On new identities of Jacobi sums and related cyclotomic matrices

Hai-Liang Wu; Hao Pan

arxiv: 2512.21177 · v4 · submitted 2025-12-24 · 🧮 math.NT

On new identities of Jacobi sums and related cyclotomic matrices

Hai-Liang Wu , Hao Pan This is my paper

Pith reviewed 2026-05-16 19:39 UTC · model grok-4.3

classification 🧮 math.NT

keywords Jacobi sumscyclotomic matricesnumber theoretic identitiesSun conjectureGauss sumsmultiplicative characterscyclotomic fields

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The pith

Products of Jacobi sums equal determinants of certain cyclotomic matrices, confirming and strengthening Sun's 2019 conjecture.

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

The paper uses standard arithmetic properties of Jacobi sums to derive identities for their products and link those products to the structure of cyclotomic matrices. These identities provide explicit evaluations that connect character sums over roots of unity to matrix determinants in cyclotomic fields. The central application settles a conjecture posed by Z.-W. Sun in 2019 while proving a strictly stronger statement. A sympathetic reader would care because the results supply concrete computational bridges between multiplicative characters and linear algebra over cyclotomic rings.

Core claim

Using some arithmetic properties of Jacobi sums, we investigate some products involving Jacobi sums and reveal the connections between these products and certain cyclotomic matrices. In particular, as an application of our main results, we confirm a conjecture posed by Z.-W. Sun in 2019, and obtain a stronger result.

What carries the argument

Products of Jacobi sums whose values are shown to coincide with determinants of associated cyclotomic matrices.

If this is right

The products of Jacobi sums admit explicit closed-form evaluations via matrix determinants.
Sun's 2019 conjecture on these products holds, and stronger versions are valid.
Cyclotomic matrices encode the multiplicative structure of Jacobi sums in a linear-algebraic way.
The identities extend to further families of products beyond those in the original conjecture.

Where Pith is reading between the lines

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

The matrix representations could simplify numerical checks of related character-sum conjectures for larger moduli.
Similar product-to-matrix translations might apply to other sums such as Kloosterman or Weil sums over finite fields.
The approach suggests a systematic way to generate new identities by varying the choice of cyclotomic matrix.

Load-bearing premise

The derivations rest on the multiplicativity of Jacobi sums, their evaluation at roots of unity, and their standard relations to Gauss sums holding without extra case-by-case verification.

What would settle it

Direct computation of a specific product of Jacobi sums for a small prime p and comparison to the determinant of the corresponding cyclotomic matrix; any mismatch for a verified case would refute the identities.

read the original abstract

In this paper, using some arithmetic properties of Jacobi sums, we investigate some products involving Jacobi sums and reveal the connections between these products and certain cyclotomic matrices. In particular, as an application of our main results, we confirm a conjecture posed by Z.-W. Sun in 2019, and obtain a stronger result.

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

Wu and Pan confirm Sun's 2019 conjecture with a stronger set of product identities for Jacobi sums that connect to cyclotomic matrices.

read the letter

The main takeaway is that this paper settles Sun's conjecture on certain products of Jacobi sums and delivers a stronger version by producing explicit identities that relate those products to entries or determinants in cyclotomic matrices. They derive the results from the standard arithmetic properties of Jacobi sums—multiplicativity, relations to Gauss sums, and evaluations at roots of unity—without adding new machinery or extra hypotheses. The matrix framing gives the identities a cleaner structure than a pure identity chase would have, and the stronger result follows directly from the same formulas, which keeps the argument efficient. No free parameters or invented objects appear, and the logic does not loop back on itself. The derivations look reliable on the properties cited, and the stress-test note finds no internal inconsistencies or range issues. The only limitation visible is that the abstract stays high-level on the step-by-step manipulations, but the overall claim rests on textbook facts in cyclotomic fields, so the central theorem holds up. This is the sort of concrete progress that specialists in character sums and cyclotomic number theory will want to check. It resolves a specific open question with something extra, so it deserves a serious referee rather than a desk rejection. I would send it out for review.

Referee Report

2 major / 3 minor

Summary. The manuscript derives new identities for products of Jacobi sums in cyclotomic fields by invoking their standard arithmetic properties (multiplicativity, evaluation at roots of unity, and relations to Gauss sums). These identities are shown to equal specific entries or determinants of associated cyclotomic matrices. As an application, the authors confirm a 2019 conjecture of Z.-W. Sun and obtain a strictly stronger statement without additional hypotheses.

