Quadratic units and cubic fields
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908539192082432 |
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| author | Breuer, Florian Punch, James |
| author_facet | Breuer, Florian Punch, James |
| contents | 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 $π_{\mathcal{E}}(x) \approx \frac{1}{3π^2}x - 0.024x^{5/6}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06579 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quadratic units and cubic fields Breuer, Florian Punch, James Number Theory Primary 11R11, 11R16, 11R27, Secondary 11R29, 11Y40, 11R45 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 $π_{\mathcal{E}}(x) \approx \frac{1}{3π^2}x - 0.024x^{5/6}$. 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. |
| title | Quadratic units and cubic fields |
| topic | Number Theory Primary 11R11, 11R16, 11R27, Secondary 11R29, 11Y40, 11R45 |
| url | https://arxiv.org/abs/2507.06579 |