The Michigan Mathematical Journal

Tree-Lattice Zeta Functions and Class Numbers

Anton Deitmar and Ming-Hsuan Kang

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We extend the theory of Ihara zeta functions to noncompact arithmetic quotients of Bruhat–Tits trees. This new zeta function turns out to be a rational function despite the infinite-dimensional setting. In general, it has zeros and poles in contrast to the compact case. The determinant formulas of Bass and Ihara hold if we define the determinant as the limit of all finite principal minors. From this analysis we derive a prime geodesic theorem, which, applied to special arithmetic groups, yields new asymptotic assertions on class numbers of orders in global fields.

Article information

Michigan Math. J., Volume 67, Issue 3 (2018), 617-645.

Received: 9 December 2016
Revised: 29 October 2017
First available in Project Euclid: 20 June 2018

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Zentralblatt MATH identifier

Primary: 20E08: Groups acting on trees [See also 20F65]
Secondary: 11N38 11N05: Distribution of primes 11R29: Class numbers, class groups, discriminants 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 22D05: General properties and structure of locally compact groups 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 57M07: Topological methods in group theory


Deitmar, Anton; Kang, Ming-Hsuan. Tree-Lattice Zeta Functions and Class Numbers. Michigan Math. J. 67 (2018), no. 3, 617--645. doi:10.1307/mmj/1529460323.

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