Gross–Koblitz formula

The Gross–Koblitz formula states that the Gauss sum τ can be given in terms of the p-adic gamma function Γ_p by

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

where

• q is a power p^f of a prime p
• r is an integer with 0 ≤ 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 digits of r in base p
• ${\displaystyle \tau _{q}(r)=\sum _{a^{q-1}=1}a^{-r}\zeta _{\pi }^{{\text{Tr}}(a)}}$, where the sum is over roots of 1 in the extension Q_p(π)
• π satisfies π^{p – 1} = –p
• ζ_π is the pth root of 1 congruent to 1+π mod π²

• 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. Springer-Verlag. pp. 383–395. ISBN 978-0-387-49893-5. Zbl 1119.11002.
• Gross, Benedict H.; Koblitz, Neal (1979), "Gauss sums and the p-adic Γ-function", Annals of Mathematics, Second Series, 109 (3): 569–581, doi:10.2307/1971226, ISSN 0003-486X, JSTOR 1971226, MR 0534763
• Robert, Alain M. (2001), "The Gross-Koblitz formula revisited", Rendiconti del Seminario Matematico della Università di Padova. The Mathematical Journal of the University of Padova, 105: 157–170, ISSN 0041-8994, MR 1834987