On some Euler products. II (Q1065865)

The result on Euler products alluded to in the previous note by the author [same volume, 335-338 (1984; Zbl 0559.12009)] is formulated in this note. The Euler products are defined in the following situation. Let be given a set P provided with a ''norm'' in a certain sense (think at the set of primes of an algebraic number field) and a map \(\alpha\) from P into some group G (think of p \(\to\) Frobenius automorphism). Then to any polynomial H(g,T) in T whose coefficients are functions of \(g\in G\), supposed to be virtual characters, and congruent to 1 modulo T is assigned the Euler product with general factor H(\(\alpha\) (p), N(p)\({}^{-s})^{-1}\). Now the author gives a sufficient condition on the datum (P,G,\(\alpha)\) in order to have the following. The Euler product corresponding to H has a meromorphic continuation to \({\mathbb{C}}\) if H is of the form \(H(g,T)=\det (1-M(g)T)\), M(g) a unitary matrix, and otherwise the Euler product has \(Re(s)=0\) as a natural boundary. The condition consists of certain properties of the L-function obtained by taking \(H(g,T)=\det (1-\rho (g)T)\), \(\rho\) an irreducible finite dimensional unitary representation of G. An idea of the proof is given, and two examples.