On the monodromy groups attached to certain families of exponential sums (Q1101486)

Let \({\mathbb{F}}\) be a finite field of characteristic p. Inspired by the results of a joint article with \textit{R. Pink} [Duke Math. J. 54, 57-65 (1987; see the review 12007)], the author shows that the monodromy groups of the \(\ell\)-adic sheaves on \({\mathbb{G}}_ m/{\mathbb{F}}\) attached to Kloosterman sums in n variables are ``as large as possible'', at least when p is larger than the constant \(2n+1\) occuring in the Feit-Thompson sharpening of Jordan's theorem on finite subgroups of \(GL_ n\). A striking application to the equidistribution of Kloosterman sums is given. Similar properties for one-variable polynomial sums are also established.