The infinitude of $\mathbb{Q}(\sqrt{-p})$ with class number divisible by $16$

TL;DR

This paper investigates the divisibility of class numbers of imaginary quadratic fields by powers of two, proving the infinitude of primes with class number divisible by 16 and describing the structure of related Hilbert class fields.

Contribution

It establishes the existence of infinitely many primes p with class number divisible by 16 and characterizes the 8-Hilbert class field for primes of specific form p = a^2 + c^4.

Findings

Infinitely many primes p with class number divisible by 16.

Existence of infinitely many primes p with class number divisible by 8 but not by 16.

Description of the 8-Hilbert class field for primes p = a^2 + c^4 with c even.

Abstract

The density of primes $p$ such that the class number $h$ of $\mathbb{Q}(\sqrt{-p})$ is divisible by $2^k$ is conjectured to be $2^{-k}$ for all positive integers $k$. The conjecture is true for $1\leq k\leq 3$ but still open for $k\geq 4$. For primes $p$ of the form $p = a^2 + c^4$ with $c$ even, we describe the 8-Hilbert class field of $\mathbb{Q}(\sqrt{-p})$ in terms of $a$ and $c$. We then adapt a theorem of Friedlander and Iwaniec to show that there are infinitely many primes $p$ for which $h$ is divisible by $16$, and also infinitely many primes $p$ for which $h$ is divisible by $8$ but not by $16$.