On the equations $x^2-2py^2 = -1, \pm 2$

TL;DR

This paper investigates the distribution of primes for which certain quadratic equations are solvable, and analyzes the structure of class groups and units in related quadratic and CM-fields, providing density results and distribution theorems.

Contribution

It improves density bounds for primes solvable in specific quadratic equations and establishes the natural density of primes with class groups containing elements of order 16, with applications to CM-fields.

Findings

Density of primes with solvable equations improved

Natural density of primes with class group elements of order 16 established

Distribution results for 16-ranks of class groups derived

Abstract

Let $E\in\{-1, \pm 2\}$. We improve on the upper and lower densities of primes $p$ such that the equation $x^2-2py^2=E$ is solvable for $x, y\in \mathbb{Z}$. We prove that the natural density of primes $p$ such that the narrow class group of the real quadratic number field $\mathbb{Q}(\sqrt{2p})$ has an element of order $16$ is equal to $\frac{1}{64}$. We give an application of our results to the distribution of Hasse's unit index for the CM-fields $\mathbb{Q}(\sqrt{2p}, \sqrt{-1})$. Our results are consequences of a twisted joint distribution result for the $16$-ranks of class groups of $\mathbb{Q}(\sqrt{-p})$ and $\mathbb{Q}(\sqrt{-2p})$ as $p$ varies.