## Higher p-adic gamma functions and Dwork cohomology.(English)Zbl 0551.12012

Cohomologie p-adique, Astérisque 119-120, 111-127 (1984).
The Gross-Koblitz formula gives a p-adic analytic interpolation of certain Gauss sums in terms of the Morita gamma function $$\Gamma_ p$$. An interpretation of the Gross-Koblitz formula in the set-up of Dwork cohomology was given by Boyarski and Dwork. In this paper we follow a suggestion of Dwork and use his theory of p-adic analytic liftings of an additive character of a finite field to construct a family of ”Dwork’s gamma functions” $$\Gamma_{D,s}$$, $$s=1,2,...$$. These functions are locally analytic in larger and larger sets and satisfy functional equations (of translation and reflection) similar to those satisfied by Morita’s $$\Gamma_ p$$. In fact $$\Gamma_{D,1}=\Gamma_ p$$. We then obtain, for each function $$\Gamma_{D,s}$$, a formula for Gauss sums similar to the Gross-Koblitz formula: the amelioration of the domain of analyticity of these formulas is obtained at the expense of simplicity of calculation of the functions $$\Gamma_{D,s}$$.
For the entire collection see [Zbl 0542.00006].

### MSC:

 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 14F30 $$p$$-adic cohomology, crystalline cohomology