The Gauss periods and cyclotomic matrices involving Gauss sums over cyclic groups

Hai-Liang Wu; Li-Yuan Wang

arxiv: 2607.02392 · v1 · pith:4E37UZJUnew · submitted 2026-07-02 · 🧮 math.NT

The Gauss periods and cyclotomic matrices involving Gauss sums over cyclic groups

Hai-Liang Wu , Li-Yuan Wang This is my paper

Pith reviewed 2026-07-03 06:28 UTC · model grok-4.3

classification 🧮 math.NT

keywords Gauss sumsGauss periodscyclotomic matrixcharacter sumscyclic groupsprime powersnumber theory

0 comments

The pith

The cyclotomic matrix of Gauss sums G_N(χ^{ki+kj}) is studied via Gauss periods when N is a prime power.

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

The paper uses the arithmetic properties of Gauss periods and character sums to examine the matrix A_k(χ) whose entries are the indicated Gauss sums over Z/NZ. The setup requires N = p^m so that the unit group is cyclic and χ generates its character group, with k dividing φ(N). A reader would care because this connects explicit character-sum evaluations to matrix constructions that appear in the study of cyclotomic extensions and related sums. The work derives structural information about the matrix directly from the known addition formulas and orthogonality relations for these sums.

Core claim

By applying the arithmetic properties of Gauss periods and character sums over cyclic groups, the cyclotomic matrix A_k(χ) = [G_N(χ^{ki+kj})]_{0≤i,j≤φ(N)/k−1} is studied, where the Gauss sum is defined by the standard sum over Z/NZ of the character times the additive character.

What carries the argument

The cyclotomic matrix A_k(χ) whose (i,j)-entry is the Gauss sum G_N(χ^{ki+kj}), which assembles scaled character values into a square array indexed by residue classes modulo k.

If this is right

The entries of A_k(χ) admit explicit expressions in terms of Gauss periods indexed by the subgroup of index k.
Linear-algebraic invariants of the matrix, such as its determinant or eigenvalues, become computable from the same period arithmetic.
The construction applies uniformly for every divisor k of φ(N) whenever N is a prime power.
Orthogonality relations among the scaled characters translate into row or column relations inside the matrix.

Where Pith is reading between the lines

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

The same matrix construction might be compared with the character table of the quotient group (Z/NZ)*/(subgroup of index k) to reveal further factorizations.
If the matrix is invertible over the cyclotomic field, its inverse could supply a change-of-basis between different bases of Gauss sums.
The approach suggests analogous matrices could be defined when the additive character is replaced by other additive characters of higher conductor.

Load-bearing premise

N must be a prime power so the multiplicative group modulo N is cyclic and χ must be a fixed generator of the full character group.

What would settle it

Explicit evaluation of all entries of A_k(χ) for N=9, k=3 and a generator χ, compared against the values predicted by the Gauss-period formulas used in the paper.

read the original abstract

In this paper, by using the arithmetic properties of the Gauss periods and character sums over cyclic groups, we study the cyclotomic matrix $$A_k(\chi)=\left[G_N(\chi^{ki+ki})\right]_{0\le i,j\le \varphi(N)/k-1},$$ where $N=p^m$ is a prime power, $\varphi(\cdot)$ is the Euler totient function, $k$ is a divisor of $\varphi(N)$, $\chi$ is a generator of character group $\widehat{(\mathbb{Z}/N\mathbb{Z})^{\times}}$, and $$G_N(\chi^{ki+kj})=\sum_{x\in\mathbb{Z}/N\mathbb{Z}}\chi^{ki+kj}(x)e^{2\pi ix/N}$$ is the Gauss sum over $\mathbb{Z}/N\mathbb{Z}$.

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

The paper sets up a specific matrix of Gauss sums indexed by multiples of a divisor k and studies its properties over prime-power moduli using classical tools.

read the letter

The punchline is that this work defines the matrix A_k(χ) with entries G_N(χ^{ki+kj}) for i,j running up to φ(N)/k-1, where N=p^m and χ generates the character group, then applies Gauss periods and character sums to it.

What is new is the concrete indexing and the choice to fix k dividing φ(N) while keeping the group cyclic. The setup is clean and the restriction to prime powers is stated up front so the cyclicity assumption is transparent.

The paper does the basic job of writing down the definition with standard notation and tying it to existing arithmetic of Gauss sums. That part is straightforward and honest.

The soft spots are that the abstract gives no hint of the actual theorems proved—no determinant, no eigenvalue formula, no relation to periods beyond the setup. Without those details the contribution looks narrow even if the calculations are correct. The scope stays inside one family of matrices in algebraic number theory, so impact outside that corner is limited.

Citation pattern follows the usual references for Gauss sums and cyclotomic fields; nothing suspicious there.

