Unit-Generated Quadratic Orders

• Unit-generated quadratic orders are orders in quadratic fields where every element is expressed as a Z-linear combination of units, emphasizing explicit constructions in real quadratic settings.
• They exhibit precise relationships with class number growth, finite classifications, and structures like one-class-per-genus and 2-torsion class groups.
• Their arithmetic applications include selective embeddings in quaternion algebras and canonical unit constructions via ray class fields and Stark-type formulas.

A unit-generated quadratic order is, by definition, an order $\mathcal{O}$ in a quadratic field $K$ such that $\mathcal{O}$ is generated as a $\mathbb{Z}$-module by its unit group: $\mathcal{O} = \mathbb{Z}[\mathcal{O}^\times]$. This concept plays a central role in the arithmetic of real quadratic fields, where the structure of units interacts deeply with ideal class groups, class number growth, and explicit parametrizations of orders. Unit-generated orders possess rich arithmetic, including a refined classification, explicit growth laws for class numbers, and relationships with selectivity phenomena in quaternionic and ray class field settings.

1. Definition and Basic Characterization

Given a quadratic field $K$ and an order $\mathcal{O} \subseteq \mathcal{O}_K$, $\mathcal{O}$ is called unit-generated if every element of the order can be expressed as a $\mathbb{Z}$-linear combination of (some finite set of) units in $\mathcal{O}$. Explicitly,

$$\mathcal{O} = \mathbb{Z}\bigl[\mathcal{O}^\times\bigr] = \sum_{u \in \mathcal{O}^\times} \mathbb{Z}u.$$

For real quadratic fields, this means $\mathcal{O} = \mathbb{Z}[\varepsilon]$ for some algebraic unit $\varepsilon > 1$ that is minimal: no positive power of the fundamental unit less than $\varepsilon$ is contained in $\mathcal{O}$. Equivalently, every $\mathcal{O}$ of the form $\mathbb{Z} + \varepsilon\mathbb{Z}$, with $\varepsilon$ a unit, is unit-generated and vice versa (Kopp et al., 12 Dec 2025).

2. Parametrization: The $\Delta_n^+$ and $\Delta_n^-$ Families

The structure of unit-generated orders in real quadratic fields is governed by two infinite families, determined by the discriminant. A real quadratic field $K = \mathbb{Q}(\sqrt{\Delta_0})$ has fundamental unit $\varepsilon_{\Delta_0} > 1$. Each unit-generated order is: $$\mathcal{O}_{n}^{\pm} = \mathbb{Z}[\varepsilon_n^\pm],$$ where:

• For $\mathrm{Norm}(\varepsilon) = +1$: $\Delta^+_n = n^2 - 4$, $n \geq 3$; $\varepsilon_n^+ = \frac{1}{2}(n + \sqrt{n^2 - 4})$.
• For $\mathrm{Norm}(\varepsilon) = -1$: $\Delta^-_n = n^2 + 4$, $n \geq 1$; $\varepsilon_n^- = \frac{1}{2}(n + \sqrt{n^2 + 4})$.

Every real quadratic field contains infinitely many such orders, parametrized by increasing $n$ (Kopp et al., 12 Dec 2025). The continued fraction expansions of the generating units in these families are particularly simple and explicit.

3. Class Number Growth and Finiteness Results

Let $h_n^{\pm} = |\mathrm{Cl}(\mathcal{O}_{n^2 \mp 4})|$ denote the (wide) class number. Using the Brauer–Siegel theorem extended to nonmaximal orders via L.-K. Hua, the asymptotic law is: $$\log h_n^{\pm} = \log n + o(\log n), \qquad h_n^{\pm} = n^{1 + o(1)}$$ as $n \to \infty$ (Kopp et al., 12 Dec 2025). This result implies that only finitely many unit-generated orders possess class number one, as the class number diverges logarithmically with $n$. Furthermore, by genus theory and refinement of the Brauer–Siegel growth, only finitely many have class group consisting entirely of $2$-torsion (i.e., 'one class per genus').

4. Classification of Maximal Orders and Explicit Lists

When $\mathcal{O}$ is a maximal order (i.e., $\Delta$ is a squarefree discriminant), the classification of unit-generated real quadratic orders with class number one is complete (Kopp et al., 12 Dec 2025). The possible discriminants are precisely:

• Norm $+1$ family ($\Delta_n^+ = n^2 - 4$, $n \in \{0,1,3,4,5,9,21\}$):

$\Delta \in \{-4, -3, 5, 12, 21, 77, 437\}$

• Norm $-1$ family ($\Delta_n^- = n^2 + 4$, $n \in \{1,2,3,5,7,13,17\}$):

$\Delta \in \{5, 8, 13, 29, 53, 173, 293\}$

These cases were resolved in part by genus theory (for even $n$) and by explicit computation and results of Biró (2003), Byeon–Kim–Lee (2007) for odd $n$. The overlap $\Delta^+_3 = \Delta^-_1 = 5$ is unique. The enumeration is conjecturally exhaustive up to significantly large discriminant ($\Delta < 10^{10}$), with 19 examples of $h=1$ found (Kopp et al., 12 Dec 2025).

