On the $16$-rank of class groups of $\mathbb{Q}(\sqrt{-8p})$ for $p\equiv -1\bmod 4$

TL;DR

This paper proves that for primes p ≡ -1 mod 4, the probability that the class group of the field Q(√-8p) contains an element of order 16 is exactly 1/16, aligning with Cohen-Lenstra heuristics.

Contribution

It introduces a variant of Vinogradov's method to rigorously establish the density of primes with class groups containing elements of order 16.

Findings

Density of such primes is 1/16.

Confirms Cohen-Lenstra heuristics predictions.

Uses novel analytic number theory techniques.

Abstract

We use a variant of Vinogradov's method to show that the density of the set of prime numbers $p\equiv -1\bmod~4$ for which the class group of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-8p})$ has an element of order $16$ is equal to $1/16$, as predicted by the Cohen-Lenstra heuristics.