Zeta functions of groups: Zeros and friendly ghosts (Q2784128)

The authors consider the global zeta function defined as the Euler product of \(W(p,p^{-s})\) over all primes \(p\), where \(W(X,Y)\) is a polynomial. The authors introduce the ghost polynomial of \(W(p,p^{-s})\), which is defined from the zeros of \(W(p,Y)\) as \(p\) ranges over all primes. The analytic behavior of the global zeta function is controlled by the ghost polynomials. In particular, it is important that the Euler product of the ghost polynomials is meromorphic; in this case we say \(W(X,Y)\) has a friendly ghost. The main results of this paper are (i) explicit formulae of the ghosts of classical reductive groups, (ii) the ghosts of classical reductive groups are friendly. The authors also consider the case of reductive exceptional groups.