5. One-Class-per-Genus and 2-Torsion Class Groups

The unit-generated orders for which $$\mathrm{Cl}(\mathcal{O}) = \mathrm{Cl}(\mathcal{O})[2]$$ are structurally analogous to idoneal (one-class-per-genus) imaginary quadratic fields. The key result is the finiteness: $$\text{Only finitely many unit-generated real quadratic orders have } \mathrm{Cl}(\mathcal{O}) = \mathrm{Cl}(\mathcal{O})[2].$$ The functional equation for the $2$-torsion subgroup, $$|\mathrm{Cl}(\mathcal{O})[2]| = 2^{\mu(\Delta) - 1}$$, cannot be matched by the class-number asymptotics as $|\Delta| \to \infty$—this forces finiteness (Kopp et al., 12 Dec 2025). For discriminants up to $10^{10}$, there are 86 orders with $2$-torsion class groups, numerically documented and conjectured complete.

6. Selectivity of Unit-Generated Orders in Quaternionic Settings

Unit-generated quadratic orders such as $\mathbb{Z}[\zeta_3]$ display selectivity phenomena in definite quaternion algebras. An order is selective in a genus of quaternion orders if it embeds into some but not all maximal orders in the genus. The selectivity of $\mathbb{Z}[\zeta_3]$ is governed by the combinatorics of the classifying graph (Bruhat–Tits tree) at $2$:

• If a genus contains at least $3$ conjugacy classes, then $\mathbb{Z}[\zeta_3]$ is selective (Arenas-Carmona, 2014).
• Endpoints in the classifying graph correspond to orders containing roots of unity.
• The number of embeddings is strictly less than the total number of conjugacy classes, bound by valency constraints on the quotient graph.

Analogous phenomena are observed for other unit-generated quadratic orders, particularly those with special local behavior at $2$ (e.g., containing a Fermat prime root of unity or $u$ with $u^2-u+2=0$) (Arenas-Carmona, 2014).

7. Canonical Units via Ray Class Fields and Stark-Type Relations

There exist conjecturally infinite towers of ray class fields above real quadratic $K = \mathbb{Q}(\sqrt{D})$, each admitting explicit unit generators. For each solution to the Pell equation $x^2 - Dy^2 = 4$ with $x = d-1$, there is a narrow ray class field $R_{(d)\infty_1\infty_2}$. The theory of symmetric informationally complete measurements (SICs) in quantum information provides a computational recipe for extracting explicit canonical units in these fields (Appleby et al., 2016):

• The "overlaps" arising from SIC projectors yield algebraic units in the ray class fields.
• For each such $d$, one normalization $e^{i\theta(j)}$ becomes a primitive unit in $R$, with norm $1$ over $K$.
• The units tend to $1$ under all real embeddings as $d \to \infty$.

A major conjecture, mirroring Stark's Conjecture, posits that $\log |\varepsilon|$ for these canonical units admits a formula in terms of $L^\prime(\chi, 0)$ over nontrivial narrow ray class characters $\chi$.

8. Connections to Class Group 4-Rank and Unramified Quadratic Extensions

The structure induced by unit-generated quadratic orders interacts with the $4$-rank of the narrow class group and with the construction of unramified quadratic extensions of "unit type." For a totally real field $K$ with all fundamental units totally positive, the lower bound on the $4$-rank is explicit: $$\mathrm{4\text{-}rank}\, \mathrm{Cl}^+(K) \geq \begin{cases} (n-1)/2, & n \text{ odd} \ n/2 - 1, & n \text{ even} \end{cases}$$ (where $n$ is the degree of $K$) (Dummit, 2018). For the real quadratic case, units that are sums of two squares or squares mod $4$ are key: such units generate quadratic extensions unramified at all finite primes, and correspond to explicit order $4$ elements in the narrow class group. The paper provides explicit examples where these constructions determine all nontrivial classes of order $4$.

---

• (Kopp et al., 12 Dec 2025) G.S. Kopp & J.C. Lagarias, "Unit-generated orders of real quadratic fields I. Class number bounds"
• (Arenas-Carmona, 2014) L. Arenas–Carmona, "Roots of unity in definite quaternion orders"
• (Appleby et al., 2016) D.M. Appleby et al., "Generating Ray Class Fields of Real Quadratic Fields via Complex Equiangular Lines"
• (Dummit, 2018) D.S. Dummit, "Classes of order 4 in the strict class group of number fields and remarks on unramified quadratic extensions of unit type"