Cyclotomic Period Matrices

• Cyclotomic period matrices are square matrices built from cyclotomic data like Gauss sums, Jacobi sums, and cyclotomic numbers, capturing key arithmetic and geometric properties.
• They feature diagonalization via Vandermonde-type factorizations and reveal determinant criteria and spectral structure that distinguish finite-field and prime-power settings.
• These matrices have practical applications in constructing explicit abelian extensions, coding theory, and combinatorial design by linking character sums with hypergeometric functions.

to=arxiv_search 天天中彩票是不是.json code ауаҩы code: {"query":"(Wu et al., 2 Jul 2026) cyclotomic matrices Gauss periods character sums cyclic groups", "max_results": 5, "sort_by": "submittedDate"} to=arxiv_search 微信天天中彩票 全民彩票天天送json {"query":"(Wu et al., 2 Jul 2026) cyclotomic matrices Gauss periods character sums cyclic groups","max_results":5,"sort_by":"submittedDate"} Cyclotomic period matrices are square matrices whose entries are organized from cyclotomic data: Gauss sums over cyclic groups, Jacobi sums and multiplicative characters over finite fields, cyclotomic numbers, or, in a geometric setting, period integrals whose values lie in a cyclotomic field. In the recent literature, the phrase does not denote a single universal object; rather, it names a family of constructions linked by character-theoretic diagonalization, Vandermonde-type factorizations, Galois symmetry, and determinant formulas with arithmetic content (Wu et al., 2 Jul 2026, Wu et al., 2021, Wu et al., 24 Dec 2025, Ahmed et al., 2018, Tadokoro, 2012).

1. Definitions and scope

A central arithmetic model is the prime-power cyclotomic period matrix introduced for $N=p^m$, with $N$ an odd prime power, $n=\varphi(N)$, $k\mid n$, and $d=n/k$. Fix a generator $\chi$ of the character group $\widehat{(\mathbb Z/N\mathbb Z)^\times}$. For each integer $r$, define the Gauss sum

$$G_N(\chi^r)=\sum_{x\in\mathbb Z/N\mathbb Z}\chi^r(x)e^{2\pi i x/N},$$

extended to zero on non-units $x$. The associated $N$0 matrix is

$N$1

This is the Wu–Wang cyclotomic period matrix built from Gauss sums over $N$2 (Wu et al., 2 Jul 2026).

A second family appears over finite fields. For $N$3 an odd prime power, a generator $N$4 of the full multiplicative-character group, and $N$5 the set of nonzero squares in $N$6, Wu–She–Wang define

$N$7

together with

$N$8

where $N$9 is the quadratic character (Wu et al., 2021). Closely related matrices in Wu–Pan are

$n=\varphi(N)$0

and, when $n=\varphi(N)$1 and $n=\varphi(N)$2,

$n=\varphi(N)$3

with $n=\varphi(N)$4 (Wu et al., 24 Dec 2025).

A third family packages cyclotomic numbers. For $n=\varphi(N)$5, $n=\varphi(N)$6, and cyclotomic numbers

$n=\varphi(N)$7

Ahmed–Tanti–Hoque assemble the array $n=\varphi(N)$8 into $n=\varphi(N)$9 matrices $k\mid n$0 or $k\mid n$1, depending on the parity of $k\mid n$2 in $k\mid n$3 (Ahmed et al., 2018).

In algebraic geometry, the term “period matrix” has its classical analytic meaning. Tadokoro studies the hyperelliptic curve

$k\mid n$4

constructs a symplectic basis of $k\mid n$5, and obtains a period matrix $k\mid n$6 whose entries lie in the cyclotomic field $k\mid n$7, $k\mid n$8 (Tadokoro, 2012). This establishes a geometric sense in which a period matrix is “cyclotomic.”

2. Prime-power Gauss-sum matrices and Gauss periods

For the matrix $k\mid n$9, the decisive auxiliary quantities are the Gauss periods. Choose a generator $d=n/k$0 of $d=n/k$1, write $d=n/k$2 with $d=n/k$3, and let $d=n/k$4 be the unique subgroup of order $d=n/k$5 in $d=n/k$6. The Gauss period of length $d=n/k$7 is

$d=n/k$8

A reindexing argument gives

$d=n/k$9

In matrix form,

$\chi$0

where

$\chi$1

This $\chi$2-decomposition is the structural core of the theory (Wu et al., 2 Jul 2026).

