Convergence of zeta functions on symplectic and metaplectic groups (Q1919628)

Let \(F\) be a totally real algebraic number field, and let \(f\) be a Hecke eigenform of integral or half-integral weight with respect to a congruence subgroup of \(G^n= \text{Sp} (n, F)\). The form \(f\) can be considered on \(G^n_\mathbb{A}\) or on the metaplectic cover \(M^n_\mathbb{A}\) of \(G^n_\mathbb{A}\), and the associated zeta function has an Euler expansion of the form \(Z(s)= \prod_{\mathfrak p} W_{\mathfrak p} (N({\mathfrak p})^{-s} )^{-1}\), where \({\mathfrak p}\) runs over all primes of \(F\), and except for finitely many \({\mathfrak p}\)'s \(W_{\mathfrak p}\) is a polynomial of degree \(2n+1\) or \(2n\) according to whether the weight is integral or half-integral. In this paper, the author proves that the zeta function \(Z(s)\) is absolutely convergent for \(\text{Re} (s)> (3n/ 2)+1\), that it has a pole at \((3n/ 2)+1\) only if the weight of \(f\) is relatively small, and that it is integral or half-integral according to whether \(n\) is even or odd. As an application, he proves that the space of holomorphic modular forms with respect to a congruence subgroup of \(G^n\) is spanned by cusp forms and Eisenstein series if the weight is not too small.