arXiv:2211.07474v2

On Some Variants Of Schinzel's Theorem For Global Function Fields

Quang-Khai Nguyen

math.NT11J6111G05

Paper metadata

Original submission: November 14, 2022
Latest revision: July 13, 2026 · version 2
Claimed MSC: 11J61, 11G05
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: Remove some parts of the paper due to the failure of Roth's theorem in positive characteristics. This paper has been superseded by the paper arXiv:2607.09002

Abstract

In this paper, we obtain global function field versions of the results of Schinzel-Postnikova for multiplicative groups, and of Hahn-Cheon for elliptic curves, which is an analog of the former result.

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Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 15, 2026

01Statements2 reported findingsCorrect

The primitive-order results for elliptic curves and one-dimensional tori over global function fields are correct. Two central proofs contain genuine written defects, but both admit direct verified repairs without changing the theorem statements.

Theorems 3.1 and 3.4Correct

Prescribed reduction orders on elliptic curves

Pages 3–7 · Theorems 3.1 and 3.4 · arXiv:2211.07474v2

For integers $n$ prime to the characteristic, local formal-group heights outside a fixed finite set are unchanged under multiplication by divisors of $n$ . If $n$ is a bad index with no reduction of exact order $n$ , this bounds $h(nP)$ by a sum of heights of $(n/r)P$ over prime divisors $r$ of $n$ , plus negligible contributions at the fixed places. Quadraticity of the canonical height and $\sum_{r\mid n,\ r\text{ prime}}r^{-2}<1/2$ then bound every bad index, proving Theorem 3.1. For an ordinary curve, adjoining a point of order $p^t$ and applying Theorem 3.1 to $P-Q$ forces the prime-to- $p$ and $p$ -primary parts of the reduction order of $P$ to be exactly $n$ and $p^t$ , respectively.

Full paper, version 2 ↗

Proposition 4.1 and Theorem 4.2Correct

Prescribed reduction orders on one-dimensional tori

Pages 7–9 · Proposition 4.1 and Theorem 4.2 · arXiv:2211.07474v2

For $\mathbb G_m$ , Möbius inversion of the valuations of $x^m-1$ shows that absence of a place where $x$ has exact order $n$ would make the divisor of $\Phi_n(x)$ consist only of the poles of $x$ , contradicting the product formula unless $x$ is constant. A one-dimensional torus becomes $\mathbb G_m$ over a finite splitting field; after extending the isomorphism over rings of integers away from finitely many places, the natural residue-field injection preserves the exact reduction order. Proposition 4.1 over the splitting field therefore descends to the stated conclusion.

Full paper, version 2 ↗

02Proofs4 reported findingsContains incorrect or incomplete proofs

The principal height and cyclotomic arguments are sound, but the proof of Theorem 3.1 starts with the wrong quantified contradiction hypothesis and the proof of Theorem 4.2 uses an invalid tensor product and lifting assertion. Each defect has a verified local repair that completes the intended proof. Several additional notation slips are typos.

Proof of Theorem 3.1Incomplete as written · verified repair

The contradiction hypothesis has the wrong quantifier

Pages 4–5 · Proof of Theorem 3.1 · arXiv:2211.07474v2

The proof assumes that every sufficiently large $n$ is bad, meaning that no place gives exact order $n$ . This is stronger than the negation of the theorem and would only exclude eventual failure, not infinitely many isolated bad indices. The repair is to fix an arbitrary bad $n$ outside the finite preliminary range. Inequality (1) uses only the defining property of that single bad index, while the estimates at the finite set $S$ are uniform for all sufficiently large $n$ . The resulting comparison is $(1-\#S\varepsilon)(n^2\widehat h(P)-c)<\frac{n^2}{2}\widehat h(P)+cn,$ which is impossible for all sufficiently large bad $n$ after choosing $\varepsilon<1/(2\#S)$ . Thus every bad index is bounded and the theorem follows. No statement change is needed.

Full paper, version 2 ↗

Proof of Theorem 4.2Incorrect as written · verified repair

The displayed base-change and arbitrary lifting argument is not defined

Pages 8–9 · Proof of Theorem 4.2 · arXiv:2211.07474v2

The formula $\mathcal O_v\otimes_KK'\cong\prod_{w\mid v}\mathcal O_w$ is not defined because $\mathcal O_v$ is not a $K$ -algebra, and an arbitrary $\mathcal O_v$ -point need not be obtained by the asserted lift. The needed fact is simpler. After enlarging $S$ , the splitting isomorphism extends to the integral models over $\mathcal O_{S'}$ . For each $w\mid v$ the natural morphism $\operatorname{Spec}\mathcal O_w\to\operatorname{Spec}\mathcal O_v$ base-changes the fixed point $x$ to $x'$ , and the injection of residue fields identifies the reduction of $x'$ with the base change of the reduction of $x$ . Exact order is preserved under that field extension. Proposition 4.1 applied to $x'$ outside $S'$ then proves the theorem exactly as stated.

Full paper, version 2 ↗

Lemmas 3.2–3.3 and Theorem 3.4Correct and complete

The remaining elliptic-curve arguments are complete

Pages 3–7 · Lemmas 3.2–3.3 and proof of Theorem 3.4 · arXiv:2211.07474v2

For multiplication by an integer prime to $p$ , the linear term in the formal-group law is a unit, so the local parameter valuation and hence the local height are unchanged. Northcott finiteness makes $h(nP)\to\infty$ , establishing the required local-height ratio. In Theorem 3.4, reduction preserves the order of the fixed torsion point outside finitely many places; elementary divisibility then forces both components of the order of $P$ to be exact.

Section 4Typos

Three unambiguous notation slips

Pages 7–9 · Proposition 4.1 and Theorem 4.2 · arXiv:2211.07474v2

In case 4 of Proposition 4.1 and its conclusion, the undefined set $P$ should be the previously defined set $W$ . In Theorem 4.2, $\mathcal O_{S'}^{\times}\subset K$ should read $\mathcal O_{S'}^{\times}\subset K'$ . Earlier, the definition of $S$ says that “ $S$ has bad reduction”; the subject is $E$ . Each replacement is fixed uniquely by the surrounding definitions and changes no argument.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2211.07474v2
Authors listed: Quang-Khai Nguyen
Audit date: August 15, 2026

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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. 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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. 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P(delta)

is proved for every sufficiently small

delta>0

and

P

is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

t=0

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...

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On Some Variants Of Schinzel's Theorem For Global Function Fields

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Prescribed reduction orders on elliptic curves

Prescribed reduction orders on one-dimensional tori

The contradiction hypothesis has the wrong quantifier

The displayed base-change and arbitrary lifting argument is not defined

The remaining elliptic-curve arguments are complete

Three unambiguous notation slips

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