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

TL;DR

This paper links the torsion part of Bowen--Franks groups derived from a graph to the minus part of class groups in cyclotomic fields, revealing structural similarities and module properties.

Contribution

It constructs a graph-based approach to relate Bowen--Franks groups to minus class groups of cyclotomic fields, establishing their cardinality and module isomorphism properties.

Findings

The torsion part of the Bowen--Franks group matches the minus class group size up to a power of p.

Both groups are Galois modules with equal isotypic component sizes after tensoring.

A graph on p-1 vertices encodes the relationship between these algebraic structures.

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 $\ell \nmid p-1$.