On the $8$-rank of narrow class groups of $\mathbb{Q}(\sqrt{-4pq})$, $\mathbb{Q}(\sqrt{-8pq})$, and $\mathbb{Q}(\sqrt{8pq})$

TL;DR

This paper investigates the 8-rank of narrow class groups in specific quadratic fields formed by products of primes, establishing new lower bounds and introducing a double-oscillation estimate for quadratic residue symbols.

Contribution

It provides new lower bounds for the proportion of quadratic fields with narrow class groups containing elements of order 8, and proves a general double-oscillation estimate for quadratic residue symbols.

Findings

Established lower bounds for 8-rank in narrow class groups.

Proved a double-oscillation estimate for quadratic residue symbols.

Analyzed families of quadratic fields of the form Q(√dpq) with primes p, q.

Abstract

Let $d \in \{-4, -8, 8\}$. We study the $8$-part of the narrow class group in the thin families of quadratic number fields of the form $\mathbb{Q}(\sqrt{dpq})$, where $p\equiv q \equiv 1\bmod 4$ are prime numbers, and we prove new lower bounds for the proportion of narrow class groups in these families that have an element of order $8$. In the course of our proof, we prove a general double-oscillation estimate for the quadratic residue symbol in quadratic number fields.