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arxiv: 2605.17595 · v1 · pith:AQCMB7UMnew · submitted 2026-05-17 · 🧮 math.AC

Elasticity of Orders from the S-relative Davenport Constant: an Arithmetic Application of a Number-Theoretic Investigation

Jared Kettinger , Grant Moles This is my paper

Pith reviewed 2026-05-19 22:17 UTC · model grok-4.3

classification 🧮 math.AC
keywords elasticityDavenport constantconductor idealnon-maximal ordersalgebraic number fieldsfactorizationquadratic orders
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The pith

The S-relative Davenport constant determines the elasticity of non-integrally closed orders when the conductor is prime or primary.

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

Orders in algebraic number fields that are not the full ring of integers are not Dedekind domains, so their ideal factorizations can vary in length in ways that are hard to control. Elasticity measures the supremum of lengths of irreducible factorizations divided by the infimum, and prior results showed the ideal class group can still influence it even without unique factorization. This paper adapts the S-relative Davenport constant, originally developed by Skałba for binary quadratic forms, and proves it supplies the exact value of elasticity for orders whose conductor ideal is prime as an ideal of the order or primary in the quadratic case. If the result holds, then elasticity becomes computable from this single invariant without needing the complete arithmetic structure of the order. The work also identifies conditions under which such an order shares the elasticity of its maximal overring.

Core claim

We develop the S-relative Davenport constant, which builds on previous work by M. Skałba. We show that this related invariant is the exact tool needed to tackle the question of elasticity in non-integrally closed orders. In particular, we investigate the elasticity of orders O whose conductor ideal I=(O:O_K) is prime as an ideal of O, as well as orders in quadratic number fields with primary conductor. We also give conditions under which O will have the same elasticity as the full ring of integers O_K.

What carries the argument

The S-relative Davenport constant, an extension of Skałba's construction for binary quadratic forms that directly supplies the elasticity value for the indicated orders.

If this is right

  • For orders with prime conductor the elasticity equals the value given by the S-relative Davenport constant.
  • The same determination holds for orders in quadratic fields whose conductor is primary.
  • There exist explicit conditions under which the order and its maximal overring have identical elasticity.

Where Pith is reading between the lines

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

  • The same reduction might apply to conductors that are powers of a prime or other restricted forms.
  • Concrete numerical checks in quadratic fields with small discriminant would test whether the predicted elasticity matches explicit factorization lengths.

Load-bearing premise

The conductor ideal I must be prime as an ideal of the order or primary when the order lies in a quadratic number field.

What would settle it

Direct computation of all irreducible factorizations in a concrete order with prime conductor that produces an elasticity value different from the number predicted by the S-relative Davenport constant.

read the original abstract

Orders in algebraic number fields have long been objects of central interest in algebraic number theory. Despite non-maximal orders failing to be Dedekind, the present authors have previously shown that the structure of the ideal class group may still contain enough information to determine elasticity. In this paper, we develop the $S$-relative Davenport constant, which builds on previous work by M. Ska{\l}ba. Although Ska{\l}ba's original construction was defined to aid in the study of binary quadratic forms, we show that this related invariant is the exact tool needed to tackle the question of elasticity in non-integrally closed orders. In particular, we investigate the elasticity of orders $\mathcal{O}$ whose conductor ideal $I=(\mathcal{O}:\mathcal{O}_K)$ is prime as an ideal of $\mathcal{O}$, as well as orders in quadratic number fields with primary conductor. We also give conditions under which $\mathcal{O}$ will have the same elasticity as the full ring of integers $\mathcal{O}_K$.

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.

Referee Report

2 major / 2 minor

Summary. The paper develops the S-relative Davenport constant, building on Skałba's construction for binary quadratic forms, and applies it to determine the elasticity of non-integrally closed orders O in algebraic number fields. It restricts attention to the case where the conductor ideal I = (O : O_K) is prime as an ideal of O, or primary for orders in quadratic fields, and derives formulas for the elasticity together with conditions under which it equals the elasticity of the maximal order O_K.

Significance. If the derivations hold, the work supplies a targeted arithmetic application of a relative Davenport constant to factorization theory for orders, extending earlier results on class groups and elasticity. The explicit restriction to prime or primary conductors permits controlled ideal-class-group behavior and yields precise statements within this class of orders.

major comments (2)
  1. [Main results on elasticity (likely §3 or §4)] The central application of the S-relative Davenport constant to elasticity relies on the primality (or primality) of the conductor I to guarantee the required splitting or unique factorization properties above I. The manuscript should include a dedicated paragraph or lemma (near the statement of the main elasticity theorem) showing precisely how this hypothesis is used in the bound or equality.
  2. [Quadratic orders subsection] For the quadratic-field case with primary conductor, the equality condition with the elasticity of O_K is stated; an explicit numerical example or small-degree verification would strengthen the claim that the relative Davenport constant computes the exact value rather than merely an upper bound.
minor comments (2)
  1. [Introduction and definitions] The notation for the S-relative Davenport constant should be introduced with a short comparison to the classical Davenport constant and Skałba's original version to aid readers unfamiliar with the construction.
  2. [Throughout] A few typographical inconsistencies appear in the use of script letters for orders and ideals; a uniform convention would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive suggestions. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Main results on elasticity (likely §3 or §4)] The central application of the S-relative Davenport constant to elasticity relies on the primality (or primality) of the conductor I to guarantee the required splitting or unique factorization properties above I. The manuscript should include a dedicated paragraph or lemma (near the statement of the main elasticity theorem) showing precisely how this hypothesis is used in the bound or equality.

    Authors: We agree that the role of the primality (or primary) hypothesis on the conductor ideal I merits explicit clarification. In the revised manuscript we will insert a short lemma immediately before the statement of the main elasticity theorem. The lemma will isolate the precise points at which the hypothesis is invoked to guarantee the required splitting behavior and unique factorization properties in the ideals lying above I, thereby justifying the direct application of the S-relative Davenport constant to obtain the stated bound or equality. revision: yes

  2. Referee: [Quadratic orders subsection] For the quadratic-field case with primary conductor, the equality condition with the elasticity of O_K is stated; an explicit numerical example or small-degree verification would strengthen the claim that the relative Davenport constant computes the exact value rather than merely an upper bound.

    Authors: We appreciate the suggestion. To make the exactness claim more transparent, we will add a brief numerical example in the quadratic-orders subsection. The example will treat a small-degree quadratic field with primary conductor and explicitly compute both the S-relative Davenport constant and the elasticity of O, confirming that the two quantities coincide and thereby verifying that the constant yields the precise value rather than only an upper bound. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper cites prior work by Skałba for the base Davenport constant construction and references the authors' own earlier results on ideal class groups containing information sufficient to determine elasticity. However, the present manuscript develops the S-relative variant as a new invariant and applies it specifically to compute elasticity for orders whose conductor is prime (or primary in the quadratic case), along with conditions for equality with the maximal order. These steps consist of independent mathematical definitions, restrictions, and derivations under the stated hypotheses rather than any reduction by construction to fitted parameters, self-definitions, or unverified self-citations. The scope is self-contained within the given conductor assumptions and does not rely on load-bearing circular premises.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No explicit free parameters, axioms, or invented entities are described in the abstract; the S-relative Davenport constant appears to be an extension of a prior definition rather than a new postulated object.

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Reference graph

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