Gross–Koblitz formula

In mathematics, the Gross–Koblitz formula, introduced by Gross and Koblitz (1979) expresses a Gauss sum using a product of values of the p-adic gamma function. It is an analog of the Chowla–Selberg formula for the usual gamma function. It implies the Hasse–Davenport relation and generalizes the Stickelberger theorem.

Boyarsky (1980) gave another proof of the Gross–Koblitz formula ("Boyarsky" being a pseudonym of Bernard Dwork), and

Robert (2001) gave an elementary proof.

Statement

The Gross–Koblitz formula states that the Gauss sum

τ

{\displaystyle \tau }

can be given in terms of the

p

{\displaystyle p}

-adic gamma function

Γ

p

{\displaystyle \Gamma _{p}}

by

τ

q

(

r

)

=

−

π

s

p

(

r

)

∏

0

≤

i

<

f

Γ

p

(

r

(

i

)

q

−

1

)

{\displaystyle \tau _{q}(r)=-\pi ^{s_{p}(r)}\prod _{0\leq i<f}\Gamma _{p}\!\left({\frac {r^{(i)}}{q-1}}\right)}

where

q

{\displaystyle q}

is a power

p

f

{\displaystyle p^{f}}

of a prime

p

{\displaystyle p}

,

r

{\displaystyle r}

is an integer with

0

≤

r

<

q

−

1

{\displaystyle 0\leq r<q-1}

,

r

(

i

)

{\displaystyle r^{(i)}}

is the integer whose base-

p

{\displaystyle p}

expansion is a cyclic permutation of the

f

{\displaystyle f}

digits of

r

{\displaystyle r}

by

i

{\displaystyle i}

positions,

s

p

(

r

)

{\displaystyle s_{p}(r)}

is the sum of the base-

p

{\displaystyle p}

digits of

r

{\displaystyle r}

,

τ

q

(

r

)

=

∑

a

q

−

1

=

1

a

−

r

ζ

π

Tr

(

a

)

{\displaystyle \tau _{q}(r)=\sum _{a^{q-1}=1}a^{-r}\zeta _{\pi }^{{\text{Tr}}(a)}}

, where the sum is over roots of unity in the extension

Q

p

(

π

)

{\displaystyle \mathbb {Q} _{p}(\pi )}

,

π

{\displaystyle \pi }

satisfies

π

p

−

1

=

−

p

{\displaystyle \pi ^{p-1}=-p}

, and

ζ

π

{\displaystyle \zeta _{\pi }}

is the

p

{\displaystyle p}

th root of unity congruent to

1

+

π

{\displaystyle 1+\pi }

modulo

π

2

{\displaystyle \pi ^{2}}

.

References

Boyarsky, Maurizio (1980), "p-adic gamma functions and Dwork cohomology", Transactions of the American Mathematical Society, 257 (2): 359–369, doi:10.2307/1998301, ISSN 0002-9947, JSTOR 1998301, MR 0552263

Cohen, Henri (2007). Number Theory – Volume II: Analytic and Modern Tools. Graduate Texts in Mathematics. Vol. 240.