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

TL;DR

This paper studies zeta functions of complexes derived from $ ext{Sp}(4)$ over non-archimedean fields, providing explicit formulas and characterizations of Ramanujan complexes via representation theory and the Riemann Hypothesis.

Contribution

It introduces two explicit rational formulas for the zeta function of complexes from $ ext{Sp}(4)$ and characterizes Ramanujan complexes through adjacency operators and the Riemann Hypothesis.

Findings

Derived explicit formulas for the zeta function as rational functions.

Characterized Ramanujan complexes via adjacency operators.

Established the connection between zeta functions and the Riemann Hypothesis.

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.