The Vandermonde-type factor $\chi$3 satisfies the orthogonality relation

$\chi$4

Consequently,

$\chi$5

These identities convert the arithmetic of Gauss periods into explicit linear-algebraic statements about $\chi$6 (Wu et al., 2 Jul 2026).

The determinant criterion is particularly sharp. If $\chi$7, then $\chi$8 contains the unique Sylow-$\chi$9 subgroup of $\widehat{(\mathbb Z/N\mathbb Z)^\times}$0, one shows $\widehat{(\mathbb Z/N\mathbb Z)^\times}$1 for every $\widehat{(\mathbb Z/N\mathbb Z)^\times}$2, and hence $\widehat{(\mathbb Z/N\mathbb Z)^\times}$3. If $\widehat{(\mathbb Z/N\mathbb Z)^\times}$4, then none of the Gauss periods vanishes, $\widehat{(\mathbb Z/N\mathbb Z)^\times}$5 is invertible, and $\widehat{(\mathbb Z/N\mathbb Z)^\times}$6 is invertible. Writing

$\widehat{(\mathbb Z/N\mathbb Z)^\times}$7

one obtains

$\widehat{(\mathbb Z/N\mathbb Z)^\times}$8

Accordingly, $\widehat{(\mathbb Z/N\mathbb Z)^\times}$9 is singular if and only if $r$0 (Wu et al., 2 Jul 2026).

Concrete instances illustrate the formalism. For $r$1, $r$2, $r$3, one has $r$4, $r$5, $r$6, and

$r$7

with rank $r$8 and determinant $r$9. For $$G_N(\chi^r)=\sum_{x\in\mathbb Z/N\mathbb Z}\chi^r(x)e^{2\pi i x/N},$$0, $$G_N(\chi^r)=\sum_{x\in\mathbb Z/N\mathbb Z}\chi^r(x)e^{2\pi i x/N},$$1, $$G_N(\chi^r)=\sum_{x\in\mathbb Z/N\mathbb Z}\chi^r(x)e^{2\pi i x/N},$$2, $$G_N(\chi^r)=\sum_{x\in\mathbb Z/N\mathbb Z}\chi^r(x)e^{2\pi i x/N},$$3, $$G_N(\chi^r)=\sum_{x\in\mathbb Z/N\mathbb Z}\chi^r(x)e^{2\pi i x/N},$$4, none of the three Gauss periods $$G_N(\chi^r)=\sum_{x\in\mathbb Z/N\mathbb Z}\chi^r(x)e^{2\pi i x/N},$$5 vanishes, and

$$G_N(\chi^r)=\sum_{x\in\mathbb Z/N\mathbb Z}\chi^r(x)e^{2\pi i x/N},$$6

(Wu et al., 2 Jul 2026).

3. Spectral structure and diagonalization

The matrix factorization $$G_N(\chi^r)=\sum_{x\in\mathbb Z/N\mathbb Z}\chi^r(x)e^{2\pi i x/N},$$7 also determines the spectrum. Since $$G_N(\chi^r)=\sum_{x\in\mathbb Z/N\mathbb Z}\chi^r(x)e^{2\pi i x/N},$$8, the matrix $$G_N(\chi^r)=\sum_{x\in\mathbb Z/N\mathbb Z}\chi^r(x)e^{2\pi i x/N},$$9 is similar, up to signs and powers of $x$0, to the diagonal matrix $x$1. Its eigenvalues are exactly

$x$2

up to an overall sign $x$3, and a full set of eigenvectors is given by the columns of $x$4. In particular,

$x$5

while $x$6 when $x$7 (Wu et al., 2 Jul 2026).

A parallel eigenvector mechanism governs the finite-field matrices. For $x$8, Wu–She–Wang diagonalize using the $x$9 independent vectors

$N$00

The $N$01-th eigenvalue is

$N$02

which splits into two Jacobi sums and can then be rewritten as a difference or sum of finite-field $N$03-values. The determinant formula for $N$04 is the product of these eigenvalues (Wu et al., 2021).

Wu–Pan formulate the same phenomenon in terms of

$N$05

for $N$06. The vectors $N$07 are eigenvectors of $N$08 with eigenvalues $N$09. When $N$10, the enlarged matrix $N$11 acquires two additional trivial eigenvectors, and its full spectrum is

$N$12

(Wu et al., 24 Dec 2025).

