On the $16$-rank of class groups of $\mathbb{Q}(\sqrt{-3p})$ for primes $p$ congruent to $1$ modulo $4$

TL;DR

This paper investigates the distribution of primes p congruent to 1 mod 4 for which the class group of the quadratic field Q(√-qp) has an order divisible by 16, confirming a specific density prediction.

Contribution

It proves that for fixed q in {3,7,11,19,43,67,163}, the density of such primes p is exactly 1/8, supporting Gerth's conjecture using Vinogradov's method.

Findings

Density of primes p with class group divisible by 16 is 1/8.

Supports a general conjecture of Gerth on class group distributions.

Employs Vinogradov's method as the main analytic tool.

Abstract

For fixed $q\in\{3,7,11,19, 43,67,163\}$, we consider the density of primes $p$ congruent to $1$ modulo $4$ such that the class group of the number field $\mathbb{Q}(\sqrt{-qp})$ has order divisible by $16$. We show that this density is equal to $1/8$, in line with a more general conjecture of Gerth. Vinogradov's method is the key analytic tool for our work.