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

2015/05/05 by Ming-Hsuan Kang, Wen-Ching Winnie Li, Kang, Ming-Hsuan +3 · 2 citations
Chemistry · Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Molecular spectroscopy and chirality #Number Theory (math.NT) #Representation Theory (math.RT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1505.00902

openalex publication_date 2015/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Zeta Functions of F1-buildings

Cited by

• Chamber zeta function and closed galleries in the standard non-uniform complex from PGL₃
• Twisted Poincare Series and Zeta functions on finite quotients of buildings

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