Gerth's heuristics for a family of quadratic extensions of certain Galois number fields

TL;DR

This paper extends Gerth's heuristics and proves lower bounds for class group averages in quadratic extensions of Galois number fields, generalizing previous results and providing new insights into class group distributions.

Contribution

The paper generalizes Fouvry and Kluners' result by establishing lower bounds for class group averages over quadratic extensions of certain Galois number fields.

Findings

Established lower bounds for average class group ratios.

Extended Gerth's heuristics to broader Galois number fields.

Provided special case bounds for fields with class number 1 where 2 splits.

Abstract

Gerth generalised Cohen-Lenstra heuristics to the prime $p=2$. He conjectured that for any positive integer $m$, the limit $$ \lim_{x \to \infty} \frac{\sum_{0 < D \le X, \atop{ \text{squarefree} }} |{\rm Cl}^2_{\Q(\sqrt{D})}/{\rm Cl}^4_{\Q(\sqrt{D})}|^m}{\sum_{0 < D \le X, \atop{ \text{squarefree} }} 1} $$ exists and proposed a value for the limit. Gerth's conjecture was proved by Fouvry and Kluners in 2007. In this paper, we generalize their result by obtaining lower bounds for the average value of $|{\rm Cl}^2_{\L}/{\rm Cl}^4_{\L}|^m$, where $\L$ varies over an infinite family of quadratic extensions of certain Galois number fields. As a special case of our theorem, we obtain lower bounds for the average value when the base field is any Galois number field with class number $1$ in which $2\Z$ splits.