The Zeta Functions of Complexes from $\Sp(4)$

Abstract

Let $F$ be a non-archimedean local field with a finite residue field. To a 2-dimensional finite complex $X_\Gamma$ arising as the quotient of the Bruhat-Tits building $X$ associated to $\Sp_4(F)$ by a discrete torsion-free cocompact subgroup $\Gamma$ of $\PGSp_4(F)$, associate the zeta function $Z(X_{\Gamma}, u)$ which counts geodesic tailless cycles contained in the 1-skeleton of $X_{\Gamma}$. Using a representation-theoretic approach, we obtain two closed form expressions for $Z(X_{\Gamma}, u)$ as a rational function in $u$. Equivalent statements for $X_{\Gamma}$ being a Ramanujan complex are given in terms of vertex, edge, and chamber adjacency operators, respectively. The zeta functions of such Ramanujan complexes are distinguished by satisfying the Riemann Hypothesis.