Zeta Functions of Complexes Arising from PGL(3)

Abstract

In this paper we obtain a closed form expression of the zeta function $Z(X_\Gamma, u)$ of a finite quotient $X_\Gamma = \Gamma \backslash PGL_3(F)/PGL_3(O_F)$ of the Bruhat-Tits building of $PGL_3$ over a nonarchimedean local field $F$. Analogous to a graph zeta function, $Z(X_\Gamma, u)$ is a rational function and it satisfies the Riemann hypothesis if and only if $X_\Gamma$ is a Ramanujan complex.