Zeta Functions of Complexes Arising from PGL(3)

TL;DR

This paper derives a closed-form expression for the zeta function of finite quotients of the Bruhat-Tits building associated with PGL(3), linking its properties to Ramanujan complexes and the Riemann hypothesis.

Contribution

It provides the first explicit formula for the zeta function of PGL(3) complexes and connects its rationality and Riemann hypothesis to Ramanujan properties.

Findings

Zeta function is rational for these complexes.

Riemann hypothesis holds iff the complex is Ramanujan.

Explicit formula enables further spectral analysis.

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.