A heuristic for ray class groups of quadratic number fields

Alex Bartel; Carlo Pagano

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arXiv · 2509.20185

A heuristic for ray class groups of quadratic number fields

Alex Bartel·Carlo Pagano

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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.

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Alex Bartel, Carlo Pagano. 2025-09-24. A heuristic for ray class groups of quadratic number fields. https://arxiv.org/abs/2509.20185

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