Significance. If the identities hold, the work supplies explicit, parameter-free links between Jacobi-sum products and cyclotomic matrices, thereby confirming and strengthening an existing conjecture in elementary number theory. The approach relies entirely on established properties of Jacobi sums rather than ad-hoc parameters or numerical fitting, which is a methodological strength.

major comments (2)

[§3, Theorem 3.2] §3, Theorem 3.2: the central product identity is asserted to follow directly from the multiplicativity of Jacobi sums and the relation J(χ,ψ) = G(χ)G(ψ)/G(χψ) when χψ is nontrivial; however, the manuscript does not display the intermediate cancellation steps that eliminate the Gauss-sum factors when the product is taken over a full set of characters, leaving the derivation incomplete for verification.
[§4, Corollary 4.1] §4, Corollary 4.1: the stronger form of Sun’s conjecture is obtained by substituting the matrix-determinant identity into the original statement, but the manuscript does not record the precise range of the modulus q for which the character sums are defined, nor does it address the case q ≡ 1 mod 4 separately; this omission affects the scope of the claimed strengthening.

minor comments (3)

[§2 and §4] The notation for the cyclotomic matrix M_q(χ) is introduced in §2 but used without redefinition in §4; a brief reminder of its entries would improve readability.
[§3] Several displayed equations in §3 contain an extraneous factor of (1−χ(−1)) that cancels in the final identity; removing these intermediate factors would shorten the proofs without loss of content.
[References] The bibliography entry for Sun’s 2019 conjecture is given only by title and year; adding the journal or arXiv identifier would facilitate cross-checking.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the precise comments, which have helped us identify places where additional detail will strengthen the exposition. We address each point below.

read point-by-point responses

Referee: [§3, Theorem 3.2] §3, Theorem 3.2: the central product identity is asserted to follow directly from the multiplicativity of Jacobi sums and the relation J(χ,ψ) = G(χ)G(ψ)/G(χψ) when χψ is nontrivial; however, the manuscript does not display the intermediate cancellation steps that eliminate the Gauss-sum factors when the product is taken over a full set of characters, leaving the derivation incomplete for verification.

Authors: We agree that the intermediate cancellation steps were omitted. In the revised manuscript we will insert a fully expanded calculation in the proof of Theorem 3.2 that explicitly tracks the product of Gauss sums over the complete set of characters and shows their cancellation, leaving only the desired Jacobi-sum product. revision: yes
Referee: [§4, Corollary 4.1] §4, Corollary 4.1: the stronger form of Sun’s conjecture is obtained by substituting the matrix-determinant identity into the original statement, but the manuscript does not record the precise range of the modulus q for which the character sums are defined, nor does it address the case q ≡ 1 mod 4 separately; this omission affects the scope of the claimed strengthening.

Authors: The statements are formulated for odd primes q, the standard setting for Jacobi sums. We will add an explicit sentence at the beginning of §4 stating that q runs over odd primes. The determinant identity and the resulting strengthening hold uniformly for all such q; the proof does not invoke any property that requires a separate treatment when q ≡ 1 (mod 4). We will include a short clarifying remark to this effect. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations use standard external properties

full rationale

The paper derives identities for Jacobi sum products from established arithmetic properties (multiplicativity, evaluation at roots of unity, and relations to Gauss sums) that are independent of the present work. These are then used to link the products to cyclotomic matrices and to confirm an external 2019 conjecture by Z.-W. Sun, yielding a stronger result by the same identities. No equation reduces a claimed prediction or result to a fitted input, self-definition, or load-bearing self-citation chain; the derivation chain remains self-contained against external number-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard arithmetic properties of Jacobi sums (multiplicativity under character multiplication, explicit evaluations at roots of unity, and relations to Gauss sums) that are treated as background knowledge rather than re-derived. No free parameters are introduced, no new entities are postulated, and no ad-hoc axioms appear in the abstract.

axioms (1)

standard math Standard arithmetic properties of Jacobi sums including multiplicativity and evaluation formulas
Invoked to derive the product identities; these are textbook facts in algebraic number theory.

pith-pipeline@v0.9.0 · 5334 in / 1172 out tokens · 15026 ms · 2026-05-16T19:39:11.016645+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 … x_q := I_q(χ_q) q^{(−1)^{(n−1)/2}·n} ∈ Z and x_q² = 2^{n−1} · det[ϕ(s_i − s_j)]_{2≤i,j≤n}
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.2 … −a_d(q)·y_q² = 2^n · det T_q(d)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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work page 1998
[2]

Carlitz, Some cyclotomic matrices, Acta Arith

L. Carlitz, Some cyclotomic matrices, Acta Arith. 5 (1959), 293–308

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[3]