This is for readers already working on explicit character-sum constructions or cyclotomic matrices. A specialist might extract a technical lemma or two, but most people will skip it.

I would send it to peer review so the calculations can be checked, though the narrow focus means it belongs in a specialized journal rather than a general one.

Referee Report

0 major / 2 minor

Summary. The paper studies the cyclotomic matrix A_k(χ) = [G_N(χ^{ki + kj})]_{0 ≤ i,j ≤ φ(N)/k − 1} where N = p^m is a prime power (ensuring (Z/NZ)* is cyclic), k divides φ(N), and χ generates the character group of (Z/NZ)*. It employs arithmetic properties of Gauss periods and character sums over cyclic groups to analyze this matrix whose entries are Gauss sums G_N(ψ) = ∑_{x ∈ Z/NZ} ψ(x) exp(2π i x / N).

Significance. If the claimed relations between the matrix entries, Gauss periods, and character sums hold, the work could clarify structural properties of Gauss-sum matrices in the cyclic case, potentially aiding computations in cyclotomic fields or character-sum estimates.

minor comments (2)

[Abstract] Abstract: the matrix entry is written as G_N(χ^{ki+ki}), which is evidently a typographical error for G_N(χ^{ki + kj}).
[Abstract] Abstract, definition of G_N: the sum is taken over all x ∈ Z/NZ, but the paper should explicitly note whether any standard normalization (e.g., division by √N) or restriction to units is applied when relating the entries to Gauss periods.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work on the cyclotomic matrix A_k(χ) and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines the matrix A_k(χ) explicitly via Gauss sums G_N(χ^{ki+kj}) for N=p^m prime power and χ a generator of the character group, then states that it studies properties of this matrix using arithmetic properties of Gauss periods and character sums. No derivation chain, prediction, or result is exhibited that reduces by construction to the inputs, fitted parameters, or self-citations; the work is an investigation of explicitly defined objects under stated hypotheses, with the central claim remaining independent of any tautological reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on the standard fact that (Z/NZ)* is cyclic when N is a prime power and on the existence of a generator character χ; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption N = p^m is a prime power, so (Z/NZ)* is cyclic
Required for χ to be a generator of the full character group; stated in the matrix definition.
domain assumption χ is a generator of the character group of (Z/NZ)*
Used to index the powers ki + kj in the Gauss sum entries.

pith-pipeline@v0.9.1-grok · 5672 in / 1297 out tokens · 28471 ms · 2026-07-03T06:28:27.446668+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 1 canonical work pages

[1]

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

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[2]

Carlitz, Some cyclotomic matrices, Acta Arith

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

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Cohen, Number theory, Vol

H. Cohen, Number theory, Vol. I. Tools and Diophantine equations, Springer, New York, 2007

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Iwaniec and E

H. Iwaniec and E. Kowalski, Analytic Number Theory, Vol. 53, American Mathematical Society, 2004

2004
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Newman, Roots of unity and covering sets, Math

M. Newman, Roots of unity and covering sets, Math. Ann. 191 (1971), 278–282

1971
[6]

H.-L. Wu, J. Li, L.-Y. Wang and C. H. Yip, On cyclotomic matrices involving Gauss sums over finite fields, Proc. Amer. Math. Soc. 153 (2025), 1411–1424

2025
[7]

Wu, L.-Y

H.-L. Wu, L.-Y. Wang and H. Pan, Onp-th cyclotomic field and cyclotomic matrices involving Jacobi sums, preprint, arXiv:2506.14316. (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 (Li-Yuan W ang) School of Physical and Mathematical Scie...

work page arXiv

[1] [1]

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

1998

[2] [2]

Carlitz, Some cyclotomic matrices, Acta Arith

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

1959

[3] [3]

Cohen, Number theory, Vol

H. Cohen, Number theory, Vol. I. Tools and Diophantine equations, Springer, New York, 2007

2007

[4] [4]

Iwaniec and E

H. Iwaniec and E. Kowalski, Analytic Number Theory, Vol. 53, American Mathematical Society, 2004

2004

[5] [5]

Newman, Roots of unity and covering sets, Math

M. Newman, Roots of unity and covering sets, Math. Ann. 191 (1971), 278–282

1971

[6] [6]

H.-L. Wu, J. Li, L.-Y. Wang and C. H. Yip, On cyclotomic matrices involving Gauss sums over finite fields, Proc. Amer. Math. Soc. 153 (2025), 1411–1424

2025

[7] [7]

Wu, L.-Y

H.-L. Wu, L.-Y. Wang and H. Pan, Onp-th cyclotomic field and cyclotomic matrices involving Jacobi sums, preprint, arXiv:2506.14316. (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 (Li-Yuan W ang) School of Physical and Mathematical Scie...

work page arXiv