A heuristic for ray class groups of quadratic number fields

2025/09/24 by Bartel, Alex, Pagano, Carlo
#11R11 #11R29 #11R65 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.20185

Abstract

We formulate a model for the average behaviour of ray class groups of real quadratic fields with respect to a fixed rational modulus, locally at a finite set S of odd primes. To that end, we introduce Arakelov ray class groups of a number field, and postulate that, locally at S, the Arakelov ray class groups of real quadratic fields are distributed randomly with respect to a natural Cohen--Lenstra type probability measure. We show that our heuristics imply the Cohen--Lenstra heuristics on class groups of real quadratic fields, as well as equidistribution results on the fundamental unit of a real quadratic field modulo an integer, and are consistent with Varma's results on average of sizes of 3-torsion subgroups of ray class groups of quardratic fields.

Citations

• 4-ranks and the general model for statistics of ray class groups of imaginary quadratic number fields
• The mean number of 3-torsion elements in ray class groups of quadratic fields
• Computing Arakelov class groups

Related

• 4-ranks and the general model for statistics of ray class groups of imaginary quadratic number fields
• Conjectures on the distribution behavior of the class numbers of certain real quadratic number fields
• The mean number of 3-torsion elements in ray class groups of quadratic fields
• Statistics of K-groups modulo p for the ring of integers of a varying quadratic number field
• Heuristics for p-class towers of imaginary quadratic fields, with an Appendix by Jonathan Blackhurst