On the 8-rank of narrow class groups of ℚ(−4pq), ℚ(−8pq), and ℚ(8pq)

Abstract

Let [Formula: see text]. We study the [Formula: see text]-part of the narrow class group in thin families of quadratic number fields of the form [Formula: see text], where [Formula: see text] 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 [Formula: see text]. In the course of our proof, we prove a general double-oscillation estimate for the quadratic residue symbol in quadratic number fields.