## Uniqueness of $$\Gamma_ p$$ in the Gross-Koblitz formula for Gauss sums.(English)Zbl 0532.12017

In this paper is determined which continuous, p-adic valued functions on $${\mathbb{Z}}_ p$$ make the Gross-Koblitz formula valid. They are found to be of the form $$\Gamma_ p G(x)/G(-\phi(-x))$$ where $$\Gamma_ p$$ is Morita’s p-adic $$\Gamma$$-function, where $$\phi$$ is defined by $$\phi(\sum^\infty_{i=0}x_ i p^ i)=\sum^\infty_{i=1}x_ i p^{i-1}$$ and where G is any continuous, non-vanishing function on $${\mathbb{Z}}_ p$$. The basic point is to prove that if F is a continuous, non-vanishing function on $${\mathbb{Z}}_ p$$ satisfying $$\prod^{n- 1}_{i=0}F(\phi^{(i)}(\epsilon))=1$$ for all positive integers n and all $$\epsilon$$ such that $$\phi^{(n)}(\epsilon)=\epsilon$$, then there exists a continuous, non-vanishing function G on $${\mathbb{Z}}_ p$$ such that $$F(x)=G(x)/G(\phi(x))$$. The author recently proves that if F is locally analytic, then G may be taken to be locally analytic too [Uniqueness of $$\Gamma_ p:$$ the locally analytic case; to appear in Astérisque].
Reviewer: G.Christol

### MSC:

 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 11L03 Trigonometric and exponential sums (general theory)
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