arXiv:2402.00739v3

p-Adically convergent loci in varieties arising from periodic continued fractions

Laura Capuano, Marzio Mula, Lea Terracini, Francesco Veneziano

math.NT11J7011D8811D09

Paper metadata

Original submission: February 1, 2024
Latest revision: July 10, 2026 · version 3
Claimed MSC: 11J70, 11D88, 11D09
arXiv categories: math.NT
Paper license: CC BY 4.0 ↗
Comments: revised version uploaded

Abstract

Inspired by several alternative definitions of continued fraction expansions for elements in $\mathbb Q_p$ , we study $p$ -adically convergent periodic continued fractions with partial quotients in $\mathbb Z[1/p]$ . To this end, following a previous work by Brock, Elkies, and Jordan, we consider certain algebraic varieties whose points represent formal periodic continued fractions with period and preperiod of fixed lengths, satisfying a given quadratic equation. We then focus on the $p$ -adically convergent loci of these varieties, characterizing the zero and one-dimensional cases.

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Audit summary

Audited against arXiv v3

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.

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Detailed mathematical audit

Generated August 15, 2026

01Statements3 reported findingsContains wrong statements

The necessary-and-sufficient convergence criterion and the principal finiteness and Pell-recurrence results are supported. Proposition 4.9(a), however, incorrectly asserts that the type $(2,1)$ locus is empty when the discriminant is $4A^2$ ; an explicit rational point contradicts that assertion. This error does not affect the proposition's finiteness conclusion or its description of the convergent locus.

Theorem 3.1Correct

The periodic convergence criterion is correct

Pages 6–9 · Theorem 3.1 · arXiv:2402.00739v3

All cyclic period matrices are conjugate and therefore have the same trace and determinant. If the trace has absolute value greater than $1$ , their eigenvalues have distinct absolute values $|\mu|_p>1>|\nu|_p$ . Condition (ii) exactly excludes a cyclic period matrix whose dominant eigenline is the wrong coordinate line. The first columns of its powers then converge to the dominant fixed point, and the identities $M_{\ell k+j-1}=M_{j-1}T_j^\ell$ make every residue class of convergents approach the same point. Conversely, the equal-modulus and exceptional triangular cases exhibited in the proof prevent convergence, establishing necessity as well as sufficiency.

Proposition 4.9(a)Incorrect

The claimed emptiness at $\Delta=4A^2$ is false

Page 12 · Proposition 4.9(a) and its proof · arXiv:2402.00739v3

The proposition states that, for $A\neq0$ , $V(F)_{2,1}(\mathcal O)$ is empty when $\Delta=4A^2$ . Take $F(x)=x^2+2x,$ so $A=1$ , $B=2$ , $C=0$ , and $\Delta=4=4A^2$ . Substitution into the three defining equations (13) shows directly that $(b_1,b_2,a_1)=(-1,0,0)\in V(F)_{2,1}(\mathcal O).$ Equivalently, equation (14) at $b_2=0$ reduces to $b_1^2+2b_1+1=0$ . Thus the printed emptiness conclusion has a formal counterexample. The point has $a_1=0$ and is not convergent, so Proposition 4.9(b) is unaffected.

Full paper, version 3 ↗

Propositions 4.18 and Theorem 4.24Correct

The pure-radical classification and finiteness conclusions are correct

Pages 18–24 · Proposition 4.18 and Theorem 4.24 · arXiv:2402.00739v3

Eliminating the middle partial quotient gives the generalized Pell equation $a_1^2-da_3^2=d-1$ and the identity $a_2=(d-1)/(a_1-da_3)$ . The convergence criterion forces $a_1,a_3$ to be $p$ -adic units in $\mathbb Z$ , and reduction of the second fraction gives exactly the displayed form $a_2=v/p^s$ . Conversely, those arithmetic conditions satisfy both parts of Theorem 3.1. The resulting Pell orbits obey the stated nondegenerate binary recurrence, so the cited perfect-power theorem yields finiteness for fixed $p$ and, when $s>1$ , finiteness as $p$ varies.

02Proofs3 reported findingsContains incorrect or incomplete proofs

The main eigenvalue argument and the Pell-recurrence arguments are correct. The proof of Proposition 4.9(a) divides by $b_2^2$ without treating $b_2=0$ , and that omitted case produces an actual counterexample to the proposition's emptiness clause. A verified local repair preserves the advertised finiteness result.

Proof of Proposition 4.9(a)Incorrect as written · verified local repair

The $b_2=0$ branch is lost before the modular obstruction is applied

Page 12 · final paragraph of the proof of Proposition 4.9(a) · arXiv:2402.00739v3

For $\Delta=4A^2$ , the discriminant of equation (14), regarded as a quadratic in $b_1$ , is $4A^2b_2^2(b_2^2+2).$ The proof divides by $4A^2b_2^2$ and applies a modulo- $4$ obstruction to $b_2^2+2=v^2$ , but it never first establishes $b_2\neq0$ . The counterexample in Part 1 lies exactly in the omitted branch. Verified repair: split into cases. If $b_2\neq0$ , the printed clearing-denominators and modulo- $4$ argument excludes solutions. If $b_2=0$ , equation (14) has the double solution $b_1=-B/(2A)$ ; equations (13) then force $a_1=0$ . Retain this one point precisely when $-B/(2A)\in\mathcal O$ . The corrected locus is still finite and its added point is nonconvergent.

Full paper, version 3 ↗

Proof of Theorem 3.1Correct and complete

Dominant-eigenline analysis

Pages 6–9 · proof of Theorem 3.1 · arXiv:2402.00739v3

The proof treats scalar, non-semisimple, diagonalizable, and coordinate-eigenline cases separately. The unit-determinant relation prevents an unaddressed eigenvalue-size regime, and the cyclic-conjugacy identities propagate the limiting fixed point through all positions in the period. No substantive case is missing.

Proofs of Proposition 4.18 and Theorem 4.24Correct and complete

Integral-point and recurrence reductions

Pages 18–24 · Sections 4.6.1–4.6.2 · arXiv:2402.00739v3

The projective completion has three points at infinity, so Siegel's theorem applies to the $S$ -integral locus. The valuation argument recovers integral outer partial quotients and the exact denominator power. Each Pell class gives a nondegenerate second-order recurrence with the hypotheses required by the quoted perfect-power theorem; absorbing the fixed prime into the finite set of allowed prime divisors proves the fixed- $p$ case, while taking the varying prime as the powered base proves the $s>1$ case.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2402.00739v3
Authors listed: Laura Capuano, Marzio Mula, Lea Terracini, Francesco Veneziano
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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For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. 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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

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p-Adically convergent loci in varieties arising from periodic continued fractions

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The periodic convergence criterion is correct

The claimed emptiness at Δ=4A2\Delta=4A^2Δ=4A2 is false

The pure-radical classification and finiteness conclusions are correct

The b2=0b_2=0b2​=0 branch is lost before the modular obstruction is applied

Dominant-eigenline analysis

Integral-point and recurrence reductions

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The claimed emptiness at $\Delta=4A^2$ is false

The $b_2=0$ branch is lost before the modular obstruction is applied