Zeta functions of \({\mathbb F}_1\)-buildings. (English) Zbl 1377.11062

Deitmar, Anton; Kang, Ming-Hsuan

J. Math. Soc. Japan 68, No. 2, 807-822 (2016).

A field of one element \({\mathbb Q}_1\) is the quotient group of a free monoid with one generator. The paper considers the group \(\mathrm{PGL}_n({\mathbb Q}_1)\) and, in particular, the Ihara zeta function associated to its Bruhat-Tits building. It is shown that the inverse of zeta function converges to a polynomial, and the analogue of Ihara’s formula which equates the zeta function to a product of (the analogue of) Langlands \(L\)-functions also holds.

Reviewer: Zhengyu Mao (Newark)

MSC:

11F70	Representation-theoretic methods; automorphic representations over local and global fields
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 in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
20E42	Groups with a \(BN\)-pair; buildings

Keywords:

field of one element; Bruhat-Tits building; Ihara zeta function

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