Hypergeometric Sheaves with Tannakian Monodromy Group G2

Based on a suggestion by Katz, we determine the Tannakian monodromy group of certain -adic hypergeometric sheaves to be the exceptional group G₂. Using the smoothing properties of the Fourier transform over the integers, known as uniformity theorems, we prove that the fourth moment is constant on an open locus of the family of hypergeometric sheaves. In our example, this implies a comparison theorem for the Tannakian monodromy groups that determines these groups if the characteristic is large.