Distribution of the bad part of class groups

TL;DR

This work probes the distribution of class groups in families of number fields beyond the standard Cohen–Lenstra–Martinet heuristics by focusing on ramification-driven structure. It exploits genus theory and the invariant part of the class group to connect ramified primes to non-random components of $ ext{Cl}_K[p^ obreak o]$, and uses Dirichlet-series techniques and Tauberian arguments to translate ramification data into field-counting asymptotics. The results are unconditional for abelian and $D_4$-extensions and conditional for general G-extensions, highlighting regimes where the heuristics fail (non-good primes) and where they still predict behavior (good primes). The paper culminates in concrete infinite-moment phenomena for $2$-torsion in $D_4$-fields and provides asymptotic counting bounds for abelian and dihedral families under standard heuristics. These findings have implications for understanding the true statistical landscape of class groups and for refining probabilistic models in algebraic number theory.

Abstract

The Cohen-Lenstra-Martinet Heuristics gives a prediction of the distribution of $\operatorname{Cl}_K[p^\infty]$ whne $K$ runs over $Γ$-fields and $p\nmid|Γ|$. In this paper, we prove several results on the distribution of ideal class groups for some $p||Γ|$, and show that the behaviour is qualitatively different than what is predicted by the heuristics when $p\nmid|Γ|$.We do this by using genus theory and the invariant part of the class group to investigate the algebraic structure of the class group. For general number fields, our result is conditional on a natural conjecture on counting fields. For abelian or $D_4$-fields, our result is unconditional.

Theorems & Definitions (71)

• Definition 1.1
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• Theorem 1.11