On the Fourier coefficients of nonholomorphic Hilbert modular forms of half-integral weight

Let f(z) = ∑ n≥1 ane 2πinz be a Hecke eigenform of half-integral weight m+1/2, and let g(z) = ∑ n≥1 bne 2πinz be the corresponding even-weight form, in the sense of [Sh 73]. In particular, g has weight 2m, and belongs to the same eigenvalues of Hecke operators as f . If n = qr with squarefree r, then an is expressible in terms of ar and the {bj}. At the end of [Sh 77], Shimura suggested that ar should be related to special values of Dirichlet series associated to g. This was borne out in [Wa 81], where Waldspurger proved the striking relation that for squarefree r, ar is essentially proportional to ∑ n≥1 φr(n)bnn −s ∣∣ s=m . Here we have twisted the standard Dirichlet series for g by a character φr obtained from the character of f and the quadratic character ( r · ) . The purpose of this paper is to derive generalizations of Waldspurger’s relation, with an explicit proportionality constant, in the case where f and g are nonholomorphic Hilbert modular forms over a totally real number field F . If F = Q, such forms are also called Maass forms. The method of proof, which ought to generalize to arbitrary number fields, follows, with some simplifications, that in [Sh 93a], which treats the case of holomorphic Hilbert modular forms. Previous investigations into this topic have included work by Kohnen and Zagier ([Ko-Za 81] and [Ko 85]) in the holomorphic case, and Katok and Sarnak ([Ka-Sa 93]) in the nonholomorphic case. Both of these treatments deal only with forms on the upper half-plane (i.e. F = Q), with some additional restrictions. Recent (not yet published) work of M. Furusawa suggests that the method in [Sh 93a] and in this paper should generalize to yield a similar formula, in the case of the correspondence between automorphic forms on Sp(n) and on O(2n+ 1). Extending Shimura’s work in [Sh 93a] to the nonholomorphic case involves two main difficulties. First, as the Fourier expansions of Maass forms involve Whittaker functions instead of exponentials, Mellin transforms and Rankin-Selberg convolutions produce more complicated “Gamma-factors” than usual; these factors must be explicitly evaluated, in order to yield precise versions of Waldspurger’s relation. Second, whereas the Fourier expansions of holomorphic forms are indexed only by totally positive elements of the field F , the expansions of Maass forms are indexed by field elements of arbitrary signature; this makes the calculations rather more delicate. Section 3 of this paper explains in explicit detail how one overcomes both of these problems in constructing a Dirichlet series from Hilbert modular forms of arbitrary integral weight. The results in section 3 are in principle known from [Ma 53], [J-L], and [W], but are not found in this form in the literature (see the discussion at the beginning of section 3). This applies particularly to the appendix to section 3, where we explicitly compute the Mellin transforms of all possible Whittaker functions at the archimedean places. Sections 1 through 4 contain nothing new, but rather set up precise definitions and normalizations of all automorphic forms, special functions, and parameters