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

TL;DR

This paper explores the relationships between Langlands L-functions and zeta functions of geodesic walks in finite quotients of apartments and buildings, generalizing Ihara's theorem and verifying the field with one element philosophy.

Contribution

It introduces new identities linking L-functions and zeta functions for finite quotients of apartments and buildings, extending previous results and including a novel identity for PGSp4.

Findings

Established a generalized Ihara's theorem for finite quotients of apartments.

Derived a new identity involving the standard L-function for PGSp4.

Verified the field with one element philosophy in the context of these identities.

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.