A common misconception is to treat all singularity criteria as identical across these constructions. The cited works show otherwise. For $N$13, singularity is governed by the divisibility condition $N$14. For $N$15, one vanishing criterion is instead $N$16 together with $N$17. For the cyclotomic-number matrices of order $N$18, $N$19 is singular while $N$20 is non-singular in the cases analyzed. The shared diagonalization machinery does not force uniform determinant behavior.

4. Jacobi sums, finite-field hypergeometric functions, and determinant formulas

In the finite-field framework of Wu–She–Wang, the determinant $N$21 admits explicit formulas controlled by Jacobi sums and Greene’s hypergeometric functions. For characters $N$22 on $N$23,

$N$24

and Greene’s finite-field binomial coefficient and hypergeometric function are defined by

$N$25

$N$26

A key identity is

$N$27

which translates the eigenvalue sums into Jacobi-sum and hypergeometric expressions (Wu et al., 2021).

The principal arithmetic cases are as follows. If $N$28 and $N$29, then

$N$30

If $N$31, then $N$32 is given by an explicit product of terms

$N$33

with the minus sign when $N$34 and the plus sign when $N$35. If $N$36 and $N$37, then

$N$38

and $N$39 is expressed as a product of differences of $N$40-values at $N$41. If $N$42 and $N$43, writing $N$44 and

$N$45

one has

$N$46

and $N$47 is expressed as a product of $N$48-values at $N$49 (Wu et al., 2021).

The determinant $N$50 of the matrix $N$51 also exhibits square-factorization phenomena. If $N$52, then there exists $N$53 such that

$N$54

If $N$55, then there is $N$56 with

$N$57

Moreover, if $N$58 is prime, then

$N$59

for some $N$60, where $N$61 is the class number of $N$62. This confirms Zhi-Wei Sun’s 2019 conjecture on the explicit form of $N$63 (Wu et al., 2021).

Wu–Pan obtain a related pair of determinant formulas from Jacobi-sum products. Define

$N$64

When $N$65, they show

$N$66

When $N$67 and $N$68 is a nonsquare, they define

$N$69

show $N$70, and prove

$N$71

where $N$72 is the Frobenius trace on the elliptic curve $N$73. This yields the stated resolution of Sun’s 2019 conjecture (Wu et al., 24 Dec 2025).

5. Cyclotomic numbers of order $N$74 and block-circulant matrices

For $N$75, the cyclotomic numbers $N$76 and Jacobi sums $N$77 are related by the discrete Fourier identities

$N$78

Thus, knowing all Jacobi sums is equivalent to knowing all cyclotomic numbers (Ahmed et al., 2018).

Ahmed–Tanti–Hoque then form the “odd-$N$79 period matrix” $N$80 and “even-$N$81 period matrix” $N$82 by

$N$83

used when $N$84, and

$N$85

used when $N$86. These matrices satisfy periodicity in both indices, and their structural symmetries depend on the parity of $N$87: $N$88

$N$89

In particular, each of $N$90 and $N$91 is block-circulant of the form

$N$92

where $N$93 are $N$94 circulant blocks (Ahmed et al., 2018).

The row-sum behavior and singularity properties are markedly different in the two parity regimes. The row sums of both $N$95 and $N$96 are all equal to $N$97. The matrix $N$98 has one zero row, hence $N$99, and in fact its minimal and characteristic polynomial is $n=\varphi(N)$00, so all eigenvalues of $n=\varphi(N)$01 vanish. By contrast, $n=\varphi(N)$02 is non-singular. For $n=\varphi(N)$03, one finds

$n=\varphi(N)$04

and the characteristic polynomial factors into $n=\varphi(N)$05 distinct real eigenvalues (Ahmed et al., 2018).

This family emphasizes that “cyclotomic period matrix” may refer not to character sums directly but to the matrix obtained after the cyclotomic numbers themselves have been determined. The Fourier relation with Jacobi sums shows that this is not a separate theory so much as a different packaging of the same cyclotomic arithmetic.

6. Cyclotomic fields and geometric period matrices

The prime-power Gauss-period matrices have an intrinsic cyclotomic-field interpretation. The field $n=\varphi(N)$06 is Galois over $n=\varphi(N)$07 with cyclic Galois group isomorphic to $n=\varphi(N)$08. Inside it lies the unique intermediate subfield of degree $n=\varphi(N)$09, namely $n=\varphi(N)$10, and the periods $n=\varphi(N)$11 are exactly the Galois conjugates of the base period $n=\varphi(N)$12. Hence

$n=\varphi(N)$13

is the minimal polynomial of the period, and the determinant formula for $n=\varphi(N)$14 links the size of the matrix to the arithmetic of the subfield $n=\varphi(N)$15 (Wu et al., 2 Jul 2026).

