INTRODUCTION Our ultimate aim is to prove several theorems of arithmeticity on the values of an Euler product Z(s) and an Eisenstein series E(z, s) at certain critical points s. We take these Z and E to be those of the types we treated in our previous book "Euler Products and Eisenstein Series," referred to as [S97] here. They are defined with respect to an algebraic group G, which is either symplectic or unitary. To illustrate the nature of our problems, let us take a CM-field K and put (0.1) G{p) =G* = {ae GLn{K) \ ap 'a ' = p}9 where p denotes complex conjugation and ip is an element of GLn(K) such that tpP = ip. This group acts on a hermitian symmetric space which we write 3^- We shall often be interested in the special case where if takes the form (0.2) ^ = [ ° o']- In this case we write Hq, or simply W, instead of 3 ^ for the symmetric space. Given a congruence subgroup r of G, a Hecke eigenform f of holomorphic type on y with respect to I"1, and a Hecke character \ of K of algebraic type, but not necessarily of finite order, we can construct a "twisted Euler product" Z(s, f, x) whose generic Euler p-factor for each rational prime p has degree n[K : Q]. Then we shall eventually prove that (0.3) Z K , f , x ) G T £ q ( f , f ) Q for T o in a certain finite subset of 2 _ 1 Z and Q-rational f. Here (f, f) is the inner product defined in a canonical way e is an integer determined by cro, the signature of y?, the weight of f, and the archimedean factor of \\ q is a certain "period symbol" determined by \ and (p. This is true for both isotropic and anisotropic p, and even for a totally definite cp. In the simplest case in which G = G1, we have q = 1. Clearly such a result requires the definition of Q-rationality of automorphic forms. If G is of type G77, then we can define the Q-rationality by the Q-rationality of the Fourier coefficients of a given automorphic form. If [K : Q] = 2, for example, then H is a tube domain of the form H = { z C^ \i{z* z) 0} , and a holomorphic automorphic form / has an expansion (0.4) f(z) = Zhh)exp(2m-tT(hz)) (zeH) with c(h) C, where h runs over all nonnegative hermitian matrices belonging to a Z-lattice in K%. Then for a subfield M of C we say that / is M-rational if c(h) e M for every h. This definition may look simplistic, but actually it is intrinsically the 1
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