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 Γ arising as the quotient of the Bruhat-Tits building X associated to Sp 4 (F) by a discrete torsion-free cocompact subgroup Γ of PGSp 4 (F), associate the zeta function Z(X Γ , u) which counts geodesic tailless cycles contained in the 1-skeleton of X Γ . Using a representationtheoretic 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.

Zeta and L-functions of finite quotients of apartments and buildings

“…Denote by O its ring of integers and ̟ a fixed uniformizer. Zeta functions counting closed geodesics contained in the 1-skeleton of finite quotient complexes arising from the Bruhat-Tits buildings of PGL 3 (F ) and PGSp 4 (F ) have been studied in [KLW10, KL14,FLW13]. Such a zeta function is expressed in two ways combinatorially: one in terms of edge adjacency operators, and the other in terms of vertex adjacency operators and the chamber adjacency operator.…”

“…Special focus is on the groups G =PGL 3 and PGSp 4 . We develop a unified approach to obtain an identity extending Ihara's theorem for such G in the apartment case (Theorem 5.3), and compare them with the results established in [KL14, KLW10,FLW13] for the building of G.…”

“…1 groups. Little was known until the recent work [KL14,KLW10] and [FLW13], which generalize Ihara's result to certain split algebraic groups of rank two, namely PGL 3 and PGSp 4 , respectively. For a rank two group, in addition to the zeta function of geodesic walks as in a rank one group, one also considers the zeta function of geodesic galleries.…”

“…For algebraic groups of higher rank, as revealed in [KL14, KLW10,FLW13], the situation is a lot more complicated. It could be illuminating to first consider the degenerate case following the philosophy of the field with one element introduced by Tits.…”

Twisted Poincaré series and zeta functions on finite quotients of buildings

“…In general, one can obtain a similar identity by comparing eigenvalues of proper parahoric operators using representation theory. See [KL14] and [FLW13] for the case of PGL 3 and GSP 4 , respectively, from the view point of this approach.…”