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

Ming Hsuan Kang, Wen Ching Winnie Li, Chian Jen Wang

Research output: Contribution to journalArticle

1 Citation (Scopus)

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 [KL14, KLW10, FLW13], 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 [FLW13] which involves the spin L-function, is also obtained.

Original languageEnglish (US)
Pages (from-to)79-117
Number of pages39
JournalIsrael Journal of Mathematics
Volume228
Issue number1
DOIs
StatePublished - Oct 1 2018

Fingerprint

L-function
Riemann zeta function
Quotient
Identity theorem
Local Field
Theorem
Walk
Geodesic
Buildings

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Kang, Ming Hsuan ; Li, Wen Ching Winnie ; Wang, Chian Jen. / Zeta and L-functions of finite quotients of apartments and buildings. In: Israel Journal of Mathematics. 2018 ; Vol. 228, No. 1. pp. 79-117.
@article{60dd7634ff304addbe1027a91979c00e,
title = "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 [KL14, KLW10, FLW13], 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 [FLW13] which involves the spin L-function, is also obtained.",
author = "Kang, {Ming Hsuan} and Li, {Wen Ching Winnie} and Wang, {Chian Jen}",
year = "2018",
month = "10",
day = "1",
doi = "10.1007/s11856-018-1756-3",
language = "English (US)",
volume = "228",
pages = "79--117",
journal = "Israel Journal of Mathematics",
issn = "0021-2172",
publisher = "Springer New York",
number = "1",

}

Zeta and L-functions of finite quotients of apartments and buildings. / Kang, Ming Hsuan; Li, Wen Ching Winnie; Wang, Chian Jen.

In: Israel Journal of Mathematics, Vol. 228, No. 1, 01.10.2018, p. 79-117.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Zeta and L-functions of finite quotients of apartments and buildings

AU - Kang, Ming Hsuan

AU - Li, Wen Ching Winnie

AU - Wang, Chian Jen

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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 [KL14, KLW10, FLW13], 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 [FLW13] which involves the spin L-function, is also obtained.

AB - 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 [KL14, KLW10, FLW13], 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 [FLW13] which involves the spin L-function, is also obtained.

UR - http://www.scopus.com/inward/record.url?scp=85051220207&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85051220207&partnerID=8YFLogxK

U2 - 10.1007/s11856-018-1756-3

DO - 10.1007/s11856-018-1756-3

M3 - Article

AN - SCOPUS:85051220207

VL - 228

SP - 79

EP - 117

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -