A heuristic for the distribution of ray class groups of number fields

Ammon, Robin (2026) A heuristic for the distribution of ray class groups of number fields. PhD thesis, University of Glasgow.

Abstract

We propose a conjecture for the distribution of the ‘good part’ of the ray class group ClK(m) of a number field K, for K running over a natural family of Galois extensions of a fixed base number field F and fixed modulus m given by an integral ideal of OF. It can be seen as a generalisation of earlier conjectures by Pagano–Sofos for the family of imaginary quadratic number fields and by Bartel–Pagano for the family of real quadratic number fields. Our conjecture is phrased in terms of the Arakelov ray class sequence of a number field introduced by Bartel–Pagano and postulates that the ‘good part’ of the latter behaves randomly in the sense of Cohen–Lenstra. To be able to state it, we develop a commensurability theory for automorphism groups of chain complexes, extending the commensurability theory of Bartel–Lenstra for automorphism groups of modules.

We show that our conjecture implies the Cohen–Lenstra–Martinet heuristics as reformulated by Bartel–Lenstra and predicts equidistribution of the reduction map O × K → (OK/m) ×. We further obtain from our conjecture a general formula for the average ℓ-torsion, ℓ a good prime, of ClK(m) in families of abelian extensions. We explicitly calculate the predicted average ℓ-torsion of ray class groups of cyclic cubic fields with fixed rational modulus for ℓ ̸= 2, 3.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
• Supervisor's Name: Bartel, Professor Alex and Sofos, Dr. Efthymios
• Date of Award: 2026
• Depositing User: Theses Team
• Unique ID: glathesis:2026-85750
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 09 Feb 2026 14:52
• Last Modified: 09 Feb 2026 14:55
• URI: https://theses.gla.ac.uk/id/eprint/85750