Zeta functions and Eisenstein series on metaplectic groups

The main objectives of this paper are: (1) to construct an Euler product from a Hecke eigenform of half-integral weight; (2) to prove its meromorphic continuation to the whole plane; (3) to examine its relationship with the Fourier coefficients of the form; and (4) to apply these results to the investigations of Eisenstein series of half-integral weight. The automorphic forms are holomorphic ones, considered on the metaplectic cover M~, of G~,, where G n = Sp(n,F) with a totally real algebraic number field F. The first three are generalizations of the corresponding results in the elliptic and Hilbert modular cases obtained in [$2] and [$6]. At the same time all four closely parallel similar results in the case of integral weight in [$9] and [S10]. To describe our results, let us take for simplicity F to be Q, in which case the weight k is half an odd positive integer. We consider a holomorphic form f of weight k, with respect to a principal congruence subgroup of Sp(n,Z) of level N, which is an eigenform in the sense that f iT(m) = 2 ( m ) f with a complex number 2(m), where N is a multiple of 4 and T(m) is a Hecke operator defined for each positive integer m in a natural way. As our first main result we shall prove (Theorem 4.4) that