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

Abstract

In this paper, we study relations between Langlands L-functions and zeta functions of geodesic walks and galleries for finite quotients of the apartments of G=PGL3 and PGSp4 over a nonarchimedean local field with q elements in its residue field. They give rise to an identity (Theorem 5.3) which can be regarded as a generalization of Ihara's theorem for finite quotients of the Bruhat-Tits trees. This identity is shown to agree with the q=1 version of the analogous identities for finite quotients of the building of G established in (KL1, KLW, FLW), verifying the philosophy of the field with one element by Tits. A new identity for finite quotients of the building of PGSp4 involving the standard $L$-function (Theorem 6.3), complementing the one in (FLW) which involves the spin L-function, is also obtained.