Arakelov class groups of random number fields

TL;DR

This paper develops a probabilistic model for Arakelov class groups of number fields, providing counterexamples to existing heuristics and revealing new insights into their Galois module structures.

Contribution

It introduces a new probabilistic framework for Arakelov class groups and constructs infinite counterexamples to Cohen--Lenstra--Martinet heuristic with non-abelian Galois groups.

Findings

Counterexamples with non-abelian Galois groups

Restrictions on Galois module structures from Omega(3) conjecture

Refinement of heuristic models for class groups

Abstract

The main purpose of the paper is to formulate a probabilistic model for Arakelov class groups in families of number fields, offering a correction to the Cohen--Lenstra--Martinet heuristic on ideal class groups. To that end, we show that Chinburg's Omega(3) conjecture implies tight restrictions on the Galois module structure of oriented Arakelov class groups. As a consequence, we construct a new infinite series of counterexamples to the Cohen--Lenstra--Martinet heuristic, which have the novel feature that their Galois groups are non-abelian.