Gross–Koblitz formula

Template:Short description In mathematics, the Gross–Koblitz formula, introduced by Template:Harvs 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. Template:Harvtxt gave another proof of the Gross–Koblitz formula ("Boyarsky" being a pseudonym of Bernard Dwork), and Template:Harvtxt 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 ${\mathrm{\Gamma }}_p$ by

${\tau }_q(r)=-{\pi }^{s_p(r)}\prod _{0\le i<f}{\mathrm{\Gamma }}_p\left(\frac{r^{(i)}}{q-1}\right)$

where

• $q$ is a power $p^f$ of a prime $p$,
• $r$ is an integer with $0\le r<q-1$,
• $r^{(i)}$ 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 $r$,
• ${\tau }_q(r)=\sum _{a^{q-1}=1}{a}^{-r}{\zeta }_{\pi }^{\text{Tr}(a)}$, where the sum is over roots of unity in the extension ${\mathbb{Q}}_p(\pi )$,
• $\pi$ satisfies ${\pi }^{p-1}=-p$, and
• ${\zeta }_{\pi }$ is the $p$th root of unity congruent to $1+\pi$ modulo ${\pi }^2$.

References

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