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.
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The research of the first author is supported by the NSC grant 103-2115-M-009-006. The research of the second author is partially supported by the NSF grant DMS-1101368 and the Simons Foundation grant # 355798. The research of the third author is supported by the NSC grant 103-2115-M-032-001. Part of the research was carried out when the authors visited the National Center for Theoretical Sciences in Hsinchu, Taiwan. They would like to thank NCTS for its hospitality and support.
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Kang, MH., Li, WC.W. & Wang, CJ. Zeta and L-functions of finite quotients of apartments and buildings. Isr. J. Math. 228, 79–117 (2018). https://doi.org/10.1007/s11856-018-1756-3
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DOI: https://doi.org/10.1007/s11856-018-1756-3