Unit-generated orders of real quadratic fields I. Class number bounds (2512.11311v1)

Abstract: Unit-generated orders of a quadratic field are orders of the form $\mathcal{O} = \mathbb{Z}[\varepsilon]$, where $\varepsilon$ is a unit in the quadratic field. If the order $\mathcal{O}$ is a maximal order of a real quadratic field, then the quadratic number field is necessarily of a restricted form, being of narrow Richaud--Degert type. However, every real quadratic field contains infinitely many distinct unit-generated orders. They are parametrized as $\mathcal{O} = \mathcal{O}{n}^{\pm}$ having quadratic discriminants $Δ(\mathcal{O}) = Δ{n}^{+} = n^2 - 4$ (for $n \geq 3$) and $Δ(\mathcal{O}) = Δ_{n}^{-} = n^2 + 4$ (for $n \geq 1$). We show the (wide or narrow) class numbers of unit-generated orders satisfy $\log \left|{\rm Cl}(\mathcal{O})\right| \sim \log \frac{1}{2}\left|Δ(\mathcal{O})\right|$ as $\left|Δ(\mathcal{O})\right| \to \infty$, using a result of L.-K. Hua. We deduce that there are finitely many unit-generated quadratic orders of class number one and finitely many unit-generated quadratic orders whose class group is $2$-torsion. We classify all unit-generated real quadratic maximal orders having class number one. We provide numerical lists of quadratic unit-generated orders whose class groups are $2$-torsion for $Δ\leq 10^{10}$, for both wide and narrow class groups, which are conjecturally complete.

• The paper establishes sharp class number bounds for unit-generated orders by proving logarithmic growth rates and effective error bounds.
• It classifies unit-generated orders with class number one and 2-torsion phenomena using genus theory and comprehensive computational data.
• The results connect arithmetic of real quadratic fields with applications in quantum information theory and cryptography.

Unit-Generated Orders of Real Quadratic Fields and Class Number Bounds

Introduction and Problem Formulation

This work addresses the global structure and distribution of class numbers in the family of non-maximal orders of real quadratic fields generated by units. Specifically, it studies quadratic orders $$\mathcal{O} = \mathbb{Z}[\varepsilon] \subseteq \mathbb{Q}(\sqrt{D})$$ where $\varepsilon$ is a unit in the quadratic field $K = \mathbb{Q}(\sqrt{D})$. Orders of this form, called unit-generated orders, are parametrized by two infinite families:

• $\Delta_n^+ = n^2 - 4$ for $n \geq 3$, with a generating unit of norm $+1$
• $\Delta_n^- = n^2 + 4$ for $n \geq 1$, with a generating unit of norm $-1$

The maximal orders among these correspond precisely to real quadratic fields of (narrow) Richaud--Degert type. The classification, asymptotics, and arithmetic constraints on the class groups and class numbers of these orders are analyzed, with particular attention to the occurrence of class number one and $2$-torsion phenomena.

Parametric Structure and Unit-Generated Orders

Every real quadratic field contains infinitely many unit-generated orders, as each field contains infinitely many units $\varepsilon > 1$, and the order $\mathbb{Z}[\varepsilon]$ is determined by the minimal polynomial of $\varepsilon$ and its trace/norm. The possible discriminants are exhausted, up to a small finite number of imaginary quadratic exceptions, by $\Delta_n^\pm$. The generating units have explicit continued fraction parametrizations, reflecting the arithmetic regularity of these families. These families are also intimately related to algebraic structures appearing in quantum information theory (notably, the parameter $\Delta^+_n$ is the discriminant associated to SIC-POVMs in dimension $n+1$).

Class Number Growth, Bounds, and Finiteness Phenomena

The main technical result is that for unit-generated orders $\mathcal{O}$ of discriminants $\Delta_n^\pm$, the class number grows logarithmically:

$$\log |\text{Cl}(\mathcal{O})| = \log n + o(\log n) \text{ as } n \to \infty.$$

This is a consequence of Hua's extension of the Brauer--Siegel theorem to orders (not just maximal orders) [Theorem 12.15.4 in Hua]. For subfamilies where the fundamental discriminant is bounded (e.g., arising in the context of a fixed quadratic field), effective error bounds proportional to $\log\log n$ are given. Thus, there can only be finitely many unit-generated orders with class number one or with $2$-torsion class group; this finitude is exceptional among real quadratic orders.

