%0 Journal Article
%T The Zeta Functions of Complexes from $\Sp(4)$
%A Yang Fang
%A Wen-Ching Winnie Li
%A Chian-Jen Wang
%J Mathematics
%D 2011
%I arXiv
%X 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.
%U http://arxiv.org/abs/1109.3854v3