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Paper metadata

Original submission
October 19, 2026
Claimed MSC
11R58, 11L05, 11J91
arXiv categories
math.NT
Paper license
CC BY 4.0 ↗

Abstract

In this paper, we establish an explicit algebra isomorphism between the normalized ∞\infty-adic arithmetic gamma values and their vv-adic counterparts. The construction relies on Thakur's arithmetic Gauss sums, which arise naturally from the Gross-Koblitz-Thakur relations. Our result demonstrates the phenomenon that Gauss sums, commonly regarded as "finite" analogs of gamma functions, serve as a bridge connecting the ∞\infty-adic and vv-adic gamma values.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues 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 October 23, 2026
01Statements2 reported findingsNo statement issues found

The stated algebra isomorphism between the normalized infinity-adic and finite-place arithmetic gamma-value algebras has the advertised domain, generators, and Gauss-sum multiplier. The finite-level formulation and its extension by the translation relations are consistent. No mathematical statement requiring correction was found.

Theorem 2.3.4No issue found

The generator assignment defines the claimed algebra isomorphism

PDF pages 8 and 15–18 · Theorem 2.3.4 and Sections 3.3–3.4 · arXiv:2610.26398v1

At level ℓ\ell, both normalized value algebras have transcendence basis indexed by d≤i<dℓd\leq i<d\ell. The strong-multiplication and Gross–Koblitz relations give the same binomial presentation on the two sides, while the cited algebraic-independence results show that this presentation has no additional kernel. Identifying the two quotient presentations therefore sends Π∞ari(y)\Pi^{\mathrm{ari}}_\infty(y) to Πvari(y)Gari(y)\Pi^{\mathrm{ari}}_v(y)G^{\mathrm{ari}}(y), and the translation relations extend the map from a fundamental interval to every permitted rational argument.

Propositions 3.2.1–3.2.3No issue found

The normalized values satisfy matching monomial relations

PDF pages 12–14 · Section 3.2 · arXiv:2610.26398v1

The normalization exponents transform under translation and carry-free base-qq multiplication exactly as required. Thakur's formula cancels the finite-place gamma factor with the arithmetic Gauss sum on the basic generators, yielding the same Gross–Koblitz product relation as on the infinity-adic side.

02Proofs2 reported findingsNo proof issues found

The proof reduces the problem to identical distribution-relation presentations and verifies the necessary translation, multiplication, and Gross–Koblitz relations before invoking the stated transcendence inputs. The rank computations, basis choices, localization at monomials, and final extension are coherent. No incorrect or materially incomplete proof was found.

Propositions 3.3.4 and 3.3.7No issue found in the reviewed proof

The two distribution groups have the required common rank

PDF pages 16–17 · Propositions 3.3.4 and 3.3.7 · arXiv:2610.26398v1

After the strong-multiplication relations reduce to the dℓd\ell digit generators, the normalization differences give an independent family of size dℓ−dd\ell-d on the infinity-adic side. On the finite-place side, the dd independent Gross–Koblitz orbit sums eliminate precisely the first dd generators. Tensoring with Q\mathbb Q supplies the rank bound, and the exhibited generators then give the asserted free bases.

Section 3.4No issue found in the reviewed proof

The quotient-ring comparison has no hidden kernel

PDF pages 17–18 · Proof of Theorem 2.3.4 · arXiv:2610.26398v1

Each evaluation factors through the same binomial ideal. The algebraically independent basis values identify the localized quotient with a Laurent polynomial algebra in dℓ−dd\ell-d variables, which makes both evaluation maps injective as well as surjective. Composing them gives the desired isomorphism, and Proposition 3.2.1 propagates it to all translates.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper
arXiv:2610.26398v1
Authors listed
Ting-Wei Chang
Audit date
October 23, 2026
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