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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