Visual properties of generalized Kloosterman sums

Abstract For a positive integer m and a subgroup Λ of the unit group ( Z / m Z ) × , the corresponding generalized Kloosterman sum is the function K ( a , b , m , Λ ) = ∑ u ∈ Λ e ( a u + b u − 1 m ) for a , b ∈ Z / m Z . Unlike classical Kloosterman sums, which are real valued, generalized Kloosterman sums display a surprising array of visual features when their values are plotted in the complex plane. In a variety of instances, we identify the precise number-theoretic conditions that give rise to particular phenomena.