Zeta functions of groups: Zeros and friendly ghosts - du Sautoy, M; Grunewald, F (2002)
We consider zeta functions defined as Euler products of W(p,p-s over all primes p where W(X, Y) is a polynomial. Such zeta functions arise naturally in the context of zeta functions of algebraic groups and nilpotent groups. The analytic behavior of such Euler products is controlled by something we call the ghost polynomial of W(X, Y). The ghost is defined from the zeros of W(p, Y) as p ranges over all primes. When the Euler product of the ghost is meromorphic on ℂ, an important property in relation to the analytic behavior of the original zeta function, we say that W(X, Y) has a friendly ghost. We give an explicit expression for the ghosts of classical reductive groups and prove that they are friendly. We also consider the ghosts of exceptional reductive groups and zeta functions of nilpotent groups. The concept of the ghost implies a new dichotomy for the representations of an algebraic group according to whether their associated zeta function has a friendly ghost or not.