The zeta functions of complexes from Sp(4)

Let F be a nonarchimedean local field with a finite residue field. To a two-dimensional finite complex X_Γ arising as the quotient of the Bruhat-Tits building X associated to Sp₄(F) by a discrete torsion-free co-compact subgroup Γ of PGSp₄(F), associate the zeta function Z(X_Γ,u) which counts geodesic tailless cycles contained in the 1-skeleton of X_Γ. Using a representation- theoretic approach, we obtain two closed-form expressions for Z(X _Γ,u) as a rational function in u. Equivalent statements for X_Γ 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.