Quadratic units and cubic fields

TL;DR

This paper studies Eisenstein discriminants in real quadratic fields, providing numerical evidence for their distribution and connections to cubic fields, supported by large-scale GPU computations.

Contribution

It offers the first extensive numerical analysis of Eisenstein discriminants up to 10^{11}, supporting a conjecture of Stevenhagen and revealing a new secondary term in their counting function.

Findings

Counting function approximates to (1/3π^2)x - 0.024x^{5/6}

Supports Stevenhagen's conjecture on Eisenstein discriminants

Reveals a secondary term similar to that in real cubic fields

Abstract

We investigate Eisenstein discriminants, which are squarefree integers $d \equiv 5 \pmod{8}$ such that the fundamental unit $\varepsilon_d$ of the real quadratic field $K=\mathbb{Q}(\sqrt{d})$ satisfies $\varepsilon_d \equiv 1 \pmod{2\mathcal{O}_K}$. These discriminants are related to a classical question of Eisenstein and have connections to the class groups of orders in quadratic fields as well as to real cubic fields. We present numerical computations of Eisenstein discriminants up to $10^{11}$, suggesting that their counting function up to $x$ is approximated by $\pi_{\mathcal{E}}(x) \approx \frac{1}{3\pi^2}x - 0.024x^{5/6}$. This supports a conjecture of Stevenhagen while revealing a surprising secondary term, which is similar to (but subtly different from) the secondary term in the counting function of real cubic fields. We include technical details of our computation method, which uses a modified infrastructure approach implemented on GPUs.