A p-adic limit formula for Gauss sums (Q802666)

The elegant Gross-Koblitz formula expresses the Gauss sum, \(g_ q(\psi_ 1,\chi_ r)\) in terms of the p-adic \(\Gamma\)-function. A proof of this theorem, due to N. Katz, is based on a ``p-adic limit formula'' for \(-g_ q\). In the paper under review, the author uses Katz's method to give a new limit formula for \(-g_ q:\) \[ -g_ q=\lim_{n\to \infty}\frac{q\beta (rq^ n)}{\beta (rq^{n+1})},\text{ where } \beta (rq^ n)=\lambda^{p(q^ n-1)r/N}\left( \begin{matrix} s/p-1\\ (q^ n- 1)r/N\end{matrix} \right). \]