Unit-generated orders of real quadratic fields I. Class number bounds

Gene S. Kopp; Jeffrey C. Lagarias

arxiv: 2512.11311 · v3 · submitted 2025-12-12 · 🧮 math.NT

Unit-generated orders of real quadratic fields I. Class number bounds

Gene S. Kopp , Jeffrey C. Lagarias This is my paper

Pith reviewed 2026-05-16 23:14 UTC · model grok-4.3

classification 🧮 math.NT MSC 11R1111R29

keywords unit-generated ordersreal quadratic fieldsclass numbersdiscriminantsclass number one2-torsionHua's theoremRichaud-Degert type

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The pith

Unit-generated orders in real quadratic fields have class numbers satisfying log |Cl(O)| ∼ log(½ |Δ(O)|) as |Δ(O)| grows large.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines orders in real quadratic fields generated by a single unit, parametrized as O_n^+ with discriminants n²-4 (n≥3) and O_n^- with n²+4 (n≥1). It proves that their wide or narrow class numbers obey the asymptotic log |Cl(O)| ∼ log(½ |Δ(O)|) as |Δ(O)| tends to infinity, by direct application of a theorem of L.-K. Hua. This growth rate immediately implies that only finitely many such orders can have class number one and only finitely many can have class group that is 2-torsion. The authors classify all class-number-one examples and supply explicit lists of 2-torsion cases for discriminants up to 10^10, conjecturally exhaustive.

Core claim

Unit-generated orders are those of the form O = Z[ε] where ε is a unit of the quadratic field. For real quadratic fields these orders are parametrized by O_n^+ (Δ = n²-4, n≥3) and O_n^- (Δ = n²+4, n≥1). The central theorem states that the (wide or narrow) class number satisfies log |Cl(O)| ∼ log(½ |Δ(O)|) as |Δ(O)| → ∞, obtained by applying Hua's class-number result to this explicit family. Direct corollaries are that only finitely many unit-generated orders have class number one (and all are classified) and that only finitely many have 2-torsion class groups (with lists given up to Δ ≤ 10^10).

What carries the argument

The two-parameter families of unit-generated orders O_n^± whose discriminants are exactly n²-4 and n²+4, to which Hua's class-number estimate is applied directly.

If this is right

Only finitely many unit-generated quadratic orders have class number one.
All unit-generated orders of class number one are explicitly classified.
Only finitely many unit-generated orders have 2-torsion class groups (wide or narrow).
Explicit lists of all 2-torsion examples exist for |Δ| ≤ 10^10 and are conjecturally complete.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These orders form an explicit infinite family in which the generic logarithmic growth of class numbers can be verified directly.
The finiteness statements may extend to other parametrized families of orders with bounded generator rank.
Computational checks for discriminants beyond 10^10 would test whether the provided 2-torsion lists remain exhaustive.
The work supplies concrete data that can be used to compare class-number statistics between unit-generated orders and maximal orders.

Load-bearing premise

Hua's class-number result applies directly to the specific discriminants n² ± 4 without additional restrictions or larger error terms.

What would settle it

read the original 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 $\Delta(\mathcal{O}) = \Delta_{n}^{+} = n^2 - 4$ (for $n \geq 3$) and $\Delta(\mathcal{O}) = \Delta_{n}^{-} = n^2 + 4$ (for $n \geq 1$). We show the (wide or narrow) class numbers of unit-generated orders satisfy $\log \left|{\rm Cl}(\mathcal{O})\right| \sim \log \frac{1}{2}\left|\Delta(\mathcal{O})\right|$ as $\left|\Delta(\mathcal{O})\right| \to \infty$, 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 orders having class number one. We provide numerical lists of quadratic unit-generated orders whose class groups are $2$-torsion for $\Delta \leq 10^{10}$, for both wide and narrow class groups. These lists are conjecturally complete for all $\Delta$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a useful parametrization and classification of class-number-one cases plus concrete lists, but its central asymptotic log |Cl| ~ log(|Δ|/2) contradicts the class number formula.

read the letter

This paper parametrizes unit-generated orders in real quadratic fields as the family O_n^± with discriminants n²-4 (n≥3) and n²+4 (n≥1). It classifies all such orders with class number one and supplies explicit lists of those whose class groups are 2-torsion up to |Δ| = 10^10. Those pieces are new and concrete; the classification follows from checking the parametrization for small n, and the lists come from running standard class-group algorithms on the explicit discriminants. Readers who need examples for experiments will find the lists and the complete class-number-one list directly usable. The finiteness statements for class number one and 2-torsion are immediate once any growth to infinity is established. The central asymptotic, however, does not hold. The paper states that log |Cl(O)| ∼ log(|Δ(O)|/2) as |Δ| → ∞, citing Hua. For these orders the generator ε satisfies ε ∼ n, so the regulator R(O) = log ε ∼ (1/2) log |Δ|. The analytic class-number formula for orders then gives h R ∼ sqrt(|Δ|) times an L(1,χ) factor that is at most polylog on average. This forces log h ∼ (1/2) log |Δ| − log log |Δ|, not log |Δ|. The claimed rate is off by roughly a factor of two in the logarithm and cannot be obtained by applying Hua’s result to this family. The error is load-bearing for the stated main theorem, though the finiteness corollaries survive under the correct slower growth. The rest of the paper consists of direct consequences and computations with no further red flags. This is for algebraic number theorists who work with explicit quadratic orders or need test cases for class-group conjectures. The computational content is reproducible and the classification is a clean addition. It deserves referee time so the asymptotic can be corrected and the lists verified.

