Journal of the Mathematical Society of Japan

Zeta functions of ${\mathbb F}_1$-buildings

Anton DEITMAR and Ming-Hsuan KANG

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The analogue of the Bruhat–Tits building of a $p$-adic group in F1-geometry is a single apartment. In this setting, the trace formula gives rise to a several variable zeta function analogously to the $p$-adic case. The analogy carries on to the fact that the restriction to certain lines yield zeta functions which are defined in geometrical terms. Also, the classical formula of Ihara has an analogue in this setting.

Article information

J. Math. Soc. Japan, Volume 68, Number 2 (2016), 807-822.

First available in Project Euclid: 15 April 2016

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Zentralblatt MATH identifier

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 11F25: Hecke-Petersson operators, differential operators (one variable) 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 14G10: Zeta-functions and related questions [See also 11G40] (Birch- Swinnerton-Dyer conjecture) 20E42: Groups with a $BN$-pair; buildings [See also 51E24]

field of one element Bruhat–Tits building Ihara zeta function


DEITMAR, Anton; KANG, Ming-Hsuan. Zeta functions of ${\mathbb F}_1$-buildings. J. Math. Soc. Japan 68 (2016), no. 2, 807--822. doi:10.2969/jmsj/06820807.

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