Davenport-Heilbronn Theorems for Quotients of Class Groups

TL;DR

This paper generalizes the Davenport-Heilbronn theorem to quotients of class groups of quadratic fields and computes average sizes of related algebraic structures across quadratic fields ordered by discriminant.

Contribution

It extends the Davenport-Heilbronn theorem to quotients of class groups by primes in a fixed set and calculates average sizes of Selmer groups and unit groups for quadratic fields.

Findings

Generalized Davenport-Heilbronn theorem for class group quotients

Computed average sizes of Selmer groups $ ext{Sel}_3^S(K)$

Determined average sizes of unit groups $ ext{O}_{K,S}^ imes/( ext{O}_{K,S}^ imes)^3$

Abstract

We prove a generalization of the Davenport-Heilbronn theorem to quotients of ideal class groups of quadratic fields by the primes lying above a fixed set of rational primes $S$. Additionally, we obtain average sizes for the relaxed Selmer group $\mathrm{Sel}_3^S(K)$ and for $\mathcal{O}_{K,S}^\times/(\mathcal{O}_{K,S}^\times)^3$ as $K$ varies among quadratic fields with a fixed signature ordered by discriminant.