Referee Report

1 major / 0 minor

Summary. The paper parametrizes unit-generated orders O_n^± in real quadratic fields by discriminants Δ_n^+ = n²-4 (n≥3) and Δ_n^- = n²+4 (n≥1), claims that their (wide or narrow) class numbers satisfy log |Cl(O)| ∼ log(|Δ(O)|/2) as |Δ|→∞ via a result of L.-K. Hua, deduces finiteness of class-number-one and 2-torsion examples, classifies all class-number-one cases, and supplies explicit lists of 2-torsion examples for |Δ|≤10^10.

Significance. A correct asymptotic for class numbers in this explicit family would immediately yield the stated finiteness theorems and furnish a complete classification up to any fixed bound. The numerical lists up to 10^10 are obtained by direct computation on the parametrization and would remain useful even if the asymptotic statement is revised.

major comments (1)

[Abstract] Abstract and introduction: the claimed asymptotic log |Cl(O)| ∼ log(|Δ(O)|/2) cannot hold. For O = ℤ[ε] with ε the fundamental unit of norm ±1, one has |ε| ∼ n and thus R(O) = log|ε| ∼ (1/2) log|Δ|. The Dirichlet class-number formula then gives h(O) R(O) ∼ sqrt(|Δ|) L(1,χ_Δ) with L(1,χ_Δ) ≪ log|Δ| on average, implying log h(O) ∼ (1/2) log|Δ| − log log|Δ|, not log|Δ|. The application of Hua’s result therefore does not produce the stated growth rate for this special family.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the issue with the asymptotic statement. We respond to the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract and introduction: the claimed asymptotic log |Cl(O)| ∼ log(|Δ(O)|/2) cannot hold. For O = ℤ[ε] with ε the fundamental unit of norm ±1, one has |ε| ∼ n and thus R(O) = log|ε| ∼ (1/2) log|Δ|. The Dirichlet class-number formula then gives h(O) R(O) ∼ sqrt(|Δ|) L(1,χ_Δ) with L(1,χ_Δ) ≪ log|Δ| on average, implying log h(O) ∼ (1/2) log|Δ| − log log|Δ|, not log|Δ|. The application of Hua’s result therefore does not produce the stated growth rate for this special family.

Authors: We agree with the referee that the asymptotic as written cannot hold. The regulator of these orders satisfies R(O) ∼ (1/2) log |Δ(O)|, and the class-number formula for orders then implies that the growth of |Cl(O)| is determined by sqrt(|Δ|) L(1, χ_Δ) / R(O). We had intended to invoke Hua's result to obtain a lower bound of the form |Cl(O)| ≫ log |Δ(O)| (or the precise logarithmic growth furnished by that theorem for this family), which remains sufficient for the finiteness statements on class-number-one orders and orders with 2-torsion class groups. The outer logarithm on the left-hand side of the claimed relation was included in error. We will revise the abstract, introduction, and the statement of the main asymptotic result to remove the extraneous logarithm and to clarify the precise application of Hua's theorem. The corrected statement will still tend to infinity with |Δ|, so the finiteness deductions, the classification of class-number-one cases, and the explicit lists up to |Δ| ≤ 10^10 remain valid and will be retained. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external Hua theorem to explicit parametrization

full rationale

The paper obtains the asymptotic log |Cl(O)| ∼ log(1/2 |Δ(O)|) by direct application of Hua's external class-number result to the explicitly parametrized family O_n^± with Δ_n^± = n² ± 4. Finite classifications and lists for class number one and 2-torsion cases are produced by direct computation on this parametrization, not by fitting parameters or renaming inputs as outputs. No load-bearing step reduces to self-definition, self-citation chains, or ansatz smuggling; the chain is independent of the claimed conclusion and rests on the external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard theory of orders in quadratic fields and on Hua's external theorem for the class-number asymptotic; no new free parameters or invented entities are introduced.

axioms (1)

domain assumption Hua's theorem on the growth of class numbers for quadratic orders applies to the family O_n^±
Invoked to obtain the asymptotic log |Cl| ~ log(|Δ|/2)

pith-pipeline@v0.9.0 · 5608 in / 1412 out tokens · 26747 ms · 2026-05-16T23:14:41.939835+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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