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 XΓ 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(XΓ,u) which counts geodesic tailless cycles contained in the 1-skeleton of XΓ. Using a representation-theoretic approach, we obtain two closed-form expressions for Z(XΓ,u) as a rational function in u. Equivalent statements for XΓ 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.