A precise classification is given for unit-generated maximal orders of class number one, fully resolving this problem by combining earlier deep work (Biró on the $n^2 + 4$ odd case, Byeon-Kim-Lee for $n^2 - 4$ odd, plus new genus-theoretic analysis for the even cases). Numerical lists of unit-generated orders of small class number and those with purely $2$-torsion class groups are provided for discriminants up to $10^{10}$, and conjectured to be complete.

Genus Theory, 2-Torsion, and One Class per Genus

The structure of the narrow and wide class groups, and their genus subgroups, are analyzed using classical genus theory. The order of the genus group, and thus the possible $2$-torsion, depends on the arithmetic of the discriminant via distinct prime divisors, following the Gauss composition law and explicit formulae for the number of genera.

The argument generalizes the finiteness result: there are only finitely many unit-generated orders whose (narrow or wide) class group is $2$-torsion, implying only finitely many unit-generated real quadratic orders have one class per genus. The classification of these orders corresponds, via analogy, to the classical idoneal number and one-class-per-genus problems for imaginary quadratic fields, with a parallel hierarchy of difficulty and open subproblems.

Computational Classification and Data

Complete lists (for all discriminants $\Delta < 10^{10}$) are provided for:

• Unit-generated orders of class number one
• Unit-generated orders whose class group is $2$-torsion
• Unit-generated orders with one class per genus

For fundamental discriminants, these lists are theoretically complete by virtue of existing finiteness results. For non-fundamental discriminants, they are conjecturally complete and exhaust the possibilities up to high computational bounds.

Relation to Richaud--Degert Fields and Applications

Maximal unit-generated orders correspond precisely to (narrow) Richaud--Degert type fields, discernible both via continued fractions and arithmetic properties of their discriminants. In quantum information—the SIC-POVM context—the appearance of certain orders in the field structure of SICs is traced to this arithmetic, and the sum of the class numbers of overorders of $\mathbb{Z}[\varepsilon]$ corresponds empirically to the enumeration of SICs in a given dimension.

Implications and Outlook

These results demonstrate that the families $\Delta_n^\pm$ of unit-generated orders display class group and class number growth rates analogous to those for imaginary quadratic fields, in stark contrast to the generic real quadratic case. The explicit and effective class number bounds, and finiteness statements for low-invariant phenomena, provide new constraints for broader heuristics and conjectures in number theory—especially on the distribution of class groups and possible extreme behaviors.

From a practical perspective, these findings can be employed in explicit arithmetic statistics, cryptography (for instance, the scarcity of class number one orders informs discriminant selection), and in problems related to the construction and field-of-definition of SIC-POVMs.

Theoretically, the work points to several natural directions:

• Extension of analogous theorems (and possible finiteness results) to unit-generated orders in other number fields, especially those of unit rank one (e.g., complex cubic, quartic CM-fields).
• Analysis of more refined distributional statistics in these parametric families (such as higher $m$-torsion, average class group structure, class field theory extensions).
• Connection with open conjectures about the completeness of the list of class number one or one-genus-per-class discriminants in the real quadratic case, paralleling the still unsolved idoneal or Euler's convenient number problems for, e.g., positive-definite quadratic forms.

Conclusion

This paper establishes a rich structural theory for unit-generated orders in real quadratic fields, providing sharp class number asymptotics, complete classifications for both class number one orders and those with $2$-torsion class groups, and a detailed arithmetic and computational framework for these families. The results highlight the exceptional nature of these parametric orders in the general landscape of real quadratic arithmetic, link them to nontrivial applications in quantum information, and motivate further investigation into unit-generated orders and their arithmetic invariants in other families of number fields.