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arXiv · 2605.01398

Bowen--Franks groups and minus class groups of cyclotomic number fields with prime conductor

Antonio Lei·Katharina Müller·Daniel Vallières
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Abstract

Let $p$ be an odd rational prime and consider the cyclotomic number field $K = \mathbb{Q}(\zeta_{p})$ of conductor $p$. We construct a directed graph $Y$ on $p-1$ vertices for which the torsion part of the corresponding Bowen--Franks group is closely related to the minus part of the class group of $K$. In particular, both groups have the same cardinality up to an explicit power of $p$. Furthermore, they are both $\mathrm{Gal}(K/\mathbb{Q})$-modules, and we prove the equality of the cardinalities of their isotypic components after tensoring them with the valuation ring of an appropriate $\ell$-adic field for $\ell \nmid p-1$.

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Antonio Lei, Katharina Müller, Daniel Vallières. 2026-05-02. Bowen--Franks groups and minus class groups of cyclotomic number fields with prime conductor. https://arxiv.org/abs/2605.01398

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