Tadokoro’s work exhibits a geometric incarnation of the same cyclotomic phenomenon. On the hyperelliptic curve

$n=\varphi(N)$16

with $n=\varphi(N)$17, the deck transformation

$n=\varphi(N)$18

has order $n=\varphi(N)$19. A basis of holomorphic differentials is

$n=\varphi(N)$20

For the loops $n=\varphi(N)$21, the normalized periods satisfy

$n=\varphi(N)$22

The raw $n=\varphi(N)$23- and $n=\varphi(N)$24-period matrices are then

$n=\varphi(N)$25

with factorizations

$n=\varphi(N)$26

where $n=\varphi(N)$27 is a classical Vandermonde matrix (Tadokoro, 2012).

Using Knuth’s formula for $n=\varphi(N)$28, Tadokoro derives the compact expression

$n=\varphi(N)$29

and writes its entries explicitly in terms of elementary symmetric polynomials in $n=\varphi(N)$30. Every entry lies in $n=\varphi(N)$31. The Galois action $n=\varphi(N)$32 corresponds to simultaneously replacing each cycle $n=\varphi(N)$33 by $n=\varphi(N)$34, so the Galois group acts by permuting rows and columns of $n=\varphi(N)$35 (Tadokoro, 2012).

A useful conceptual distinction follows. In arithmetic papers such as Wu–Wang, Wu–She–Wang, and Wu–Pan, “cyclotomic period matrix” refers to a matrix whose entries are Gauss sums, characters, or cyclotomic numbers. In Tadokoro’s geometric setting, it refers to a genuine period matrix of a Riemann surface whose entries belong to a cyclotomic field. The shared adjective “cyclotomic” points to the governing root-of-unity structure, not to identical matrix definitions.

7. Applications, lineage, and conceptual significance

The prime-power theory of Wu–Wang places these matrices within the arithmetic of abelian extensions. The determinant formula and period decomposition are described as useful in constructing explicit integral bases in abelian extensions of $n=\varphi(N)$36, analyzing the weight distributions of certain cyclic codes, designing symmetric block designs via difference sets, and understanding circulant Hadamard matrices in combinatorial design theory (Wu et al., 2 Jul 2026). This suggests that cyclotomic period matrices function as an interface between explicit character-sum identities and structured linear algebra.

The finite-field determinant formulas have already been applied to conjectural problems. Wu–She–Wang confirm Sun’s 2019 conjecture on the explicit form of $n=\varphi(N)$37 by expressing the determinant as $n=\varphi(N)$38 times a square, up to a sign factor involving $n=\varphi(N)$39 in the prime case (Wu et al., 2021). Wu–Pan likewise resolve Sun’s 2019 conjecture by connecting $n=\varphi(N)$40 to the Frobenius trace $n=\varphi(N)$41 on the elliptic curve $n=\varphi(N)$42 and to the Jacobi-sum product $n=\varphi(N)$43 (Wu et al., 24 Dec 2025).

Historically, the recent prime-power results are positioned as extending older work on cyclotomic matrices. In particular, when $n=\varphi(N)$44, the Wu–Wang construction recovers and extends results of Carlitz and of Wu–Li–Wang–Yip on $n=\varphi(N)$45-th-order cyclotomic matrices; for $n=\varphi(N)$46, it gives the first systematic treatment of Gauss-sum matrices over the non-field ring $n=\varphi(N)$47 (Wu et al., 2 Jul 2026). The finite-field papers, by contrast, connect the subject to Greene’s hypergeometric formalism and to Jacobi-sum identities.

A final point of terminology is essential. “Cyclotomic period matrix” is best understood as a thematic label rather than a rigid standard definition. The common ingredients are roots of unity, multiplicative characters, character sums, Vandermonde or circulant structure, and Galois symmetry. The precise matrix depends on context: Gauss sums over $n=\varphi(N)$48, character evaluations over $n=\varphi(N)$49, cyclotomic numbers of order $n=\varphi(N)$50, or period integrals on a hyperelliptic curve. The literature therefore supports a plural reading of the subject: cyclotomic period matrices form a family of related constructions unified by cyclotomic arithmetic rather than by a single canonical model.