Iwasawa theory for branched $\mathbb{Z}_{p}$-towers of finite graphs and Ihara zeta and $L$-functions

Rusiru Gambheera; Daniel Vallières

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arXiv · 2508.07373

Iwasawa theory for branched $\mathbb{Z}_{p}$-towers of finite graphs and Ihara zeta and $L$-functions

Rusiru Gambheera·Daniel Vallières

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Abstract

We revisit the theory of Ihara $L$-functions in the context initially studied by Bass and Hashimoto and more recently by Zakharov. In particular, we study if the Artin formalism is satisfied by these $L$-functions. As an application, we give a proof of the analogue of Iwasawa's asymptotic class number formula for the $p$-part of the number of spanning trees in branched $\mathbb{Z}_{p}$-towers of finite connected graphs using Ihara zeta and $L$-functions. Moreover, we relate a generator for the characteristic ideal of the finitely generated torsion Iwasawa module that governs the growth of the $p$-part of the number of spanning trees in such towers with Ihara $L$-functions.

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Rusiru Gambheera, Daniel Vallières. 2025-08-10. Iwasawa theory for branched $\mathbb{Z}_{p}$-towers of finite graphs and Ihara zeta and $L$-functions. https://arxiv.org/abs/2508.07373

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