On cyclotomic matrices involving Gauss sums over finite fields
Authors: Wu Hai-Liang , Li Jie , Wang Li-Yuan , Yip Chi Hoi 
Published: December 2024

Inspired by the works of L. Carlitz [Acta Arith. 5 (1959), pp. 293–308] and Z.-W. Sun [Finite Fields Appl. 56 (2019), pp. 285–307] on cyclotomic matrices, in this paper, we investigate certain cyclotomic matrices involving Gauss sums over finite fields, which can be viewed as finite field analogues of certain matrices related to the Gamma function.

For example, let

q
=

p
n

q=p^n

be an odd prime power with

p
p

prime and

n
∈

Z

+

n\in \mathbb {Z}^+

. Let

ζ
p

=

e

2
π

i

/

p

\zeta _p=e^{2\pi \mathbf {i}/p}

and let

χ
\chi

be a generator of the group of all multiplicative characters of the finite field

F

q

\mathbb {F}_q

. For the Gauss sum
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript q Baseline left-parenthesis chi Superscript r Baseline right-parenthesis equals sigma-summation Underscript x element-of double-struck upper F Subscript q Baseline Endscripts chi Superscript r Baseline left-parenthesis x right-parenthesis zeta Subscript p Superscript normal upper T normal r Super Subscript double-struck upper F Super Sub Subscript q Super Subscript slash double-struck upper F Super Sub Subscript p Super Subscript Superscript left-parenthesis x right-parenthesis Baseline comma">

G
q

(

χ

r

)
=

∑

x
∈

F

q

χ

r

(
x
)

ζ
p

T
r

F

q

/

F

p

(
x
)

,

\begin{equation*} G_q(\chi ^{r})=\sum _{x\in \mathbb {F}_q}\chi ^{r}(x)\zeta _p^{\mathrm {Tr}_{\mathbb {F}_q/\mathbb {F}_p}(x)}, \end{equation*}

we prove that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="det left-bracket upper G Subscript q Baseline left-parenthesis chi Superscript 2 i plus 2 j Baseline right-parenthesis right-bracket Subscript 0 less-than-or-equal-to i comma j less-than-or-equal-to left-parenthesis q minus 3 right-parenthesis slash 2 Baseline equals left-parenthesis negative 1 right-parenthesis Superscript alpha Super Subscript n Superscript Baseline left-parenthesis StartFraction q minus 1 Over 2 EndFraction right-parenthesis Superscript StartFraction q minus 1 Over 2 EndFraction Baseline 2 Superscript StartFraction p Super Superscript n minus 1 Superscript minus 1 Over 2 EndFraction Baseline comma">

det

[

G
q

(

χ

2
i
+
2
j

)
]

0
≤
i
,
j
≤
(
q
−
3
)

/

2

=
(
−
1

)

α
n

(

q
−
1

2

)

q
−
1

2

2

p

n
−
1

−
1

2

,

\begin{equation*} \det \left [G_q(\chi ^{2i+2j})\right ]_{0\le i,j\le (q-3)/2}=(-1)^{\alpha _n}\left (\frac {q-1}{2}\right )^{\frac {q-1}{2}}2^{\frac {p^{n-1}-1}{2}}, \end{equation*}

where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha Subscript n Baseline equals StartLayout Enlarged left-brace 1st Row 1st Column 1 2nd Column a m p semicolon if n identical-to 1 left-parenthesis mod 2 right-parenthesis comma 2nd Row 1st Column left-parenthesis p squared plus 7 right-parenthesis slash 8 2nd Column a m p semicolon if n identical-to 0 left-parenthesis mod 2 right-parenthesis period EndLayout">

α
n

=

{

1

a
m
p
;
if 
n
≡
1

(
mod

2
)
,

(

p
2

+
7
)

/

8

a
m
p
;
if 
n
≡
0

(
mod

2
)
.

\begin{equation*} \alpha _n= \begin {cases} 1 & \text {if } n\equiv 1\pmod 2,\\ (p^2+7)/8 & \text {if } n\equiv 0\pmod 2. \end{cases} \end{equation*}