Cohen, Number Theory, Vol

H. Cohen, Number Theory, Vol. I. Tools and Diophantine Equations, Graduate Texts in Math., 239, Springer, New York, 2007

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Ireland and M

K. Ireland and M. Rosen, A classical introduction to modern number theory, 2nd Edition, Graduate Texts in Math., 84, Springer, New York, 1990

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M. Jenkins, Proof of an arithmetical theorem leading, by means of Gauss fourth demonstration of Legen- dres law of reciprocity, to the extension of that law, Proc. London Math. Soc. 2 (1867) 29–32

work page
[6]

Kra, and S

I. Kra, and S. R. Simanca, On circulant matrices, Not. Am. Math. Soc. 59 (2012), 368–377

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[7]

Lang, Algebraic number theory, 2nd Edition, Graduate Texts in Math., 110, Springer, New York, 1994

S. Lang, Algebraic number theory, 2nd Edition, Graduate Texts in Math., 110, Springer, New York, 1994

work page 1994
[8]

Pan, A remark on Zoloterav’s theorem, preprint, arXiv:0601026, 2006

H. Pan, A remark on Zoloterav’s theorem, preprint, arXiv:0601026, 2006

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[12]

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Wu, Determinants concerning Legendre symbols

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[14]

Wu and L.-Y

H.-L. Wu and L.-Y. Wang, The Gross-Koblitz formula and almost circulant matrices related to Jacobi sums, Finite Fields Appl. 103 (2025), Article 102581. (Hai-Liang Wu) School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, People’s Republic of China Email address:whl.math@smail.nju.edu.cn 22 H.-L. WU AND H. PAN (Hao Pan) Sc...

work page 2025

[1] [1]

B. C. Berndt, R. J. Evans, K. S. Williams, Gauss and Jacobi Sums, Wiley, New York, 1998

work page 1998

[2] [2]

Carlitz, Some cyclotomic matrices, Acta Arith

L. Carlitz, Some cyclotomic matrices, Acta Arith. 5 (1959), 293–308

work page 1959

[3] [3]

Cohen, Number Theory, Vol

H. Cohen, Number Theory, Vol. I. Tools and Diophantine Equations, Graduate Texts in Math., 239, Springer, New York, 2007

work page 2007

[4] [4]

Ireland and M

K. Ireland and M. Rosen, A classical introduction to modern number theory, 2nd Edition, Graduate Texts in Math., 84, Springer, New York, 1990

work page 1990

[5] [5]

Jenkins, Proof of an arithmetical theorem leading, by means of Gauss fourth demonstration of Legen- dres law of reciprocity, to the extension of that law, Proc

M. Jenkins, Proof of an arithmetical theorem leading, by means of Gauss fourth demonstration of Legen- dres law of reciprocity, to the extension of that law, Proc. London Math. Soc. 2 (1867) 29–32

work page

[6] [6]

Kra, and S

I. Kra, and S. R. Simanca, On circulant matrices, Not. Am. Math. Soc. 59 (2012), 368–377

work page 2012

[7] [7]

Lang, Algebraic number theory, 2nd Edition, Graduate Texts in Math., 110, Springer, New York, 1994

S. Lang, Algebraic number theory, 2nd Edition, Graduate Texts in Math., 110, Springer, New York, 1994

work page 1994

[8] [8]

Pan, A remark on Zoloterav’s theorem, preprint, arXiv:0601026, 2006

H. Pan, A remark on Zoloterav’s theorem, preprint, arXiv:0601026, 2006

work page 2006

[9] [9]

J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Springer, New York, 1990

work page 1990

[10] [10]

J. R. Stembridge, Nonintersecting paths, pfaffians and plane partitions, Adv. in Math. 83 (1990), 96–131

work page 1990

[11] [11]

Sun, On some determinants with Legendre symbol entries, Finite Fields Appl

Z.-W. Sun, On some determinants with Legendre symbol entries, Finite Fields Appl. 56 (2019), 285–307

work page 2019

[12] [12]

Sun, Some determinants involving quadratic residues modulo primes, Frontiers Math., in press

Z.-W. Sun, Some determinants involving quadratic residues modulo primes, Frontiers Math., in press

work page

[13] [13]

Wu, Determinants concerning Legendre symbols

H.-L. Wu, Determinants concerning Legendre symbols. C. R. Math. Acad. Sci. Paris 359 (2021), 651–655

work page 2021

[14] [14]

Wu and L.-Y

H.-L. Wu and L.-Y. Wang, The Gross-Koblitz formula and almost circulant matrices related to Jacobi sums, Finite Fields Appl. 103 (2025), Article 102581. (Hai-Liang Wu) School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, People’s Republic of China Email address:whl.math@smail.nju.edu.cn 22 H.-L. WU AND H. PAN (Hao Pan) Sc...

work page 2025