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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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