## The Michigan Mathematical Journal

### Tree-Lattice Zeta Functions and Class Numbers

#### Abstract

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

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

Dates
Revised: 29 October 2017
First available in Project Euclid: 20 June 2018

https://projecteuclid.org/euclid.mmj/1529460323

Digital Object Identifier
doi:10.1307/mmj/1529460323

Mathematical Reviews number (MathSciNet)
MR3835566

Zentralblatt MATH identifier
06969986

#### Citation

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. https://projecteuclid.org/euclid.mmj/1529460323

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