ratio of the square of Fourier coefficients \(a(n)\) at a fundamental discriminant \(n\) of modular forms \[ f(z)=\sum_{\varepsilon(-1)^kn\equiv 0,1(4),n>0} a(n)e[nz] \] of weight \((2k+1)/2\) belonging to the Kohnen's space \(S_{(2k+1)/2}(N,\chi)\) by the central value of the zeta function of the modular form \(F\) which is the image of \(f\) under the Shimura correspondence \(\Psi\), where \(\chi\) is any primitive character modulo \(N\) and \(n\) is equal to \(4\tau\) with a square-free integer \(\tau\) satisfying \(\tau\equiv 2,3\pmod 4\). The purpose of Part II is to remove these conditions about \(\chi\) and \(n\) and to derive a generalization of results in Part I in the case where \(N\) is any odd integer, \(\chi\) is an arbitrary primitive character modulo \(N\) and \(n\) is an arbitrary positive fundamental discriminant. Though the method of our proof is more complicated, it is the same as that of Part I. To obtain these results the author needs to modify slightly its method. It is mentioned that the results give a generalization of some results in Kohnen-Zagier and Kojima and the method can be applied to the case of Maass wave forms of half integral weight.
modular cusp forms of half integral weight
scientific article; zbMATH DE number 1333302
scientific article