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

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.


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
Full Text: DOI arXiv Euclid


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