Gross–Koblitz formula explained

In mathematics, the Gross–Koblitz formula, introduced by 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. gave another proof of the Gross–Koblitz formula ("Boyarsky" being a pseudonym of Bernard Dwork), and gave an elementary proof.

Statement

The Gross–Koblitz formula states that the Gauss sum

\tau can be given in terms of the p -adic gamma function \Gamma _{p} by\tau _{q(r)} = - \pi ^{s _{p(r)}} \prod _{0\leq} \Gamma _{p\left( (r ^{(i)})/(q -1)} \right)

where

q is a power p ^{f} of a prime p ,r is an integer with 0 \leq r < q -1 ,r ^{(i)} is the integer whose base-p expansion is a cyclic permutation of the f digits of r by i positions,s _{p(r)} is the sum of the base-p digits of r ,\tau _{q(r)} = \sum _{a ^{q-1} =1} a ^{-r} \zeta ^{Tr (a)} _{\pi} , where the sum is over roots of unity in the extension Q _{p(\pi)} ,\pi satisfies \pi ^{p} = - p , and\zeta _{\pi} is the p th root of unity congruent to 1 + \pi modulo \pi ^{2} .

References

• • Book: Cohen, Henri . Henri Cohen (number theorist)

. Henri Cohen (number theorist) . 2007 . Number Theory – Volume II: Analytic and Modern Tools. 978-0-387-49893-5. Springer-Verlag. Graduate Texts in Mathematics. 240 . 1119.11002 . 383–395 .